5th Conference on Algebraic Combinatorics and Graph …conferences2012.kashanu.ac.ir/11.pdf · 5th...

5th Conference on Algebraic Combinatorics and Graph

Theory

Extended Abstract Booklet

University of Kashan

Kashan, Iran

July 3-4, 2012

[email protected]

The Members of Scientific Committee

S. Akbari Sharif University of Technology

A. R. Ashrafi University of Kashan

M. A. Iranmanesh Yazd University

G. R. Baradaran Khosroshahi IPM

R. Jahanipour University of Kashan

H. Daghigh University of Kashan

G. R. Safakish Bu Ali Sina University

B. Taeri Isfahan University of Technology

A. Abdollahi University of Isfahan

G. H. FathTabar University of Kashan

E. Ghorbani K. N. Toosi University of Technology

D. Kiani Amirkabir University of Technology

H. R. Maimani Shahid Rajaee Teacher Training University

H. YousefiAzari University of Tehran

The Members of Organizing Committee

A. R. Ashrafi H. Khadashenas

J. Asgari A. Mohebbi

M. Bahramian M. J. Nadjafi-Arani

B. Bazigaran M. Pourbabaee

H. Daghigh Gh. Rahmanimehr

G. H. Fath-Tabar A. A. Rezaei

R. Jahani-Nezhad A. Saadatmandi

R. Jahanipour H. R. Tabrizidoz

Content INVITED SPEAKERS

A. Amini………….………………………………………………………...…………......…..…..1 ON A GRAPH OF IDEALS R. B. Bapat…………………………………………………………………………….……….…2 THE PRODUCT DISTANCE MATRIX OF GRAPH B. Davvaz…………………………………………………………………………….……...........3 RELATIONS ON GROUPS, POLYGROUPS AND HYPERGROUPS A. Dilek Maden (GÜNGÖR)…………………………………………………..………………...4 RELATIONS ON GROUPS, POLYGROUPS AND HYPERGROUPS G. Y. Katona…………………………………………………….……………………………..…5 KIRCHHOFF INDEX AND RESISTANCEDISTANCE ENERGY K. Khashyarmanesh, M. Afkhami and Z. Barati……………………………………………...6 RING GRAPH AND OUTERPLANAR GRAPHS H. R. Maimani and S. Kiani…..………………………………………………………………....7 ON THE DOMINATION NUMBERS OF UNIT GRAPH OF COMMUTATIVE RINGS Gh. R. Omidi, Gh. Raeisi and Kh. Tajbakhsh…………………………………........................8 ON COLORING NUMBER OF GRAPHS ORAL PERESENTATION A. Abdolghafurian and Mohammad A. Iranmanesh………………………...………………10 ON DIVISIBILITY GRAPH FOR SYMMETRIC GROUPS S. Akbari, D. Kiani and M. Mirzakhah………………………………………………….…...12 CHARACTERIZING GRAPHS BY MEANS OF SOME COEFFICIENTS OF THEIR CHARACTERISTIC POLYNOMIALS Saeid Alikhani, GeeChoon Lau, and Saeed Mirvakili……...……………………………….13 ON THE K-EDGE MAGIC GRAPHS Masoud Ariannejad…………………………………………….………………………………14 SOME NEW KINDS OF SPANNING TREES

S. Bahramian……..……………………………………………………………………………..15 COMPLETE MULTIPARTITE GRAPHS AND THEIR NULL SET H. Damadi and F. Rahmati……………………………………….……………………………16 EDGE SINGULARITY OF GRAPHS M. Davoudi Monfared, E. S. Haghi……………………………………………………………17 NON-COMMUTING GRAPH TO PRIME GRAPH IN ALTERNATING GROUP OF DEGREE 16 M. Emami and O. Naserian………………………………………..…………………………..18 ON THE LARGE SETS OF T-DESIGNS OF SIZE NINE Zeinab Foruzanfar and Zohreh Mostaghim………………………..……………………..….19 A NOTE ON CONJUGACY CLASS GRAPHS Mohsen Ghasemi, Pablo Spiga……………………………………….………………………..20 4-VALENT ONE REGULAR GRAPHS OF ORDER 6P2

Mostafa Khorramizadeh………………………….……………………………………………21 AN APPLICATION OF BIPARTITE GRAPHS IN COMBINATORIAL SCIENTIFIC COMPUTING Asghar Madadi, Rashid ZaareNahandi…………………………...…………………………22 PERFECT MATCHING OF 3UNIFORM HYPERGRAPHS R.Manaviyat………………………………………………………….…………………………24 ON THE INCLUSION IDEAL GRAPH OF A RING M. Mirzargar…………………………………………………………..………………………..25 SURVEY ON THE COMMUTING GRAPH OF A FINITE GROUP M. Mogharrab………………………………………………………….……………………….26 SOME PROPERTIES OF FRUCHT GRAPH Sirous Moradi…………………………………………………………..……………………….27 ECCENTRICITY IN THE KRONECKER PRODUCT OF GRAPHS Marcel Morales, Ali Akbar Yazdan Pour andRashid ZaareNahandi……………………..28 CASTELNUOVO-MUMFORD REGULARITY UNDER REDUCTION PROCESSES ON GRAPHS AND HYPERGRAPHS M. J. Nadjafi-Arani and Sandi Klavžar…………………………………………………….30 CUT METHOD AND DJOKOVIĆ-WINKLER'S RELATION R. Nasiri-Gharghani and G. H. Fath-Tabar………………………………………………….31 THE SECOND MINIMUM OF THE IRREGULARITY OF GRAPHS

M. R. Nezadi-Niasar and G. H. Fath-Tabar…………………………………………………..32 FIBONACCI NUMBER OF LINEAR ALKANES CNH2N+2 S. M. Seyyedi, F. Rahmati and M. Saeedi……………………………………………………..34 SHELLABLE AND COHENMACAULAY COMPLETE TPARTITE GRAPHS Reza Sharafdini…………………………………………………………………………………35 SEMISIMPLICITY OF THE ADJACENCY ALGEBRA OF COHERENT CONFIGURATIONS M. Tavakoli, F. Rahbarnia, M. Mirzavaziri and A. R. Ashrafi……………………………...36 FURTHER RESULTS ON THE THIRD ZAGREB INDEX OF GRAPHS Zahra Yarahmadi, Sirous Moradi and Tomislav Doslic …………………………………….37 ECCENTRIC CONNECTIVITY INDEX OF GRAPHS WITH SUBDIVIDED EDGES R . ZAFERANI and C . ADIGA ………………………………………………………………..38 ON SUM DEGREE ENERGY OF A GRAPH POSTER PERESENTATION Z. AkbarzadehGhanaie and A. R. Ashrafi……………………………………………………40 RANDIC INDEX, DIAMETER AND THE AVERAGE DISTANCE OF GRAPHS Z. Alem, A. Iranmanesh and Y. Pakravesh…………………………………………....……..41 SHELL POLYNOMIAL OF SOME GRAPHS S. Alikhani, R. Hasni and S. Mirvakili……………………………………………...………...42 ON CHROMATIC UNIQUENESS OF COMPLETE CERTAIN PARTITE GRAPHS S. M. Anvariyeh and Z. Mirzaei …………………………………………………...……….…43 GRAPHS, THE WAY TO THE ZERO-DIVISORS OF RINGS H. Azanchilar, Z.Assadi-Golzar and Kh.Shamsi…………………………………………..…45 THE LARGEST CLIQUES AND APPLICATIONS M. Bahramian………………………………………………………………………...……..….46 THE JACOBIAN OF GRAPHS M. Bahramian and G. H. Fath-Tabar…………………………………………………………47 The Szeged Eigenvalues OF THE PATH A. Chatrazar, N. Shahbaznejad, M. Shayan and A. Chatrazar……………………...……...48 REGULAR OF CIRCUIT GRAPHS OF UNIFORM MATROIDS

A. Chatrazar, M. Shayan, A. Chatrazar and N. Shahbaznejad………………………...…...49 THE NUMBERS OF EDGES IN CIRCUIT GRAPH OF MATROID U1;N

Hassan Daghigh and Somayeh Didari………………………………………………….……..50 ON THE RANK OF A SPECIAL FAMILY OF ELLIPTIC CURVES T. DehghanZadeh and A. R. Ashrafi………………………………………………………...51 THE MAXIMUM RANDIC INDEX OF TRICYCLIC AND TETRACYCLIC GRAPHS S . D Jafari and A . R . Ashrafi…………………………………………..…..………………..….53 FURTHER RESULTS ON THE INNER POWER OF GRAPHS Z. Fallahzadeh and M. A. Iranmanesh……………………………….….…………………....55 ON THE ENERGY OF THE COMPLEMENT KTH ITERATED LATIN SQUARE LINE GRAPHS A. Farokh, S. M. Babamir,M. M. Morovati.and. H. Banki………………………………….57 RESOLVING SHORTEST PATH PROBLEM IN WEIGHTED GRAPH USING HIGH PERFORMANCE COMPUTER Khadijeh Fathalikhani and Hassan YousefiAzari………………………………………….59 DISTANCEBASED GRAPH INVARIANT AND THE ECCENTRIC CONNECTIVITY INDEX OF GRAPH OPERATIONS L. Ghanbari and S. Mikaeyl Nejad……………………………...…………………………….60 APPLICATIONS OF GRAPH THEORY PROBLEMS IN BIOINFORMATICS F. Gilasi and A. R. Ashrafi………………………………………………………...……….….62 THE ENERGY OF CAYLEY GRAPHS ON DIHEDRAL GROUPS Z. Gholam-Rezaei and G. H. Fath-Tabar ………………………………………...……….…63 ON THE DIFFERENCE OF WINER AND DETOUR INDEX OF GRAPHS F. Gholami and F. Pashaie ………………………………………………………...…………..65 AN APPLICATION OF CRYSTAL GRAPHS ON LIE SUPERGROUP F. Gholaminezhad and A. R. Ashrafi………………………………………………..…….….67 THE ORBITS OF THE AUTOMORPHISM GROUP ACTION OF HAMMING GRAPHS M. Ghorbani and M. Songhori………………………………………………………..…….…69 NOTE ON EXTENDED DOUBLE COVER GRAPHS Modjtaba Ghorbani and Farzaneh Nowroozi Larki……………………………………...….70 NONCOMMUTING GRAPH AND POWER GRAPH OF FINITE GROUPS MODJTABA GHORBANI …………………………..……….…………………………….………71 ECCENTRIC DISTANCE SUM OF COMPOSITE GRAPHS

MODJTABA GHORBANI AND MAHIN SONGHORI………………..……………………....………72 A note on Eccentric Distance Sum E. Haghi and A. R. Ashrafi………………………………………………………...……….….73 Q-CONJUGACY CHARACTER TABLE OF DIHEDRAL GROUP M. HakimiNezhaad and A. R. Ashrafi…………………………………………...………….74 COMPUTING SPECTRAL, LAPLACIAN SPECTRAL AND SEIDEL SPECTRAL DISTANCES OF HYPERCUBES, ITS COMPLEMENT AND THEIR LINE GRAPHS A. Hamzeh and A. Iranmanesh………………………………….……………………….……76 ON THE SPLICE AND LINK OF SOME GRAPHS A. Heydari……………………………………………………………………………………...77 HYPER-WIENER INDEX AND SCHULTZ INDEX OF GENERALIZED BETHE TREES M. A. Hosseinzadeh, A. Iranmanesh and T. Doslic………………………………………..…78 NARUMI-KATAYAMA INDEX OF GRAPHS S. Hossein-Zadeh, A. Iranmanesh, A. Hamzeh and M. A. Hosseinzadeh…………………..79 THE COMMON NEIGHBORHOOD OF COMPOSITE GRAPHS S. Irandoost and G. H. Fath-Tabar……………………….…………………………………..80 SOME BOUNDS ON THE SPECTRAL MOMENTS OF GRAPHS M. A. Iranmanesh and S. M. Shaker………………………………………………………….82 ON PROPER POWER GRAPHS M. Javanbakht………………………………………………………………………...…..…..84 KEY PRIDISTRIBUTION SCHEME FOR HIERARCHICAL WIRELESS SENSOR NETWORKS USING COMBINATORIAL DESIGNS M. Kadkhoda………………………………………………………………………...…………85 COMPLEXITY ANALYSIS OF NESTED IF ORDERS BY REWRITING SYSTEMS: A COMBINATORIAL SOLUTION M. Kadkhoda…………………………………………………………………………………...86 LABELING KNUTH-BENDIX ORDER FOR PROVING TERMINATION F.Kamali and S. Davodpoor……………………………………….…………………………..87 CIRCULANT WEIGHING MATRICES M. Khanzadeh, M. A. Iranmanesh……………………………………...………………..…...89 SOME NEW UPPER BOUNDS FOR SKEW LAPLACIAN ENERGY OF AN (N, M)-DIRECTED GRAPH

F. KoorepazanMoftakhar and A. R. Ashrafi………………………………….……….……91 FIXING NUMBER OF FULLERENES A. Loghman……………………………………………………………..……………………....93 A NOTE ON EDGE-COLOURING OF REPLACEMENT PRODUCTS E. Mahfooz and G. H. Fath-Tabar………………………………………………………..…...94 WINER INDEX OF TWO CONNECTED GRAPHS S. Malekpoor and G. H. Fath-Tabar ……………………………………………………..…..96 SOME BOUNDS ON THE DETOUR INDEX OF GRAPHS M. R. Mollaei and A. Gholami……………………………………………………...……..…..97 PERMUTATION SYMMETRY OF FULLERENE ISOMERS OF C88 A. Moraveji, S. Madani and H. Shabani…………………………………………...……....…98 ESTRADA INDEX OF (3,6)CAGE AND 6CAGE. Z.Mostaghim and M.Zakeri……………………………………………………...…………..100 SOME RESULTS ON CONJUGACY CLASS GRAPHS OF P-SINGULAR ELEMENTS A. R. Naghipour and M. Rahmati……………………………………………………..…......101 SOME DIGRAPHS ARISING FROM NUMBER THEORY Y. Pakravesh and A. Iranmanesh…………………………………..……………….……….102 ON THE MATCHING POLYNOMIAL OF SOME GRAPHS M. Pourbabaee, H. R. Tabrizidooz and A. R. Ashrafi…………………….………………..103 THE MAXIMUM RANDIC INDEX OF PENTACYCLIC GRAPHS Z. Sadri-Irani and A. Karbasion ………...…………...………………………………….…..105 SEARCHING NUMBER IN GRAPH A. Saeidi and S. Zandi………………..………………………………………………..……...107 A GRAPH RELATED TO THE ELEMENTS OF FINITE GROUPS H. Shabani…………………………………………………………………………..…………108 FIXING NUMBER OF NONCOMMUTING GRAPHS S. Sheikhi and S. M. Babamir………………………………………….………………….....109 AN APPROACH FOR STATE PREDICTION BY HIDDEN MARKOV MODEL GRAPH Z. Shiri-Barzoki, G. H. Fath-Tabar and A. R. Ashrafi……………….………………..…...111 THE CHROMATIC POLYNOMIAL OF LINEAR PHENILENES

A. Soltani and A. Iranmanesh………………………………………………………..………113 THE EDGE WIENER TYPE TOPOLOGICAL INDICES F. Taghvaee and A. R. Ashrafi…………………………………………………………....….114 ORDERING GENERALIZED PETERSEN GRAPHS WITH RESPECT TO SPECTRAL MOMENTS A. Asghar-Talebi and Sh. Safarizade………………………………………………….…….115 SOME PROPERTIES OF PRINCIPAL IDEAL GRAPH OF THE RING ZN A. A. Talebi and N. Mehdipoor…………………………………………………….………...117 CLASSIFYING CUBIC S-REGULAR GRAPHS OF ORDERS 36P AND 36P2 A. Zolfi and A. R. Ashrafi………………………………………………………..….………..118 THE TOP TEN MINIMUM AND MAXIMUM VALUES OF NARUMI-KATAYAMA INDEX IN UNICYCLIC GRAPHS PERSIAN PAPARES

120.....................................................................................................................................اسالمی زیبا و بابامیر سادات فائزه مالزم بی سیم بی حسگر شبکه در اعتماد قابل و مفید هاي داده پایداري اساس بر شبکه گذاري کد

121...............................................................................................................).........اکبري سعید با مشترك کار( بهمنی اصغر

هاي هادامارد تعمیمی از ماتریس

122.....................................................................................................................................................................زهرا تند پور مینیمم مجموع رنگ آمیزي گراف ها

123...............................................................................................................................................................سمیه غنی یار لو

همبنددودیی - 4قضیه شکافنده براي مترویدهاي

124..........................................................................................................................................فرزانه کتابی و علیرضا اشرفی ضلعی رشدیافته هاي شش وینر دستگاهشاخص

125...................................................................................................................................................فاطمه کوره پزان مفتخر

نظریه ي شمارشی پولیا

[1]

5th Conference on Algebraic Combinatorics and Graph Theory, July 34, 2012

ON A GRAPH OF IDEALS

Afshin Amini

Department of Mathematics, College of Sciences, Shiraz University, Shiraz 71457, Iran

[email protected]

Abstract

In this work, we associate a graph Γ+(R) to a ring R whose vertices are nonzero proper right

ideals of R and two vertices I and J are adjacent if I + J = R. Then we try to translate

properties of this graph into algebraic properties of R and vice versa. For example, we

characterize rings R for which Γ+(R) respectively is connected, complete, planar or a forest.

Also we find the dominating number of Γ+(R).

Keywords: graph of ideals; planarity; complete graph. 2010 Mathematics Subject Classification: 05C25, 05C10.

[2]


THE PRODUCT DISTANCE MATRIX OF GRAPH

R. B. Bapat

Indian Statistical Institute, New Delhi, India

This is joint work with S. Krishnan

Let G be a strongly connected weighted directed graph. The weight of a directed path is

defined as the product of the corresponding edge weights. The distance from vertex i to j is

defined to be the minimum weight of an ijpath. The product distance matrix of G is then

defined in the usual way. We obtain a formula for the determinant and the inverse of the

product distance matrix, analogous to a classical result of Graham, Hoffman and Hosoya for

the classical distance. The edge orientation matrix of a directed tree is defined. It is a matrix

with rows and columns indexed by the edges of a directed tree. The (e,f)entry is 1 or 1

according as the edges e and f are similarly or oppositely oriented. We obtain a formula for

the determinant and the inverse, when it exists, of the edge orientation matrix.

[3]


RELATIONS ON GROUPS, POLYGROUPS AND HYPERGROUPS

B. Davvaz

Department of Mathematics, Yazd University, Yazd, Iran

Email: davvaz@yazduni. ac. ir

Abstract

Hyperstructure theory both extends some wellknown group results and introduce new

topics leading us to a wide variety of applications, as well as to a broadening of the

investigation fields. In a classical algebraic structure, the composition of two elements is

an element, while in an algebraic hyperstructure, the composition of two elements is a

set. By using a certain type of equivalence relations, we can connect semihypergroups to

semigroups and hypergroups (polygroups) to groups. These equivalence relations are

called strong regular relations. In this paper, we review some strong regular relations

on hyperstructures and we give some their applications.

[4]


KIRCHHOFF INDEX AND RESISTANCEDISTANCE ENERGY

A. DILEK MADEN (GÜNGÖR)

Department of Mathematics, Faculty of Science, Selcuk University, Campus, Konya, 42075, Turkey

E-mail: [email protected]

Abstract

In this talk, we report lower bounds for the Kirchhoff index of a connected (molecular) graph

in terms of its structural parameters such as the number of vertices (atoms), the number of

edges (bonds), maximum vertex degree (valency), second maximum vertex degree and

minimum vertex degree. Also we give the NordhausGaddum-type result for Kirchhoff

index. In here we define the resistance distance energy as the sum of the absolute values of

the eigenvalues of the resistance distance matrix and also we obtain lower and upper bounds

for this energy.

2010 Mathematics Subject Classification: 05C50, 15C12. Keywords and phrases: Kirchhoff index, ResistanceDistance energy. References

[1] W. Xiao, I. Gutman, Resistance distance and Laplacian spectrum, Theor. Chem. Acc. 110

(2003), 284289.

[2] H. Chen, F. Zhang, Resistance distance and the normalized Laplacian spectrum, Discr.

Appl. Math. 155 (2007), 654661.

[3] D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993), 81-95.

[4] B. Zhou, N. Trinajstić, A note on Kirchhoff index, Chem. Phys. Lett. 455 (2008),

120123.

[5]


EXTENSION OF PATHS AND CYCLES FOR HYPERGRAPHS

Gyula Y. Katona

Department of Computer Science and Information Theory, Budapest University of Technology and Economics, 1521 Budapest Pob. 91, Hungary

Abstract

In [1] we defined the hamiltonian cycle in hypergraphs in a new way. The definition can

be extended to paths and cycles as well. There are many results for hypergraphs that

uses the traditional path and cycle definition of Berge, so almost all of them can be

considered with the new definition. In the present talk, I will deal with such results.

Keyword: hamiltonian cycle, tight path, tight cycle. References

[1] G.Y. Katona and H.A. Kierstead, Hamiltonian chains in hypergraphs, J. Graph Theory, (1999) 30(3): 205--212

[6]


RING GRAPH AND OUTERPLANAR GRAPHS

Kazem Khashyarmanesh, Mojgan Afkhami and Zahra Barati

1Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 115991775, Mashhad, Iran

2Department of Mathematics, University of Neyshabur, P. O. Box 91136899, Neyshabur, Iran

3Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 115991775, Mashhad, Iran

Abstract

In this talk, we investigate when the unit, unitary and total graphs are ring graphs and also we

characterize finite commutative rings such that their comaximal graph (or zerodivisor graph)

are ring graphs. Moreover, we study the case that they are outerplanar.

Keywords: Ring graph, Outerplanar, unit graph, unitary graph, total graph, Comaximal

graph, Zerodivisor graph.

[7]


ON THE DOMINATION NUMBERS OF UNIT GRAPH OF

COMMUTATIVE RINGS

HAMID REZA MAIMANI AND SIMA KIANI

Abstract

In this talk, we investigate the domination number and total domination number of unit graph

corresponding to a finite commutative ring with nonzero identity R which is obtained by

setting all the elements of R to be the vertices and defining distinct vertices x and y to be

adjacent if and only if x + y is a unit of R. Bounds are obtained in general and exact results

are obtained when R R1 × ·×Rk where (Ri , mi) is a local ring such that ii mR / ≥ k + 1 for

any k ≥ 2. Also we characterize the rings whose domination number of unit graphs are 1, 2, 3

and 4.

[8]


ON COLORING NUMBER OF GRAPHS

Gholam Reza Omidi, Ghafar Raeisi and Khosro Tajbakhsh

1Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 2School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

3Mathematics Department, Faculty of Science, Shahrekord University, Shahrekord, Iran 4Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University,

Teharn, Iran

Abstract

A graph G is kdegenerate if every subgraph of G contains a vertex of degree at most k. The

coloring number of a graph G, col(G), is the smallest integer k such that G is

(k1)degenerate. In this paper, we give some upper bounds for the coloring numbers of

Hminor free graphs, H {K5 , Kr,s}, for r ≤ 2 and also (r,s) {(3,3), (3,4), (4,4)}.

Moreover, we investigate a new version of the Brooks' theorem for coloring number with

some extensions. These results generalize some known results and give some new results on

group choice number, group chromatic number and the choice number of graphs with much

shorter proofs. In addition, we prove that for each pair k, m of natural numbers there exists a

natural number f(k,m) such that every graph with coloring number at least f(k,m) contains a

kconnected subgraph of coloring number at least m.

Keyword: Chromatic number; Coloring number.

Oral Presentations of the

5th Conference on Algebraic

Combinatorics and Graph Theory

July 34, 2012,


[10]


ON DIVISIBILITY GRAPH FOR SYMMETRIC GROUPS

A. Abdolghafurian and Mohammad A. Iranmanesh

Department of Mathematics, Yazd University, Yazd, 89195-741, Iran

Abstract

In this paper we consider the divisibility graph for symmetric groups. The number of

connected components of this graph for symmetric groups is calculated.

Introduction

There are several graphs associated to various algebraic structures, especially finite groups,

and many interesting results have been obtained recently, for excellent survey see for

example [5].

Let 푋 be a set of positive integers and 푋∗ = 푋 − {1}. Several graphs have been associated

to 푋:

1) The common divisor graphΓ(푋) is a graph with vertex set 푉 Γ(푋) = 푋∗ and edge

set 퐸 Γ(푋) = {푥, 푦}: gcd(푥,푦) ≠ 1 .

2) The prime vertex graphΔ(푋) is a graph with vertex set 푉 Δ(푋) = 휌(푋) =

⋃ 휋(푥)∈ , where 휋(푥) is the set of primes dividing 푥and edge set퐸 훥(푋) =

{{푝, 푞}:푝푞|푥, 푥 ∈ 푋}.

3) The bipartite divisor graph퐵(푋) is a graph with the vertex set 푉(퐵(푋) = 휌(푋) ∪

푋∗ and the edge set 퐸 퐵(푋) = {푝,푥}: 푝 ∈ 휌(푋), 푥 ∈ 푋∗, 푝 푥 .

Recently A. R. Camina and R. D. Camina in [3] introduced a new directed graph using

the notion of divisibility of positive numbers.

4) The divisibility digraph퐷(푋) has 푋∗as the vertex set and there is an arc connecting

(푎,푏) with 푎,푏 ∈ 푋 whenever 푎 divides 푏.

[11]


Let 푋 = 푐푠(퐺). For properties of Γ(cs(퐺)), Δ(cs(퐺)) and 퐵 cs(퐺) , we refer to [1, 2, 3,

4]. In this paper we investigate the graph 퐷 푐푠(퐺) , where 퐺is the symmetric group 푆 .

Result

Theorem: Let σ ∈ S . If σ be a p-cycle where p ≥ n − 1, then the corresponding vertex to σ is

an isolated vertex of underlying graph of D(cs(G)). The other vertices are in a connected

component.

References

[1] E. A. Bertram, M. Herzog, and A. Mann, On a graph related to conjugacy classes of

groups. Bull. London Math. Soc. 22 (1990) 569–575.

[2] D. Bubboloni, S. Dolfi, M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs

for group conjugacy class sizes. J. Pure Appl. Algebra 213 (2009) 1722–1734.

[3] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure

of finite groups: a survey. AEJM 4 (2011) 559–588.

[4] C. Casolo and S. Dolfi, The diameter of a conjugacy class graph of finite groups. Bull.

London Math. Soc. 28 (1996) 141–148.

[5] M. L. Lewis. An overview of graphs associated with character degrees and conjugacy

class sizes in finite groups. Rocky Mountain J. Math. 38 (2008) 175–211.

[12]


CHARACTERIZING GRAPHS BY MEANS OF SOME COEFFICIENTS OF

THEIR CHARACTERISTIC POLYNOMIALS

S. Akbaria, c, D. Kiani b, c and M. Mirzakhahb, c

aDepartment of Mathematical Sciences, Sharif University of Technology,

Tehran, Iran.

bDepartment of Pure Mathematics, Faculty of Mathematics and Computer Science Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran.

cSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM),

P.O. Box 19395-5746, Tehran, Iran.

Abstract

Considering all graphs of order n, “Can we determine graphs by means of some coefficients

of their characteristic polynomial of the corresponding matrices?” is the main problem

investigated in this talk. For instance, we show that the complete graph is determined by its

adjacency coefficients a3 or a4. In order to solve this problem, suppose that T is a tree and G

is a graph on n vertices, for which L(T, x) L(G, x) = cx ≠ 0, where L(T, x) is the Laplacian

characteristic polynomial of T. We want to know which properties of G are determined by

this condition.

Keywords: Laplacian matrix of a graph, Characteristic polynomial of a graph, Coefficients

of generalized adjacency matrices.

[13]


ON THE K-EDGE MAGIC GRAPHS

Saeid Alikhani1 , GeeChoon Lau 2 and Saeed Mirvakili 3

1 Department of Mathematics, Yazd University, 89195-741, Yazd, Iran 2 Faculty of Computer and Mathematical Sciences, UniversitiTeknologi Mara, Johor

85009 Segamat, Malaysia 3 Department of Mathematics, Payame Noor University, 19395-4697 Tehran, I.R. Iran

Abstract

Let G be a graph with vertex set V and edge set E such that p|V| and q|E| . For integer

0k , define an edge labeling }1,...,1,{: qkkkEf and define the vertex sum for a

vertex v as the sum of the labels of the edges incident to v . If such an edge labeling induces

a vertex labeling in which every vertex has a constant vertex sum (mod p),then G is said to be

k -edge magic ( k -EM). In this paper, we consider maximal outerplanar graphs and obtain

some results for these kind of graphs to be k -EM.

AMS subject Classification 2010:05C78

Keyword and phrases: k-edge-magic, outer planer graph

[14]


SOME NEW KINDS OF SPANNING TREES

Masoud Ariannejad

Department of Mathematics, University of Zanjan, Iran

Abstract

The Spanning tree packing number of a graph G or STP(G) is the maximum number of edge-

disjoint spanning tree contained in G. Let ST(G) be the set of all spanning trees of G. We say

that a spanning tree T of G is of type STi if STP(G\T)=i-1, where 1 ≤ i ≤ STP(G). The set of

all STi for all 1≤ I ≤ STP(G) give a partition for ST. In this note we study these new kinds of

spanning trees through enumerating and characterizing them in some specific graphs.

[15]


COMPLETE MULTIPARTITE GRAPHS AND THEIR NULL SET

S. BAHRAMIAN

[email protected]

Semnan Branch, Islamic Azad University, Semnan, Iran

Abstract

For any non-trivial abelian group A (written additively) let A* = A \ {0}. A functionl ∶

E(G) → A∗is called a labeling of G. Any such labeling induces a map l ∶ V(G) → A, define

by

l (v) = l(uv).∈ ( )

If there exits a labeling lwhich induce a constant label con V(G), we say that lis an A-

magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural numbers

h ∈ Nsuch that Ghas a Zh-magic labeling with index 0. In 2007, E. Salehi determined the null

set of complete bipartite graphs. In this talk we generalize this result by obtaining the null set

of complete multipartite graphs.

[16]


EDGE SINGULARITY OF GRAPHS

H. Damadi and F. Rahmati

Faculty of Mathematics and Computer Science, Amirkabir University of Technology, P. O. Box 158754413, Tehran, Iran.

E-mail address: [email protected]

Abstract

We introduce edge singular and edge smooth graphs and study the edge singularity and edge

smoothness for graphs. We prove that complete graphs are edge smooth and introduce two

conditions such that G is edge singular if and only if satisfies this conditions.

Keywords: Binomial edge ideal, Edge smooth, Edge singular.

[17]


Non-commuting Graph to Prime Graph in

Alternating Group of Degree 16

M. Davoudi Monfared 1;2

Department of Mathematics, Tafresh Branch, Islamic Azad University, P. O. Box 39515-164 Tafresh, I. R. Iran

E. S. Haghi 3

Department of Mathematics, Tafresh Branch, Islamic Azad University

Tafresh, I. R. Iran

Abstract

Let G be a finite non-abelian group. We define a graph G , called the non-commuting graph of G, with vertex set G Z(G) such that two vertices x and y are adjacent if and only if xy yx. Another graph associated to a finite group G is the prime graph GK(G) with vertex set (G), the set of all prime divisors of |G|. Two distinct primes p and q are adjacent if and only if G contains an element of order pq. In this article we show that for a group G, if A16 G, then GK(A16) = GK(G) where A16 is the alternating group of degree 16. Keywords: non-commuting graph, prime graph, alternating group. 1 The first author would like to thanks Tafresh Branch Islamic Azad University for financial support. 2 Email: [email protected] 3 Email: [email protected]

[18]


ON THE LARGE SETS OF T-DESIGNS OF SIZE NINE

M. Emami and O. Naserian

Department of Mathematics, Faculty of Mathematical Sciences, University of Zanjan, Zanjan 45195313, I. R. Iran

Email: [email protected]

Abstract

We investigate the existence of some large sets of size nine. We take advantages of the

recursive and direct constructing method to show that in case LS[9](2, 5, v) the trivial

necessary conditions are also sufficient. In particular, LS[9](2, 5, 29) is constructed.

This fills a missing gap.

AMS Subject Classification 2010: Primary: 05B05.

Keyword and Phrases: t-designs, large sets of t-designs.

[19]


A NOTE ON CONJUGACY CLASS GRAPHS

Zeinab Foruzanfar and Zohreh Mostaghim

DEPARTMENT OF MATHEMATICS, IRAN UNIVERSITY OF SCIENCE AND TECHNOLOGY, TEHRAN, IRAN

Abstract

Let G be a nonabelian finite group, Γ(G) the attached graph related to its conjugacy classes.

The vertices of Γ(G) are represented by the non-central conjugacy classes of G, and connect

two vertices C and D with an edge if |C| and |D| have a common prime divisor. In this paper

we obtain some finite groups that graphs related to their conjugacy classes are complete

graphs.

[20]


4-VALENT ONE REGULAR GRAPHS OF ORDER 6P2

MOHSEN GHASEMI and PABLO SPIGA

Department of Mathematics, Urmia University,Urmia 57135, Iran


Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy


Abstract:

A graph is said to be one-regular if its automorphism group acts regularly on the set of its

arcs. In this paper we classify the 4-valent one-regular graphs having 6p2 vertices, with p a

prime.

2000 Mathematics Subject Classification. 20B25, 05C25. Key words and phrases: one-regular graphs, 4-valent.

[21]


AN APPLICATION OF BIPARTITE GRAPHS IN COMBINATORIAL

SCIENTIFIC COMPUTING

Mostafa Khorramizadeh

Faculty member Shiraz University of Technology

71555-313, Siraz, Iran

Email: [email protected]

Abstract:

In many applications, we are concerned with matrices with many zero entries. These matrices

are called sparse matrices. The main difficulty with algorithms regarding sparse matrices is

that they do not preserve sparsity during their steps. In this manuscript we propose an

algorithm designed to deal with sparse matrices and discuss some ideas to preserve the

sparsity during its steps. To do so, we first make use of the concepts, relating to the theory of

bipartite graphs especially the Delmuge-Mendelson decomposition, to obtain a sparse matrix,

the columns of which are linearly independent and span the linear space of all vectors whose

product with an original sparse matrix is the zero vector. We then compare the resulting

algorithm with an efficient algorithm, which uses the concepts of circuits of graphs with

respect to sparsity and the computing time. The numerical results show that our proposed

algorithm is comparable with that algorithm and in some cases is more efficient than that

algorithm. Both implemented algorithms are applied to some standard test problems arising

from linear programming problems.

Keywords: Bipartite graph, Delmuge-Mendelsohn decomposition, Sparse matrices, Combinatorial scientific computing

[22]


PERFECT MATCHING OF 3UNIFORM HYPERGRAPHS

ASGHAR MADADI

Department of Mathematices, University of Zanjan, Zanjan, Iran

RASHID ZAARENAHANDI

Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

ABSTRACT

A hypergraph F consists of a collection of vertices V and a collection of subsets of V which

are called hyperedges and denoted by HE(F). A d-uniform hypergraph is a hypergraph that all

hyperedges have cardinality equals to d. Let F be a 3uniform hypergraph. Any 2-subset of a

hyperedge is called an edge. A 3uniform hypergraph is called 3partite if there is a partition

푀 , 푀 , 푀 of V such that for all 푎, 푏 ∈ 푀 , 1 ≤ 푖 ≤ 3, there is no any hyperedge containing

a,b. In a hypergraph, a perfect matching is a collection of disjoint hyperedges which covers

all vertices of V. A set 퐴 = {푥 , … , 푥 } which is subset of V is called independent in F if there

is no any hyperedge with all vertices in A. This set is called strongly independent if for all

푥 ,푥 ∈ 퐴 there is no c ∈HE(F) such that 푥 ,푥 ∈ 푐. The hypergraph F is called well-covered

if all its maximal independent sets have the same cardinality. The hypergraph F is called

strongly well-covered if all its maximal strongly independent sets have the same cardinality.

In this note F denotes a 3-uniform 3-partite hypergraph with parts 푀 ,푀 ,푀 which has a

perfect matching of graphs between each two parts. Suppose that e={a,b} be an edge in F.

The set N(e) consists of all elements that their union with e is a hyperedge in F. If A is a

subset of edges in F then N(A)={N(e):e ∈ A}. All the edges that lies in 푀 ∪푀 is denoted

by E(푀 ,푀 ).

The maximum number of disjoint hyperedges in F is denoted by 훽 (퐹). A subset A of

V is a minimal vertex cover for F if (i) every hyperedge of F is incident with one vertex in A,

[23]


and (ii) there is no proper subset of A with the first property. If A satisfies condition (i) only,

then A is called a vertex cover for F. The minimal cardinality of vertex covers is denoted by

훼 (퐹).

In graph theory Hall's theorem [2,3] states that for a bipartite graph G with vertex set

V the following conditions are equivalent:

a) G has a perfect matching.

b) |A| ≤ |N(A)| for each independent set A which is subset of V.

We give a similar equivalent condition for a 3-uniform hypergraph. König's theorem

states that for a bipartite graph, the maximum number of independent edges is equal to the

smallest number of minimal vertex cover [3]. Analogous theorem for 3-uniform hypergraphs

is proved.

Let G be the class of graphs with some disjoint maximal cliques covering all vertices.

The second author (ZaareNahandi) in [4] conjectured that if G be a spartite well-covered

graph with all maximal cliques of size s then G is in the class G. We give analogous result to

this conjecture in hypergraph F which states that if F is wellcovered then there is a perfect

matching in F.

[24]


[25]


SURVEY ON THE COMMUTING GRAPH OF A FINITE GROUP

M. Mirzargar

Department of Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

Abstract

The commuting graph of a finite group is a graph whose vertex set is the group elements, two distinct

elements x and y being adjacent if xy = yx, this graph is denoted by Δ(G). In this paper we consider

the structure of the commuting graph and the automorphism group of this graph, Aut(Δ(G)). We

prove that Aut(Δ(G)) is a nonabelian group such that its order is not prime power and square-free

number.

Keywords: Commuting graph, automorphism group, ACgroup.

[26]


SOME PROPERTIES OF FRUCHT GRAPH

M. Mogharrab

Department of Mathematics, Persian Gulf University, Bushehr-75169, I. R. Iran

Abstract

A graph is called asymmetric graph if it has no nontrivial symmetries. The smallest

asymmetric cubic graph is the twelve vertex Frucht graph that is a 3regular graph with 12

vertices, 18 edges, and has no nontrivial symmetries. It was first described by Robert Frucht

in 1939. In this paper, we present some property of this graph and compute its topological

indices containing Wiener, edge PadmakarIvan, vertex PadmakarIvan, vertex Szeged,

edge Szeged, eccentric connectivity, the family of GeometricArithmetic and the Zagreb

group for the Frucht graph.

Keywords. Frucht graph, Asymmetric graph, Hamiltonian graph, Topological index.

[27]


ECCENTRICITY IN THE KRONECKER PRODUCT OF GRAPHS

Sirous Moradi 1

Department of mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, I. R. Iran

Email: [email protected] Abstract

Let G and H be graphs. The kronecker product GH of G and H has vertex set V (GH) = V (G) V (H) and edge set E(G H) = {(a; b)(c; d)|ac E(G) and bd E(H)}. In this paper, We present explicit formula for eccentricity of each vertex of kronecker product G H.

Keywords: Eccentricity, Kronecker Product, Radius, Diameter, Center.

[28]


CASTELNUOVO-MUMFORD REGULARITY UNDER REDUCTION

PROCESSES ON GRAPHS AND HYPERGRAPHS

Marcel Morales1, Ali Akbar Yazdan Pour1,2, Rashid ZaareNahandi2

1Department of Mathematics, Universite de Grenoble I - Institute Fourier (UJF),

Grenoble, France 2Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran

Abstract

Let 푆 = 퐾[푥 , . . . ,푥 ] be the polynomial ring over a field K and I a homogeneous ideal of S.

We say that I has a d-linear resolution if I is generated by elements of degree d and there is a

graded minimal free resolution for I, such that 훽 , (퐼) = 0, for all 푗 ≠ 푑. That is, the

graded minimal free resolution of I is of the form

0 → 푆 (−푑 − 푠) → ⋯ → 푆 (−푑 − 1) → 푆 (−푑) → 퐼 → 0. Proving that a class of ideals has a d-linear resolution, is difficult in general. It is

worth to note that classification of homogeneous ideals with linear resolution is equal to

classification of square-free monomial ideals.

Classification of square-free monomial ideals with 2-linear resolution, was

successfully done by Fröberg [4] in 1990. Fröberg observed that the circuit ideal of a graph G

has a 2-linear resolution if and only if G is chordal, that is, G does not have an induced cycle

of length >3.

In this paper, we introduce some reduction processes on graphs and hypergraphs

which preserve the regularity of related circuit ideals. By these operations, we transform a

(hyper)graphG to a smaller (hyper)graph G, while the Castelnuovo-Mumford regularity does

not change under these operations. As consequences of these reductions, we give an

alternative proof for Fröberg’s theorem [4] on linearity of circuit ideals of chordal graph, as

well as, linearity of circuit ideals of generalized chordal graphs as defined in [2]. Moreover, a

large class of hypergraphs such that their circuit ideals have a linear resolution is introduced.

Finally, a formula for regularity of circuit ideals of decomposable hypergraphs will be given.

[29]


Keywords: Castelnuovo-Mumford regularity, circuit ideal, simplicial, hypergraph.

[30]


CUT METHOD AND DJOKOVIĆ-WINKLER'S RELATION

M. J. Nadjafi-Arani

Faculty of Mathematical Sciences, University of Kashan

Kashan 87317-51167, I. R. Iran

Sandi Klavžar

Faculty of Mathematics and Physics, University of Ljubljana

SI-1000 Ljubljana, Slovenia

Abstract

Let Ө* be the transitive closure of the Djoković-Winkler's relation Ө. It is proved that the

Wiener index of a weighted graph (G,w) can be expressed as the sum of the Wiener index of

weighted quotient graphs with respect to an arbitrary combination of Ө*-classes. A related

result for edge-weighted graphs is also given and a class of graphs studied in [8], is

characterized as partial cubes. We will compute distance polynomial functions on graphs

with transitive Djoković Winkler's relation.

Keyword: Wiener index, weighted graph, Djoković-Winkler's relation, partial cube,

Isometric embedding.

[31]


THE SECOND MINIMUM OF THE IRREGULARITY OF GRAPHS

R. Nasiri-Gharghani1 and G. H. Fath-Tabar2

1Department of Mathematics, University of Qom, Qom, Islamic Republic of Iran 2Department of Pure Mathematics, Faculty of Mathematical Sciences,

University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

The word graph refers to a finite, undirected graph without loops and multiple edges. For a

graph G, Albertson has defined the irregularity of G as irr(G)=∑ |푑(푢)− 푑(푣)| where

d(u) is the degree of vertex u. Recently, this graph invariant gained interest in chemical graph

theory. In this work, we present some new result on the second minimum of the irregularity

of graphs.

Theorem 1. Let G be a graph then Irr(G) is only even number.

Corollary 2. The second minimum of the irregularity of a graph is 2.

Theorem 3. There are 25 types of graphs with irregularity of 2.

References

1. M. O. Albertson, The irregularity of a graph, Ars Comb. 46 (1997) 219225.

2. G. H. FathTabar, B. Furtula and I. Gutman, A new geometric-arithmetic index, J. Math.

Chem. 47 (2010) 477486.

3. G. H. FathTabar, Old and new Zagreb index, MATCH Commun. Math.Comput. Chem.

65 (2011)7984.

[32]


FIBONACCI NUMBER OF LINEAR ALKANES CNH2N+2

M. R. Nezadi-Niasar and G. H. Fath-Tabar

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

Let G =(V, E) be a graph. The Fibonacci number of graph G, F(G) is defined as 퐹(퐺) =

∑ 푞 (퐺, 푟) where 푞 (퐺, 푟)the number of independent sets with r elements in G. In this

paper, The Fibonacci number of alkanes C H is presented. Keywords: Graph, Fibonacci number. Introduction and Results

Suppose G = (V, E) is a simple connected graph with vertex and edge sets V(G) and E(G),

respectively. For a graph G, its Fibonacci number simply denoted by F(G) is defined as the

number of subsets of V(G) in which no two vertices are adjacent in G, i.e. in graph-

theoretical terminology, the number of independent sets of G, including the empty set.

Although there is in [1] the algorithm to calculate the Fibonacci number desired in any given

tree, but we present here a closed formula to calculate the Fibonacci number of an arbitrary

linear alkanes. In this paper we denote the number of independent sets with r elements by

푞 (퐺, 푟)thus F(G) = ∑ q (G, r).

[33]


Figure 1. Some Linear Alkanes.

Lemma 1. The maximum number of independent sets in CnH2n+2 is 2n + 2.

Theorem 2.

F(C H ) = 3n + 3

+n− k − 1

k2n− 2k + 2

r − k

+ 2n− k − 2

k2n− 2k − 1

r − k − 1 +n− k − 3

k2n − 2k− 4

r − k − 2 .

References

[1] X. Li, Z. Li, and L. Wang, The inverse problems for some topological indices in

combinatorial chemistry, J. Computation Biology, 10(1) (2003) 4755.

[34]


SHELLABLE AND COHENMACAULAY COMPLETE

TPARTITE GRAPHS

S. M. Seyyedi, F. Rahmati and M. Saeedi

Faculty of Mathematics and Computer Science, Amirkabir University of Technology, P. O. Box 15875-4413, Tehran, Iran.


Abstract

Let G be a simple undirected graph. We find the number of maximal independent sets in

complete t-partite graphs. We will show that vertex decomposability and shellability are

equivalent in these graphs. Also, we obtain an equivalent condition for being Cohen-

Macaulay in complete t-partite graphs.

[35]


SEMISIMPLICITY OF THE ADJACENCY ALGEBRA OF COHERENT

CONFIGURATIONS

Reza Sharafdini

Department of Mathematics, Faculty of Sciences, Persian Gulf University, IRAN

Abstract

It is known that that the Frame number characterizes them semisimplicity of adjacency algebra of a association scheme. In this paper, we confirm this fact to coherent configurations which are a generalization of association scheme. This is the first step to study modular representation of coherent configurations.

[36]


FURTHER RESULTS ON THE THIRD ZAGREB INDEX OF GRAPHS

M. Tavakoli1, F. Rahbarnia1, M. Mirzavaziri1 and A. R. Ashrafi2

1Department of Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, I. R. Iran

2Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

Suppose G is a simple graph. The third Zagreb index of G is defined as M3(G) =

e=uvE(G) |deg(u)−deg(v)| [1, 2]. In this paper, it is proved that the third Zagreb index of a

graph is an even non-negative integer. Moreover, any even non-negative integer is the third

Zagreb index of a given caterpillar. We also prove that if T is an arbitrary tree then M3(T)

4. Finally the maximum and minimum of this graph invariant in the classes of all tricyclic

and tetracyclic graphs are computed.

Keywords: Third Zagreb index, tricyclic graph, tetracyclic graph.

References

[1] G. H. Fath-Tabar, Old and new Zagreb index, MATCH Commun. Math. Comput. Chem.

65 (2011), 79–84.

[2] A. Astaneh-Asl and G. H. Fath-Tabar, Computing the First and Third Zagreb

Polynomialsof Cartesian Product of Graphs, Iranian J. Math. Chem. 2 (2011), 73–78

[37]


ECCENTRIC CONNECTIVITY INDEX OF GRAPHS WITH SUBDIVIDED EDGES

Zahra Yarahmadi Department of Mathematics, Faculty of Science, Khorramabad Branch,

Islamic Azad University, Khorramabad, I. R. Iran

Sirous Moradi Department of mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, I. R. Iran

Tomislav Doslic

Faculty of Civil Engineering, University of Zagreb, Kaciceva 26, 10000 Zagreb, Croatia

Abstract

We consider four classes of graphs arising from a given graph via di_erent types of edge

subdivisions. We present explicit formulas expressing their eccentric connectivity index in

terms of the eccentric connectivity index of the original graph and some auxiliary invariants. Keywords: Eccentric connectivity index, subdivided graph.

[38]


ON SUM DEGREE ENERGY OF A GRAPH

R . ZAFERANI1 and C . ADIGA2

1Nemone Dolati Shahid Esfahani,High Scool & Pre-Univesity , BABOL , IRAN 2 Department of Studies in Mathematics, University of Mysore, MYSORE - 570 006 , INDIA

E-mail : [email protected] and [email protected]

Abstract

In this paper , we introduce the sum degree matrix SD(G) of a simple graph G of order n and

obtain a bound for eigenvalues of SD(G). The energy of G is defined as the sum of the

absolute values of the eigenvalues of the graph G and is denoted by ESD(G) . We prove that

the energies of certain classes of graphs are less than that of ESD(Kn). We also compute the

energy of certain graphs .

Keywords : sum degree matrix , eigenvalues , energy of graph.

2000 Mathematics subject classification : 05C50 , 58C40.

Poster Presentations of the

5th Conference on Algebraic

Combinatorics and Graph Theory

July 34, 2012,


[40]


RANDI퐜 INDEX, DIAMETER AND THE AVERAGE DISTANCE OF GRAPHS

Z. Akbarzadeh Ghanaie1 and A. R.Ashrafi2

1Mathematics Department, Young Research’s Society of Shahid Bahonar University of

Kerman, Kerman, Iran 2Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan,

Kashan 87317-51167, Iran

Abstract

The Randic index R(G) of a graph G is defiend as the sum of over all edges uv of G,

where 푑 and 푑 are the degrees of vertices u and v, respectively. Let D(G) be the diameter

and µ(G) be the average distance of a graph G. Aouchiche, Hansen and Zheng [1] and

Fajtlowicz [2] proposed the conjectures on the relationship between the Randic index,

diameter and average distance.Yang and Lu [3] proved an stronger theorem about the first

conjecture. In this paper, we continue this program to find new relationship on the

Randic index R(G) to the average distance µ(G).

Keywords: Randic index; diameter; average distance. AMS Subject Classification (2000): 05C12, 05C35, 92E10.

References

[1] M. Aouchiche, P. Hansen and M. Zheng, Variable neighborhood search for extremal

graphs 19: Further conjectures and results about the Randic index, MATCH Commun. Math.

Comput. Chem. 58 (2007), 83–102.

[2] S. Fajtlowicz, On conjectures of Graffiti, Discrete Math. 72 (1988), 113–118.

[3] Y. Yang and L. Lu, The Randic index and the diameter of graphs, arXiv: 1104.0426v1,

[math.CO], April 5, 2011.

[41]


ON THE SHELL POLYNOMIAL OF SUBDIVISION GRAPHS

Z. Alem and A. Iranmanesh

Department of Pure Mathematics, Faculty of Mathematic Science, Tarbiat Modares University, P. O. Box: 14115-137, Tehran, Iran

Keyword and phrases: Shell polynomial, Cluj Tehran index, Subdivision graph.

[42]


ON CHROMATIC UNIQUENESS OF COMPLETE CERTAIN PARTITE GRAPHS

S. Alikhani1 , R. Hasni 2 and S. Mirvakili 3

1 Department of Mathematics, Yazd University, 89195-741 Yazd, Iran

2 Department of Mathematics, Faculty of Science and Technology, Universiti Malaysia Terengganu, 21030 Kuala Terengganu, Malaysia

3 Department of Mathematics, Payame Noor University, 19395-4697 Tehran, I.R. Iran

Abstract

Let G be a simple graph and N . A mapping },{1,2,)(: GVf is called a -

colouring of G if )()( vfuf whenever the vertices u and v are adjacent in G . The

number of distinct -colourings of G , denoted by ),( GP is called the chromatic

polynomial of G . Two graphs G and H are said to be chromatically equivalent, denoted

G~H, if P(G, λ) = P(H, λ). We write [G] = {H|H ~G}. If [G] = {G}, then G is said to be

chromatically unique. In this paper, we first characterize certain complete 6-partite graphs G

with 6n + i vertices for i = 0, 1, 2 according to the number of 7-independent partitions of G.

Using these results, we investigate the chromaticity of G with certain star or matching

deleted. As a by-product, many new families of chromatically unique complete 6-partite

graphs G with certain star or matching deleted are obtained.

Mathematical Subject Classification: 05C15

Keywords: Chromatic Polynomial; Chromatically Closed; Chromatic uniqueness, 6-Partite

graph.

[43]


GRAPHS, THE WAY TO THE ZERO-DIVISORS OF RINGS

Seyed Mohammad Anvariyeh and Zohreh Mirzaei

Department of Mathematics, Yazd Univercity, Yazd, Iran

Abstract

The set of zero-divisors of a ring, which is denoted by 푍(푅), is not closed under addition and

so it does not have any algebraic structure. In this paper, we are on it to describe and examine

some conditions that may create some ideals from the set of zero-divisors. In this way, we

apply the zero-divisor graph associated to a ring and its properties to explore the nature of

푍(푅). Also, we try to equip the set of zero-divisors to an algebraic structure to encourage

motivated mathematicians to work on this topic.

Introduction

A zero-divisor graph of a ring R , is a graph whose vertices are nonzero zero-divisors of R

and two distinct vertices x and y are adjacent if and only if 0.xy

Theorem 1: Let 푅 be a finite commutative ring with unity. Then 푍(푅) is an ideal if and only

if there exists a vertex in its zero-divisor graph which is connected to all other vertices of the

graph.

Theorem 2: Let 푅 be a commutative ring and 푎 is a cut-vertex of its zero-divisor graph.

Then {0,푎}is an ideal of 푅.

Theorem 3: Let 푅 be a non-local, commutative ring and 퐴 is a cut-set of its zero-divisor

graph. Then 퐴 ∪ {0}is an ideal of 푅.

[44]


References

[1] Anderson, David F. and Livingston, Philips, The zero-divisor graph of a commutative

ring, J. Algebra, 217(1999), 434447.

[2] Axtell, M., Baeth, N., and Stickles, J., Cut vertices in zero-divisor graphs of finite

commutative rings, Communications in Algebra, 39 (6), (2011), 21792188.

[3] Cote, B., Ewing, C., Huhn, M., Plaut, C. M., Weber, D., Cut-sets in zero-divisor graphs of

finite commutative rings, Communication in Algebra, (2011), 28492861.

[45]


THE LARGEST CLIQUES AND APPLICATIONS

퐇.퐀퐙퐀퐍퐂퐇퐈퐋퐀퐑ퟏ, 퐙.퐀퐒퐒퐀퐃퐈 퐆퐎퐋퐙퐀퐑ퟐ, 퐊퐇.퐒퐇퐀퐌퐒퐈ퟑ∗

Abstract

In this paper we have researched about cliques in graphs and have presented an algorithm for

finding maximum cliques. In order to show some applications of cliques we have found the

maximum clique in air lines for some cites in Iran. At last we have applied the properties of

cliques for computing elements of the factors of elementary p-groups.

Let (G;V) be a simple graph and (H;U) be the subgraph of G. If each two distinct

elements of H are always connected in G by an edge of it, then this subgraph H of G spanned

by U is called a clique of G. Maximum cliques, which are the largest among all cliques in a

graph and maximal cliques, whose vertices are not a subset of the vertices of a larger clique

have been studied over the last three decades.

In this paper, the maximum cliques are studied. The maximum clique problem is

computationally equivalent to some other important graph problems, for example, the

maximum independent set problem and the minimum vertex cover problem.

[46]


THE JACOBIAN OF GRAPHS

M. Bahramian

Department of Pure Mathematics, Faculty of Mathematical Sciences,


Abstract

Let G = (V,E) be a graph and Div(G) be the free abelian group generated by V(G). The

elements of Div(G) are called divisors on G. For a divisor D, the coefficient of (v) in D is

denoted by D(v). For the divisor D, define degree of D, the sum of nonzero coefficients of D

and denote by deg(G) and let Div^0(G) be the subgroup of Div(G) consisting of divisors of

degree zero and let Princ(G) be the group of principal divisors. J(G) = Div0(G)/Princ(G) is

well-defined and is a finite abelian group. This group called the Jacobian of the graph G. In

this paper we find the structure of Jacobian group in some graphs.

Keywords: Graph, Jacobian of a Graph, Smith Normal form.

[47]


THE SZEGED EIGENVALUES OF THE PATH

M. Bahramian and G. H. Fath-Tabar



Abstract

Let G=(V,E) be a graph. For 푒 = 푢푣 ∈ 퐸, 푛 (푒) is the number of vertices of G closer to u

than to v, and 푛 (푒) is defined analogously. For a given graph G its Szeged weighting is

defined by 푤(푒) = 푛 (푒)푛 (푒), where e = uv is an edge of G, where 푛 (푒) is the number of

vertices of G closer to u than v and 푛 (푒) is defined analogously. The adjacency matrix of

this weighted graph is called Szeged matrix of G, and denoted by SzM(G). In this paper we

determine the eigenvalues of 푆푧푀(푃 ) where 푃 is the path with n vertices, and calculate the

eigenvalues of the 푚 × 푛 lattice by using it.

Keywords: Sz-Matrix, eigenvalue.

[48]


REGULAR CIRCUIT GRAPHS OF UNIFORM MATROIDS

Asiyeh Chatrazar1, Najibeh Shahbaznejad2, Madineh Shayan1 and Azam Chatrazar3

1Urmia University, Urmia, Iran 2Zanjan University, Zanjan, Iran

3Shiraz University, Shiraz, Iran

Abstract

In this paper, we prove circuit graphs of uniform matroids are regular and we will be obtain degree of every vertex and the number of edges in circuit graphs of uniform matroids .

Keywords: Matroid, circuit graph of matroid, uniform matroid.

[49]


THE NUMBERS OF EDGES IN CIRCUIT GRAPH OF MATROID U1,N

Asiyeh Chatrazar1, Madineh Shayan1, Azam Chatrazar3 and Najibeh Shahbaznejad2

1Urmia University, Urmia, Iran 2Zanjan University, Zanjan, Iran

3Shiraz University, Shiraz, Iran

Abstract

In this paper, we prove that numbers of edges in circuit graph of matroid 푈 , are (푛(푛 −

1)(푛 − 2))/2, for all 푛 ≥ 2.

Keywords: Matroid, circuit graph of matroid.

[50]


ON THE RANK OF A SPECIAL FAMILY OF ELLIPTIC CURVES

Hassan Daghigh and Somayeh Didari

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan , Iran

Abstract

By the MordellWeil theorem, the group of rational points on an elliptic curve is a finitely

generated abelian group. There is no known algorithm for finding the rank of an elliptic

curve. This paper gives a survey of rank of elliptic curves over rational numbers. In particular

we investigate the rank of special families of elliptic curves over the field of rational

numbers.

Keywords: Elliptic curves, Mordell Weil Group, Selmer Group, Birch and Swinnerton Dyer

Conjecture, Parity Conjecture.

[51]


THE MAXIMUM RANDIC INDEX OF TRICYCLIC AND TETRACYCLIC

GRAPHS

T. DehghanZadeh and A. R. Ashrafi

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731751167, I. R. Iran

Abstract

Let G be a simple and connected graph with n vertices and m edges. The cyclomatic number

of G is defined as c(G) = m n + 1. The graphs with c = 3, 4 are called tricyclic and

tetracyclic, respectively. Throughout this paper G is such a graph. The Randić index of G is

defined as R(G) = u,v[d(u)d(v)]1/2, where d(u) denotes the degree of a vertex u and the

summation runs over all edges uv of G. This topological index was first proposed by Randić

[1] in 1975. It is suitable for measuring the extent of branching of the carbonatom skeleton

of saturated hydrocarbons. In this paper, the first and second maximum of the Randić index

in the class of all tricyclic and tetracyclic nvertex connected graphs are computed. It is

proved that, the first and second maximum of this topological index in the class of tricyclic

graphs are 26

625 n

and23

625 n

, respectively. Moreover, the first and second

maximum of this topological index in the class of tetracyclic graphs are 26

625 n

and

221266 n

, respectively. The graphs with these extremal values are depicted in

Figure 2.

Keywords: Randić index, tricyclic graph, tetracyclic graph.

[52]


References

1. M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97

(1975) 6609–6615.

[53]


FURTHER RESULTS ON THE INNER POWER OF GRAPHS

S . D jafari and A . R . Ashrafi

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 8731751167, I. R. Iran

Abstract

All graphs in this paper are finite without multiple edges . A graph invariant is any function

on a graph that does not depend on a labeling of its vertices . If a graph invariant has

application in chemistry , it is called topological index. Among topological indices two that

are known under various names, but mostly as Zagreb indices . Due to their chemical

relevance they have been subject of numerous papers in chemical literature while the first

Zagreb index , also attracted a significant attention of mathematicians. Let G be a graph with

vertex and edge sets V(G) and E(G), respectively . For every vertex u V(G) , the edge

connecting u and v is denoted by uv and degG(u) denotes the degree of u in G . We will omit

the subscript G when the graph is clear from the context.

The first and second Zagreb indices were originally defined as M1(G) =

∑ deg(u) ( )2 and M2(G) = ∑ [deg(u)deg(v)] ( ) , respectively . The first Zagreb

index can be also expressed as a sum over edges of G, M1 (G) = ∑ [deg(u) + ( )

deg(v)], We refer the reader to [9] for the proof of this fact .The readers interested in more

information on Zagreb indices can be referred to [1, 3, 6, 8, 9, 10 ] and references therein.

For the sake of completeness we state the exact definition of graph power . Given a

graph G , and a positive integer k , the kth inner power of G is the graph G(k) defined as

follows:

V(G(k)) = { (x0 , x1 , ... , xk-1) | xi \in V(G) for 0 ≤ i < k},

E(G(k)}) ={ (x0 , x1 , … , xk-1)(y0 , y1 , … , yk-1) | xi yi±1 E(G) for 0 ≤ i < k},

where arithmetic on the indices is done modulo [4].

[54]


A graph G with the vertex set V(G) is bipartite if V(G) can be partitioned into two

subsets V1 and V2 such that all edges have one endpoint in V1 and the other in V2 . The

smallest number of edges that have to be deleted from a graph to obtain a bipartite spanning

subgraph is called the bipartite edge frustration of G and denoted by )(G . It is easy to see

that )(G is a topological index and G is bipartite if and only if )(G = 0.

Throughout this paper our notation is standard . For terms and concepts not defined

here we refer the reader to any of several standard monographs. We will prove some new

extremal values of Zagreb coindices, degree distance and reverse degree distance over some

special classes of graphs are determined . We begin by Zagreb coindices of graphs.

[55]


ON THE ENERGY OF THE COMPLEMENT K

TH ITERATED LATIN SQUARE

LINE GRAPHS

Z. Fallahzadeh and Mohammad A. Iranmanesh

Department of Mathematics, Yazd Univercity, Yazd, 89195-741, Iran

Abstract

In this paper we consider the enrgy of of the complement kth iterated Latin square line graphs

and we will obtain an upper bound for this energy.

INTRODUCTION

The energy of a graph is sum of the absolute values of its eigenvalues. A Latin square is an 푛

-by- 푛 grid, each entry of which is a number between 1 and 푛, such that no number appears

twice in any row or column. A Latin square graph has 푛 vertices, one for each cell in the

square. Two vertices are joined by an edge if the following holds.

1. They are in the same row,

2. They are in the same column, or

3. They hold the same number. MAIN RESULTS

Theorem 1. Let G be a regular graph of order 푛 and of degree r ≥ 3.

Let 푛 and 푟 be the order and degree of the kth iterated line graph L (G), k ≥ 2,

퐸 퐿(퐺) = (푛 푟 − 4)(2푟 − 3) − 2 = (2푛 − 4)(푟 − 1)− 2.

Corollary 2. Let G be a Latin square graph of order n such that 푛 ≥ 4 푇ℎ푒푛,

퐸 퐿(퐺) = (푛 푟 − 4)(2푟 − 3) − 2 = (2푛 − 4)(푟 − 1)− 2.

REFERENCES

[56]


[1] N. Biggs, Algebraic graph theory, Cambridge University Press 1974.

[2] H. B. Walikar, H. S. Ramane , I. Gutman, S. B. Halkarni, On equienergetic graphs and

molecular graphs, Kragujevac J. Sci. 29 (2007) 73-84.

[3] D. Cvetkovic, M. Doob, H. Sachs, Spectra of graphs - Theory and Applications,

Academic Press, New York, 1980.

[57]


RESOLVING SHORTEST PATH PROBLEM IN WEIGHTED GRAPH USING

HIGH PERFORMANCE COMPUTER

Azam Farokh, Seyed Morteza Babamir, Mohamad Mehdi Morovati, Hoda Banki

Faculty of Engineering, University of Kashan, Iran

Abstract

Traveling Salesman Problem (TSP) is a classical NP-hard problem in computer science. For n

cities, TSP is the problem of finding a tour visiting all cities exactly once and returning to the

starting city such that the sum of the distances between cities becomes minimum. While there

is no known algorithm that solves the TSP in general travelled, many optimizations and

limitations can lead to finding a satisfying solution. For solving TSP, a weighted graph is

considered that vertices represent cities and edges weight represents the distance between

cities. The shortest path in the graph passes through every vertex exactly once and then back

to the starting city.

There are many methods for solving TSP and the most direct solution would to try for

all permutations (ordered combinations) in order to which one is the cheapest. The running

time for this approach lies within the polynomial factor of O(n!), which n indicates the cities,

So this solution becomes impractical 푛 ≥ 15.

This paper aims to present a parallel and distributed method for solving the TSP in a

weighted graph using a high performance computer (HPC) cluster and Microsoft Windows

HPC Server 2008 R2 platform. Microsoft Windows HPC has been designed for programs that

require clusters with capability of high performance computing. A cluster is a technique in

which many computers are connected to each other by a network for parallel processing to

achieve high performance computing. High performance computing is a branch of computer

science that concentrates on processing large amount of data. Using the proposed method, the

problem can be solved for many cities. Also by taking advantages of parallelism and

distribution, the time of solving the problem is reduced and thus the performance is increased.

[58]


In this method, head node divides problem space into smaller regions that can be like the

Figure 1. Then head node distributes these regions between compute nodes equally.Compute

nodes perform computations of their related regions in parallel. Each node sends its results to

the head node and at last the head node combines these results and determines the appropriate

route.

Figure1- The problem space which is divided into smaller regions

[59]


A DISTANCEBASED GRAPH INVARIANT AND THE ECCENTRIC

CONNECTIVITY INDEX OF GRAPH OPERATIONS

KHADIJEH FATHALIKHANI1 AND HASSAN YOUSEFIAZARI2

1Department of Mathematics, University of Kashan, Kashan, Iran

2School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran

Abstract

A topological index of a graph G is a number Top(G) which is invariant under graph

isomorphism. In the late 1940s, the most well known parameter, Wiener index, was

introduced to analyze the chemical properties of paraffins (alkanes). This is a distance-based

index, whose mathematical properties and chemical applications have been widely

researched. Numerous other indices have been defined. One of the recent ones is the eccentric

connectivity index which is a topological model. This index gives a high degree of

predictability of pharmaceutical properties, and may provide leads for the development of

safe and potent anti-HIV compounds. The aim of this paper is to investigate some

mathematical properties of the eccentric connectivity index by defining a distance-based

invariant.

Consider a simple connected graph G, and let V (G) and E(G) denote its vertex and

edge sets, respectively.The distance between u and v in V (G), dG(u,v), is the length of the

shortest u - v path in G. The eccentricity of a vertex uV(G), ecG(u), is the maximum distance

between u and any other vertex in G. The eccentric connectivity index of a graph G, )(Gc , is

defined as )( )(deg)()( GVv GGc vvecG . In this paper, the eccentric connectivity index of

some graph operations like the Cartesian product, Join, Symmetric difference and tensor

product is computed.

Keywords: Eccentric connectivity index, graph operations, eccentricity

[60]


APPLICATIONS OF GRAPH THEORY PROBLEMS IN

BIOINFORMATICS

Leila Ghanbari1 and Soghra MikaeylNejad2

1Department of Computer Science, Kashan University, Kashan, Iran, [email protected] 2Department of Computer, Payame Noor University, Tabriz, Iran, [email protected]

Abstract

The Graph-Theory is one of the most applicable fields of mathematics in other sciences.

Graphs are excellent structures for storing, searching, and retrieving large amounts of data.

Discussing in such approaches is very attractive, because using of such tool helps researchers

to think simpler. Graphs can be used to represent networks embodying many different

relationships among data. By reducing an instance of a problem to a standard graph problem,

we may be able to use well-known graph algorithms to provide an optimal solution for the

main problem. In this paper we want to introduce some of applications of most popular

problems of graph theory in Bioinformatics in brief. Bioinformatics is one of the modern

sciences, which searches for some ways to extracts information from biological molecules,

same as DNA, RNA, Protein and etc. The definition of this science in brief is conceptualizing

biology in terms of molecules (in the sense of Physical chemistry) and applying informatics

techniques (derived from disciplines such as applied math, computer science and statistics) to

understand and organize the information associated with these molecules, on a large scale. In

short, bioinformatics is a management information system for molecular biology and has

many practical applications [7]. The amount of data about these molecules or macro

molecules is growing very fast. Extracting information and reach to the knowledge about

biological molecules need to have powerful tools for modeling the information. One of such

[61]


tools is graph. There are too many applications such this for graph theory in bioinformatics,

which can enumerate some of them in brief as bellow:

Using Hamiltonian cycle and Euler circuit for sequencing DNA/shortest superstring

problem [1],

Clique based data mining for related genes in biomedical database and computing

protein interaction modules [2,8],

Using graph matching method for global alignment of protein-protein interaction

networks [12].

References

[1] Blazewicz J. and Kasprzak M.,"Graph reduction and its application to DNA sequence

assembly", Bulletin of the Polish Academy of Sciences, Technical Sciences 56, 6570,

(2008).

[2] Dinga Ch., Wangb Ch., Yanga Q. and Holbrooka S., "Computing Protein Interaction

Modules via Clique Finding based on Generalized MotzkinStrauss Formalism", a

Lawrence Berkeley National Laboratory, Berkeley, CA 94720, Stanford University,

Stanford, CA 94305 (2007).

[3] Eisenberg E. and Levanon E. Y., "Preferential attachment in the protein network

evolution", Phys Rev Lett, 91 (13), 138701 (2003).

[4] Fera D., Kim N., Shiddelfrim N., Zorn, J., Laserson, U., Gan, H. H. and Schlick, T.,

"RAG: RNAAsGraphs web resource", BMC Bioinformatics, 5:88 (2004).

[5] Forman J. J., Clemons P. A., Schreiber S. L. and Haggarty S. J., “Spectral NET an

Application for Spectral Graph Analysis and Visualization”, BMC Bioinformatics, 19:6

260, (2005).

[6] Haggarty S., Clemons P. and Schreiber, S., "Chemical genomic profiling of biological

networks using graph theory and combinations of small molecule perturbations", J Am.

Chem. Soc. 125:1054310545 (2003).

[62]


THE ENERGY OF CAYLEY GRAPHS ON DIHEDRAL GROUPS

F. Gilasi1, and A. R. Ashrafi2

1Department of Mathematics, Sahand University of Technology, Tabriz, Iran

2Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I. R. Iran

Abstract

The Cayley graph Cay(G; S) on a finite group G relative to an nonempty subset S of

G\{1},with property if s S, then s-1 S, is the graph with the vertex set G and gh is an edge

if and only if h−1g S [1]. Suppose is a simple graph with n vertices and adjacency matrix

A(). Let eigenvalue set of A() is 1, 2, … ,n, such that 1 ≥ 2 ≥ … ≥ n, that called the

spectrum of , denoted by Spec(). The energy E() of a graph is defined as the sum of

the absolute values of its eigenvalues [2]. In this paper, the energy of Cayley graphs on a

dihedral group is computed.

References

1. Stefko Miklavic and Primoz Potocnik: Distance-regular Cayley graphs on dihedral

groups. J. Comb. Theory, Ser. B 97(1): (2007)14-33.

2. Thirugnanam Tamizh Chelvam, Sekar Raja and Ivan Gutman, Strongly Regular

Integral Circulant Graphs and their Energies, BULLETIN OF IMVI 2 (2012), 9-16.

Corresponding author (Email: [email protected]).

[63]


ON THE DIFFERENCE OF WIENER AND DETOUR INDICES OF GRAPHS

Z. Gholam-Rezaei and G. H. Fath-Tabar


Abstract

The Wiener and detour index of connected graph G are defined as 푊(퐺) =

∑ 푑(푢, 푣){ , }⊂ ( ) and 휔(퐺) = ∑ 퐷(푢,푣){ , }⊂ ( ) where d(u,v) is the

distance of vertices u and v and D(u,v) is the length of the longest path

between u and v. In this paper, some results on the difference between

Winer index and detour index of graphs are presented.

Keywords: Wiener index, detour index, graph.

Introduction and Results

Let G = (V,E) be a simple connected graph with vertex and edge sets V(G) and E(G),

respectively. The distance between vertices u and v of G is denoted by dG(u, v) and detour u

and v, D(u,v) is defined as the number of edges in a maximal path connecting u and v. A

topological index is a numerical quantity related to a graph that is invariant under all graph

isomorphisms. The Wiener index W(G) was the first distance-based topological index, it is

defined as the sum of all distances between vertices of G [1] and detour index of G is defined

as 휔(퐺) = ∑ 퐷(푢, 푣){ , }⊂ ( ) . Hosoya [2] was the first scientist to introduce the name

‘‘topological index’’. In this paper, we present some formula for the difference of Winer

index and detour index λ(G)=훚(G)-W(G) of graphs. Theorem 1.λ(G)=0 if and only if G is a tree. There is any graph with λ(G)=1, λ(G)=2,

λ(G)=4 and λ(G)=10. λ(G)=3 if and only if G is complete graph on 3 vertices.

[64]


Theorem 2. Let G be a unicycle graph with cycle Cn and m cut edges then

휆(퐺) =

⎩⎪⎨

⎪⎧ 푛(푛 − 1)

4 +푚(푛 − 1)

2 + 푚 푚 |푛 − 2푗 + 2푖| 푖푓 푛 푖푠 표푑푑

푛 (푛 − 2)4 +

푚푛(푛 − 2)2 + 푚 푚 |푛 − 2푗 + 2푖| 푖푓 푛 푖푠 푒푣푒푛

,

where mi is the number of cut edges on vertex vi∈V(Cn).

References

1. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69

(1947) 17–20.

2. H. Hosoya, Topological index, a newly proposed quantity characterizing the topological

nature of structure isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (1971) 2332–

2339.

[65]


AN APPLICATION OF CRYSTAL GRAPHS ON LIE SUPERGROUP

F. Gholami

Department of Basic Science, Shahrood branch, Islamic Azad University, Shahrood, Iran

F. Pashaie

Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, Iran

Abstract

The aim of this talk paper is to explain a graph theoric representation of the Cauchy

decomposition for Liesuperalgebras which has been developed in [1]. Mainly, by the crystal

graph theory , the set of all (m, n)-hook semistandard tableaux of shape λ, can be denoted as a

colored oriented graph, which is called glm|n-crystal. We give well-behavioritiesof glm|n-

crystal, with respect to some well known operations. A real Lie supergroup is a real

supermanifoldG = (G,AG) where the supercoalgebra퐴∗ has the structure of a super co-

commutative Hopfsuperalgebra such that multiplication and antipode are push-forwards of

morphisms of real supermanifolds. The set of all left-invariant superderivations on a Lie

supergroup G defines a Lie superalgebra related to G.

Let 푔=푔 ⊕ 푔 be a simple Lie superalgebra with highest weight휆. Let 휋denote a

finitedimensional representation of 푔. The crystal graph of the representation 휋(g) is a

colored graph Γ = (푉,퐸), where, V is the partially ordered set of all weights of g and E

consisted by edges relating vertices v , v' ∈V if there exists a simple root vector Y of g s.t.휋

(X)v = v'.

We explain the notion and primary results on crystal graphs for the general linear

supergroup [2], and extend the Cauchy decomposition to a semi-infinite type and point out a

relation between the semi-infinite type and a special Cauchy type identity.

[66]


Keywords: Crystal graphs, Lie supergroup, Cauchy decomposition.

References

1. G. Benkart, S. J. Kang and M. Kashiwara, “Crystal bases for the quantum superalgebra

Uq(gl(m, n))”, J. Amer. Math. Soc. 13 (2) (2000), 295–331.

2. J. H. Kwon, ''Crystal graphs for Lie superalgebras and Cauchy Decomposition'', J. Alg.

Comb. 25 (2007) 57–100.

[67]


THE ORBITS OF THE AUTOMORPHISM GROUP ACTION OF HAMMING

GRAPHS

F. Gholaminezhad and A. R. Ashrafi

Department of Nanocomputing, Institute of Nanoscience and Nanotechnology, University of

Kashan, Kashan 87317-51167, I R IRAN

E-mail: corresponding author's [email protected]

Abstract

An automorphism of a graph G is a permutation 휎 in V(G) with the property that ∀ 푢,푣 ∈

푉(퐺), u is adjacent to v if and only if 휎(푢) is adjacent to 휎(푣). The set of all automorphism

of a graph G is the group 퐴푢푡(퐺). It’s proved by Peter J. Cameron [3], that any graph and its

complement have the same automorphism group and 퐴푢푡 (퐾 ) = 푆 .

The direct product of two groups H and H′ is H × H′ such that (h , h ) h′ , h′ =

(h′ h′ , h h′ ). we define the semi-direct product of H′and H by H′ ⋊ H = {(h′, h) h′ ∈ H′, h ∈

H} and (h , h ) h′ , h′ = (h′ h (h ), h h ). Suppose we have a group H acting on a set X.

Then consider the Cartesian product 푌 = 푋 × 푋 × ⋯ × 푋 , where each Xi = X. If 퐻′ is a

group of permutations acting on the labels {1,2,⋯ ,푛}, the semi-direct product of 퐻 by 퐻′ is

the wreath product 퐻 ≀ 퐻′.

The Cartesian product of two graphs 퐺 and 퐺 ′is the graph 퐺□퐺 ′ where 푉(퐺□퐺 ′) =

푉(퐺) × 푉(퐺 ′) and (푢,푢′) is adjacent to (푣,푣 ′) if and only if either 푢 = 푣 and 푢′푣 ′ ∈ 퐸(퐺′)

or 푢′ = 푣′and 푢푣 ∈ 퐸(퐺). For a given n, denote the complete graph and H(n,m) the Hamming

graph obtained by Cartesian product of n complete graphs 퐾 . The Hypercube 푄 = 퐻(푛, 2)

is the special class of Hamming graph in which each vertex is the sequence

푢 = (푢 , 푢 ,⋯ ,푢 ) where 푢 = 0 표푟 1, and two vertices u and v are adjacent if and only if

the sequences u and v differ in exactly one place. 푄 has 2n vertices, 2n-1n edges and is n -

regular.

[68]


In this paper we would obtain the orbits of automorphism group action of Hypercube 푄

and Hamming graphs H(m,n).

Keywords: Hamming Graphs, Hypercube, Automorphism group Action of Graph

References

[1] F. A. Chaouche, A. Berrachedl, Automorphisms group of generalized Hamming Graphs,

Electronic Notes in Discrete Mathematics, 24 (2006) 915.

[2] M. R. Darafsheh, Computation of Topological Indices of Some Graphs, Acta Appl. Math.

110 (2010) 1225-1235.

[3] Peter J. Cameron, Topics in Algebraic Graph Theory (ed L W Bemeke, R. J. Wilson)

Cambridge, 2005.

[69]


NOTE ON EXTENDED DOUBLE COVER GRAPHS

MODJTABA GHORBANI AND MAHIN SONGHORI

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, I. R. Iran

ABSTRACT

For a simple graph G with vertex set V = {v1,v2,…,vn}, the extended double cover of G,

denoted G*, Is the bipartite graph with bipartition (A,B) where A = {a1,a2,…,an},

B={b1,b2,…,bn}, in which aiand bjare adjacent if and only if i=j or viand vjare adjacent in G.

In this paper we study some properties of extended double cover graphs.

Keywords: Cover graphs, Wiener index, Szeged index, weighted PI index.

[70]


NONCOMMUTING GRAPH AND POWER GRAPH OF FINITE GROUPS

Modjtaba Ghorbani and Farzaneh Nowroozi Larki

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785 136, I. R. Iran

Abstract

Let G be a finite group. The power graph of G is a graph whose vertex set is the elements of

G, two elements x; y G being adjacent if x = ym, for some integer m. The non-commuting

graph of G is defined as the graph whose vertex set is G \ Z(G), and two distinct vertices x

and y are joined by an edge whenever xy yx. We denote this graph by (G) correspondence

with group G. In this paper we obtain the power graph and non-commuting graph of some

well-known groups.

Keywords: non-commuting graph, power graph, Dihedral group.

[71]


ECCENTRIC DISTANCE SUM OF COMPOSITE GRAPHS

MODJTABA GHORBANI

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, I. R. Iran

Abstract

The eccentric distance sum is one of the topological discriptor has been recently used in

many studies in graph theory. In this paper we study how it behaves under several binary

operations on graphs. Our main results are explicit formulas for the eccentric distance sum of

most common classes of composite graphs.

Keywords: eccentric distance sum, distance sum, eccentricity, Cartesian product, composition of

graphs, join of graphs.

[72]


A NOTE ON ECCENTRIC DISTANCE SUM

Modjtaba Ghorbani and Mahin Songhori

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran,

16785 136, I. R. Iran

Abstract

A graph invariant namely, the eccentric distance sum is defined as X as

)( )(deg)()( GVv GGc vvecG , where ecG(v) is the eccentricity of a vertex v in G and DG(v)

is the sum of distances of all vertices in G from v. In this paper, we compute the eccentric

distance sum of Volkmann tree and then we obtain some results for vertex transitive graphs.

Keywords: eccentricity, eccentric distance sum, Volkmann tree.

[73]


QCONJUGACY CHARACTER TABLE OF DIHEDRAL GROUP

E. Haghi and A. R. Ashrafi

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I R IRAN

E-mails: [email protected] & [email protected]

Abstract

The QConjugacy character table are proposed for finite groups and applied to combinatorial

enumeration. Thus, the maturity of an irreducible representation is related to the maturity of a finite

group by means of the relationship between the inherent automorphism of the group and its inner

portion. As a result, a character table is transformed into a more concise form called a Qonjugacy

character table. Matured characters are defined as dominant-class functions on the basis of such a

Qconjugacy character table. Thereby, a matured character is represented by a linear combination of

Qconjugacy characters. In [2], S. Fujita defined the Qconjugacy character table and conjectured for

cyclic group. In this article, we consider the dihedral groups and get a conjecture for Qconjugacy

character table of these groups. Keywords: Q-conjugacy character table, dihedral group.

References

1. W. Burnside, Theory of Groups of Finite Order, The University Press, Cambridge,

1987.

2. S. Fujita, Markaracter tables and q-conjugacy character tables for cyclic groups, an

application to combinatorial enumeration, Bull. Chem. Soc. Jpn., 71, (1998), 1587-

1596.

[74]


COMPUTING SPECTRAL, LAPLACIAN SPECTRAL AND SEIDEL SPECTRAL

DISTANCES OF HYPERCUBES, ITS COMPLEMENT AND THEIR LINE

GRAPHS

Mardjan HakimiNezhaad and Ali Reza Ashrafi

Department of Pure Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, I R Iran

Abstract

Suppose M1 and M2 are two n n matrices with eigenvalues

),M(...)M()M( ini2i1 i = 1, 2. The spectral distance between M1 and M2 is

defined as:

n

1i 2i1i21 .|)M()M(|)M,M(

The Seidel adjacency matrix of a graph Γ with adjacency matrix A is the matrix S

defined by AIJS 2 . The ndimensional hypercube Qn is the simple graph whose

vertices are the thentuples with entries in {0,1} and whose edges are the pairs of ntuples

that differ in exactly one position, the number of its vertices and edges are 2n and n2n−1,

respectively. In this paper the spectral, Laplacian spectral and Seidel spectral distances of the

hypercube and its complement, as well as the line graphs of hypercube and the line graph of

the complement of hypercube are computed.

References

1. N. Alon, Eigenvalues and expanders, Combinatorica 6(1986), 83–96.

2. A. E. Brouwer, W. H. Haemers, Spectra of graph( www.win.tue.nl/~aeb/2WF02/

spectra.pdf)

3. X. Chen, Energy of a Hypercube and its Complement , Int. J. Algebra, Vol. 6, 2012,

no. 16, 799 – 805.

[75]


4. Xu Jin and Qu Ruibin, The spectra of hypercubes. Gongcheng Shuxue Xuebao 16

(1999), no. 4, 15.

5. Z. Chen, Spectra of extended double cover graphs, Czechoslovak Math. J. 54 (2004),

1077–1082.

[76]


ON THE SPLICE AND LINK OF SOME GRAPHS

A. Hamzeh and A. Iranmanesh

Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box: 14115-137, Tehran, Iran

Abstract

Suppose G and H are graphs with disjoint vertex sets. Following T. Doslic, for given vertices

u 2 V (G) and v 2 V (H) a splice of G and H by vertices u and v, (G:H)(u; v), is dened by

identifying the vertices u and v in the union of G and H. Similarly, a link of G and H by

vertices u and v is defined as the graph (G H)(u; v) obtained by joining u and v by an edge in

the union of these graphs. In this paper, we obtain some results over splice and link of some

graph.

Keywords : Splice, Link, Graph operation.

[77]


HYPERWIENER INDEX AND SCHULTZ INDEX OF GENERALIZED

BETHE TREES

Abbas Heydari

Department of Mathematics, Islamic Azad University, Arak Branch, Arak, Iran

Abstract

A topological index of a graph is a real number related to structure of the molecular graph. It

does not depend on the labeling or pictorial representation of the graph. Let Tbe a tree, u and

v are arbitrary pairs of vertices of T. These vertices are joined by a unique path which we

denote by p. Denote by nu(p) the number of vertices of T lying on one side of the path p,

closer to vertex u. Denote by nv(p) the number of vertices of T lying on the other side of the

path p, closer to vertex v. Then the hyper-Wiener index of T is defined as

푊푊(푇) = 푛 (푝)푛 (푝).

In which the summation goes over all paths of T. The Schultz index of a molecular graph G

was introduced by Schultz in 1989 for characterizing alkanes by an integer as follow:

MIT(G)= ∑ deg(푖) [푑(푖, 푗) + 퐴(푖, 푗)], ,

where deg(i) is vertex degree of i and A(i, j) is the (i, j) entry of the adjacency matrix of G.

A generalized Bethe tree Bk is an unweighted rooted trees with k levels such that in

each level the vertices have equal degree. For example factorial trees, dendrimer trees and

starlike trees in which all of the pendent vertices have equal distances from central vertex

(regular Starlike trees) can be considered as generalized Bethe tree. If all of the non pendent

vertices of tree have equal degree tree is called dendrimer tree. Denote by Dp,r the dendrimer

tree with r + 1 levels in which degree of non pendent vertices is p + 1. In this paper we

compute the hyper-Wiener index and Schultz index of generalized Bethe tree in term of

number of levels and degrees of vertices of the tree. As application of introduced methods the

hyper-Wiener and Schultz index of dendrimer and regular starlike trees will be calculated.

[78]


NARUMI-KATAYAMA INDEX OF GRAPHS

M. A. Hosseinzadeh1, A. Iranmanesh1, T. Doslic2

1Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box: 14115-137, Tehran, Iran

2Faculty of Civil Engineering, University of Zagreb, Kaciceva 26, 10000 Zagreb, Croatia.

Abstract

The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the

degrees of the vertices of G. In this paper we compute this index for Splice and Link of two

graphs. At least with use of Link of two graphs, we compute this index for a class of

dendrimers. With this method, the NK index for other class of dendrimers can be computed

similarly.

Keywords : Narumi-Katayama Index, Splice, Link, Chain graphs, Dendrimers.

[79]


THE COMMON NEIGHBORHOOD OF COMPOSITE GRAPHS

S. Hossein-Zadeh and A. Iranmanesh

Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O.Box:14115-137, Tehran, Iran

Abstract

In this paper, we only consider simple and connected graphs. Let G be a connected graph

with vertex and edge sets V (G) and E(G), respectively. The Cartesian product G H of

graphs G and H is a graph such that V (G H) = V (G) V (H), and any two vertices (a; b)

and (u; v) are adjacent in G H if and only if either a = u and b is adjacent with v, or b = v

and a is adjacent with u. The join G = G1 + G2 of graphs G1 and G2 with disjoint vertex sets

V1 and V2 and edge sets E1 and E2 is the graph union G1⋃G2 together with all the edges

joining V1 and V2. The composition G = G1[G2] of graphs G1 and G2 with disjoint vertex

sets V1 and V2 and edge sets E1 and E2 is the graph with vertex set V1 V2 and u = (u1, v1)

is adjacent with v = (u2, v2) whenever (u1 is adjacent with u2) or (u1 = u2 and v1 is adjacent

with v2). In this paper, we compute the common neighborhood of these graph operations. Keywords : Cartesian product, Composition, Join, Common neighborhood graph.

[80]


SOME BOUNDS ON THE SPECTRAL MOMENTS OF GRAPHS

S. Irandoost and G. H. Fath-Tabar


Abstract

Suppose G is a graph and λ1, λ2,..., λn are the eigenvalues of G. The kth spectral moments of

G, Sm(G, k) is defined as the sum of λik, i=1, 2,…, n. In algebraic graph theory is proved the

equality of Sm(G, k) and the number of all closed walks with length k in G. We present some

new bounds for the spectral moments of graphs.

Keywords: Wiener index, detour index, graph.

Introduction and Results The adjacency matrix of graph G is a 0, 1 matrix A(G) = [aij], where aij is the number of

edges connecting vi and vj. The spectrum of G is the set of eigenvalues of A(G), together with

their multiplicities. It is a wellknown fact in algebraic graph theory that a graph of order n

has exactly n real eigenvalues λ1, λ2,·· , λn. The basic properties of graph eigenvalues can be

found in the famous book of Cvetkovic et al. The kth spectral moments of G, Sm(G,k) is

defined as the sum of λik, i=1, 2,…, n. The Laplacian Spectral moments L Sm(G,k) is defined

by Laplacian spectrum of G as Sm(G, k).

Theorem 1. Let G be a graph with n vertices and m edges then

푆푚(퐺, 2푘) ≤ (푛 − 푛 )(2푚

푛− 푛 )

where n0 is the number of zero eigenvalues of G.

[81]


Theorem 2. Let G be a connected graph with n vertices and m edges then

LSm(G, k) ≤ (n − 1)( ) .

References

1. D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs-Theory and Application, Third

ed., Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

[82]


ON PROPER POWER GRAPHS

Mohammad A. Iranmanesh and Seyed M. Shaker .

Department of Mathematics,

Yazd University, Yazd, 89195-741, Iran

[email protected] [email protected]

ABSTRACT

Let G be a group. The power graph P(G) is a graph with vertex set G and in which two distinct vertices x and y are joined if one is a power of the other. The proper power graph P (G) is a subgraph of P(G) obtained by omitting the identity element and its adjacent edges. In this paper we conclude the number of connected component of S for 2 ≤ n ≤ 7.

Keywords: Power graphs, Proper power graphs, Connected component.

INTRODUCTION

Let G be a group. The power graph P(G) was defined by Chakrabarty, Ghosh and Sen, as a

graph with vertex set G and two distinct vertices x and y are joined if one is a power of the

other. Until now authors obtained some results about P(G), (see [1, 2, 3]), for example they

proved that for every finite group G, the power graph P(G) is connected because the identity

element 1 is adjacent with every elements of G. But what happened if we omit the vertex

1 ? For answer to this question we consider the proper power graph P (G) as a subgraph of

P(G) obtained by omitting the vertex 1 and its adjacent edges. By c (G) we denote the

number of the connected components of P (G), which is not always equal to 1. In this paper

we prove the following theorem.

MAIN RESULT

Theorem 1. In the following table, c (S ) is calculated for 2 ≤ n ≤ 7.

[83]


7 6 5 4 3 2 n 226 185 31 13 4 1 c (S )

REFERENCES

[1] P. J. Cameron, The power graph of a finite group II, J. Group theory 13, 779-783 (2010).

[2] I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups,

Semigroup Form 78, 410-426 (2009).

[3] M. Mirzargar, A. R. Ashrafi, M. J. Nadjafi-Arani, On the power graph of a finite group,

To appear in Filomat.

[84]


KEYPRIDISTRIBUTION SCHEME FOR HIERARCHICAL WIRELESS SENSOR

NETWORKS USING COMBINATORIAL DESIGNS

M. Javanbakht and H. Haj Seyyed Javadi

Mathematics and Computer Science, Shahed University, Tehran, Iran

Abstract

Combinatorial designs are powerful mathematical tools that have had very deep impact on

applied fields such as coding theory, communications and cryptography in recent years. We

review one of these applications, key predistribution in wireless sensor networks (WSNs),

and develop it from homogeneous WSNs to Hierarchical WSNs.

Keywords: Combinatorial design, Security wireless sensor network, Key predistribution,

Transversal design.

[85]


COMPLEXITY ANALYSIS OF NESTED IF ORDERS BY REWRITING

SYSTEMS: A COMBINATORIAL SOLUTION

Mohammad Kadkhoda

Mathematics and Informatics Research Group, ACECR, Tarbiat Modares University, P. O. Box: 14115343, Tehran, Iran

Abstract

The complexity of programs is necessary for their efficiency comparing and recognition of

their requirement spaces and time. Obtaining of run times for some program as nested if

orders is sophisticate. In this paper, we analyse the computation paths of nested if orders by

rewriting systems as occure in fact. We propose a combinatorial solution for complexity of

such segment of programs.

Keywords: Complexity, nested orders, rewriting system, combinatorics.

[86]


LABELING KNUTH-BENDIX ORDER FOR PROVING TERMINATION

Mohammad Kadkhoda, Saeeid Jalili and Mohammad Izadi

Mathematics and Informatics Research Group, ACECR, Tarbiat Modares University, P. O. Box: 14115343, Tehran, Iran

Abstract

Knuth-Bendix order (KBO) is one of many orders that establish termination of term rewriting

systems (TRSs). Semantic labeling (SL) is an advanced method for proving termination. Its

semantic part is given by a quasi-model of the rewrite rules, or a quasi-model of the usable

rules. In this paper we proposed labeling Knuth-Bendix order (ℓKBO) to create a quasi-model

for semantic labeling with natural numbers. This order is based on using labels for computing

weight of terms. A compatible ℓKBO can be found for labelled rewrite systems when the

formula of conditions of ℓKBO is satisfiable.

Keywords: Knuth-Bendix order, semantic labeling, term rewriting system, termination.

[87]


CIRCULANT WEIGHING MATRICES

F. Kamali and S. Davodpoor

Shahid Rajaee Teacher Training University, Tehran, Iran

Abstract

A weighing matrix W = W(n, k) of order n with weight k is a square matrix W of order n and

entries 푤 , ∈ {−1, 0, +1} such that 푊푊 = 푘퐼 . A circulant weighing matrix, denoted

CW(n, k) is a right cyclic shift of the previous row. In [3] a theorem with a structure was

presented that states if there exist CW(푛 , k) and CW(푛 , k) with gcd(푛 , 푛 ) = 1 then there

exist a CW(푛 푛 , 푘 ). The structure: if CW(푛 , k) = circ(푎 ,푎 , … , 푎 ) and CW(푛 , k) =

circ(푏 , 푏 , … , 푏 ) then

CW(푛 푛 ,푘 ) = circ(푎 푏 ,푎 푏 , … ,푎 푏 , … ,푎 푏 ,푎 푏 , … , 푎 푏 ).

The above structure doesn’t produce circulant weighing matrix. Counterexample : We can

show that the matrices circ(-1,1,1, 0,1, 0, 0) and circ(-1,1,1,1) are CW(7,4) and CW(4,4)

respectively. But the given structure produces

circ(1,−1,−1,−1,−1,1,1,1,−1,1,1,1,0,0,0,0,−1,1,1,1,0,0,0,0,0,0,0,0)

that isn’t CW(28,16).

References

[1] K. T. Arasu and A. J. Gutman, Circulant weighing matrices, Cryptogr. Commun. 2

(2010), 155171.

[2] M. H. Ang, K. T. Arasu, S. L. Ma and Y. Strassler, Study of proper circulant weighing

matrices with weight 9, Discrete Mathematics, 308 (13) (2008) 28022809.

[3] K. T. Arasu and J. Seberry, Circulant weighing designs, J. Combin. Designs 4 (1996)

437447.

[88]


[4] P. Eades and R. M. Hain, On circulant weighing matrices, Ars. Combin. 2 (1976)

265284.

[5] K. T. Arasu, K. H. Leung, S. L. Ma, A. Nabavi and D. K. Ray Chaudhuri, Determination

of all possible orders of weight 16 circulant weighing matrices, Finite Fields Appl. 12 (2006)

498538.

[89]


SOME NEW UPPER BOUNDS FOR SKEW LAPLACIAN ENERGY OF AN

(N,M)-DIRECTED GRAPH

M. Khanzadeh and Mohammad A. Iranmanesh

Department of Mathematics, Yazd University, Yazd, 89195-741, Iran

ABSTRACT

An (n, m)-directed graph is a simple directed graph with n vertices and m arcs. In this paper we will find some upper bounds for skew Laplacian energy of an (n, m)-directed graph.

INTRODUCTION

Let 퐺 be a simple directed graph with vertex set 푉(퐺) = {푣 ,⋯ ,푣 }. Let 퐷(퐺) be the diagonal matrix whose ith diagonal entry is the degree of the vertex v (1 ≤ i ≤ n). Let 푆(퐺) be the adjacency matrix of 퐺. Then 퐿 = 퐷 − 푆 is the Laplacian matrix of 퐺. Also the eigenvalues of 퐿 and 퐿 = 퐷 + 푆 say µ , µ , … , µ and µ , … , µ are said to be the skew Laplacian and sign less skew Laplacian eigenvalues of G respectively. We also define the skew Laplacian energy of graph G as follows:

E (G) = µ −2mn .

The skew sign less Laplacian energy of graph G is define as

E (G) = µ −2푚푛

MAIN RESULATS

Theorem 1: Let G be an (n, m)-directed graph. Then, we have

E (G) ≤ E (G) + ∑ |d − |.

Theorem 2: Let G be an (n, m)-directed graph. Then, we have

|E (G)− E (G)| ≤ 2E (G).

[90]


Theorem 3: Let H be an induced subgraph of a simple graph G. Suppose H~ denotes the

union of H and all vertices ofG − H (as isolated vertices). Then,

E (G) − E (H~) ≤ E G − E(H) ≤ E (G) + E (H~)

where, E(H) is the edge set of H.

RERERENCES

[1] C. Adiga and Z. Khoshbakht, On some inequalities for the skew Laplacian energy

of digraphs. Journal of Inequalities in Pure and Applied Mathematics, Vol. 10 (2009), Issue

3, Article 80, 6pp.

[91]


FIXING NUMBER OF FULLERENES

F. KoorepazanMoftakhar and A. R. Ashrafi


University of Kashan, Kashan, I R Iran

Abstract

The fixing number of a graph is the minimum cardinality of a set S of vertex set of such

that every nonidentity automorphism of G moves at least one member of S. In this case, it is

easy to see that the automorphism group of the graph obtained from by fixing every node

in S is trivial [1,2].

The fullerenes are an allotropic form of the carbon, for the first time evidenced in

1985 by Kroto et al. [3]. The molecular graphs of these molecules are constructed from

pentagons and hexagons. If F is a fullerene and p, h, n and m are the number of pentagons,

hexagons, carbon atoms and bonds between them. Then a classical result in fullerene

chemistry states that p = 12, v = 2h + 20 and e = 3h + 30 [4]. The (r,s)fullerenes are the

most studied generalization of fullerene graphs. They are fully constructed from r and

sgons. The ordinary fullerenes are simply (5,6)fullerene. It is obtained from Euler’s

formula that the (r,6)fullerene graph exists if and only if r = 3, 4 or 5.

The aim of this paper is to compute the fixing number of (r,6)fullerenes. It is

proved that the fixing number of an (4,6) and (5,6)fullerenes is equal to 1 or 2, but there

is a class Fn of (3,6)fullerenes with fixing number n + 1.

References

1. F. Harary, Methods of Destroying the Symmetries of a Graph, Bull. Malaysian Math.

Sc. Soc. (Second Series) 24 (2001) 183 191.

[92]


2. D. Erwin and F. Harary, Destroying automorphisms by fixing nodes, Discrete

Mathematics 306 (2006) 3244 – 3252.

3. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60:

buckminsterfullerene, Nature,318 (1985)162 163.

4. P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Oxford Univ. Press,

Oxford, 1995.

[93]


A NOTE ON EDGE-COLOURING OF REPLACEMENT PRODUCTS

Amir Loghman

Department of Mathematics, Payame Noor Universtiy, P O BOX 19395-3697 Tehran, IRAN

Abstract

Let G be an (n,m)-graph (n vertices and m-regular) and H be an (m,d)-graph. Then the

replacement product G®H of graphs G and H has vertex set V(G®H)=V(G)×V(H) and there

is an edge between (v,k) and (w,l) if v=w and klE(H) or vwE(G) and kth edge incident on

vertex v in G is connected to the vertex w and this edge is the lth edge incident on w in G,

where the numberings k and l refers to the random numberings of edges adjacent to any

vertex of G. In this work, we describe necessary conditions on graphs G and H, that G®H is

d+1-edge-colourable (or (G®H) = d+1).

[94]


WIENER INDEX OF 2CONNECTED GRAPHS

E. Mahfooz and G. H. Fath-Tabar


Abstract

Let G be a connected graph. The Wiener index of graph G is defined as 푊(퐺) = ∑ 푑(푢, 푣){ , }⊂ ( ) where d(u,v) is the distance of vertices u and v. In this paper, some new upper bounds for Wiener index of two connected graphs are presented.

Keywords: Graph, Wiener index, 2connected graph.

1. Introduction

In this article, G = (U, V) is a simple connected graph with vertex and edge sets V(G) and

E(G), respectively. As usual, the distance between vertices u and v of G is denoted by dG(u,v)

and it is defined as the number of edges in a minimal path connecting u and v. A topological

index is a numerical quantity related to a graph that is invariant under all graph

isomorphisms. A topological index related to the distance function d(−,−) is called a

‘‘distance-based topological index’’. The Wiener index W(G) was the first distance-based

topological index, it is defined as the sum of all distances between vertices of G [1]. The

Wiener index has noteworthy applications in chemistry and the references there in for the

mathematical properties and chemical meaning of this index. Hosoya [2] was the first

scientist to introduce the name ‘‘topological index’’, and to reformulate the Wiener index in

terms of the distance function d(−,−). In this paper, we present some new bounds for Wiener

index and edge Wiener index of 2connected graphs.

2. Results

In this section we present some inequalities for winer index of two connected graphs.

[95]


Lemma 1. Let G be a two connected graph and Vi={x∈V(G)| d(x,v)=i}. Then |Vi|>1.

Theorem 1. Let G be a two connected graph then 푊(퐺) ≤ with equality if and only if

G is cycle Cn.

References

1. H. Wiener, Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69

(1947) 17–20.

2. H. Hosoya, Topological index, a newly proposed quantity characterizing the topological

nature of structure isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn. 44 (1971) 2332–

2339.

[96]


SOME BOUNDS ON THE DETOUR INDEX OF GRAPHS

S. Malekpoor and G. H. FathTabar


Abstract

Suppose G = (V,E) is a connected graph with vertex and edge sets V(G) and E(G). A topological index of a graph is a numerical quantity related to the graph that is invariant under all graph isomorphisms. The detour index of graph G is defined as ω(G) = ∑ D(u, v),{ , }⊂ ( ) where D(u , v) is the length of the longest path between vertices u and v. In this paper, some new lower and upper bound for detour index of graphs are presented.

Keywords: Graph, detour index.

[97]


PERMUTATION SYMMETRY OF FULLURENE ISOMERS OF C88

Mohammad Reza Mollaei and Ahmad Gholami

Department of Mathematics, Faculty of Mathematical Sciences,

University of Qom, P. O. Box: 371614661, Qom, I. R. Iran

Abstract

Fullerenes are molecules in the form of polyhedral closed cages made up entirely of n three

coordinate carbon atoms and having 12 pentagonal and (n/2-10) hexagonal faces, where n is

equal or greater than 20. The symmetry of these molecules are important in the problem of

counting their isomers. In this article the permutation symmetry group of the isomers of a C88

fullerene are calculated.

Keywords: Fullurenes, Permutation, Isomer, Symmetry.

[98]


ESTRADA INDEX OF (3,6) AND 6CAGES

A. Moraveji, S. Madani and H. Shabani

Department of Mathematics, Faculty of Mathematical Sciences, .University of Kashan, Kashan 87317 – 51167, I. R. Iran

Abstract

Let λ1, λ2, . . .λn be an eigenvalues of the simple graph G. the Estrada

index of G is the summation of exponent λi, 1 ≤ i ≤ n, and denoted by

EE(G). In this paper, we computed the Estrada index of (3,6)cage and 6-

cage.

Keywords: Estrada index, (3,6)-Cage, 6-Cage.

Introduction and Main Results

A (3,6)cage is a 3-regular polyhedron each face of which is either a hexagon or a triangle.

We apply some results of [2,3] to compute the Estrada index of a (3,6)cage C. To do this,

we assume that C has parameters r; s and t and = (1,2) is the solution of key equation that

presented in [3]. By solving this key equation, the spectra of C is obtained. Then there are

integers m and n such that 0 m 2s – 1, 0 n 2r – 1, 1 = rm, 2 = qm+sn, q = (r + t)/2 and

= ei/rs. On the other hand, the set of complex solutions of key equations can be divided into

two subsets * = *(C) and its complex conjugates. Then the Estrada index of C is as

follows:

Corresponding author ([email protected]).

[99]


퐸퐸(퐶) = 푒 + 3푒 + 푒±| |

∈ℜ∗( )

There is a close relationship between a (3,6)cage C and its toroidal cover S(C). In

[3], an algorithm was introduced by which one can construct a toroidal cover from a

(3,6)fullerene. Suppose DG denotes the duplex of G. By substitution of C as G and the fact

that the 6-cage S(C) is the duplex of C, we have:

퐸퐸 푆(퐶) = 3푒± + 푒± + 푒±| |

∈ℜ∗( )

References

1. E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett. 319 (2000)

713–718.

2. P. W. Fowler, P. E. John, H. Sachs, (3,6)Cages, hexagonal toroidal cages, and their

spectra, DIMACS Ser. Discrete Math. Theoret .Comput. Sci. 51 (2000) 139174.

3. P. E. John, H. Sachs, Spectra of Toroidal graphs, Discrete Math.309 (2009)

[100]


SOME RESULTS ON CONJUGACY CLASS GRAPHS OF PSINGULAR

ELEMENTS

Z. Mostaghim and M. Zakeri

Department of Mathematics, Iran University of Science and Technology, Tehran, Iran

Abstract

In this paper we consider some family of groups and study conjugacy class graphs of their

psingular elements.

[101]


SOME DIGRAPHS ARISING FROM NUMBER THEORY

A. R. Naghipour and M. Rahmati

Faculty of Mathematical, Shahrekord University, P.O. Box: 115, Shahrekord , Iran

Abstract

In this paper, we study the properties of the digraph ( )n for which there is a directed edge

from a to b if 5 (mod )a b n for , {0,1, , -1}a b n . We consider two sub-digraphs of ( )n .

Let 1( )n be induced by the vertices which are co-prime to n and 2( )n be induced by the

vertices which are not co-prime to n . The conditions for regularity and semiregularity of the

subdigraph 1( )n are presented. It is shown that every component of the digraph 1( )n is

cycle if and only if 5 does not divide the Euler function ( )n and n is square free. We give

a formula for the number of fixed points of the digraph 1( )n .

Keywords: Digraph, Charmichael function, Group theory.

[102]


ON THE MATCHING POLYNOMIAL OF SOME GRAPHS

Yaghub Pakravesh and Ali Iranmanesh

Department of Pure Mathematics, Faculty of Mathematic Science, Tarbiat Modares University, P. O. Box: 14115-137, Tehran, Iran

Abstract

A simple graph is denoted by G = (V, E) where 푉 is the set of vertices and 퐸 is the set of

edges. A matching 푀 is any set of independent edges in 퐺, i.e., no pair of edges in 푀 have a

vertex in common. A 푘-matching is a matching on 푘 edges and a perfect matching is a

matching that covers all the vertices in 퐺. The matching polynomial of a graph 퐺 on 푛

vertices is defined as

휇(퐺; 푥) = (−1)

[ ]

푝(퐺,푘)푥

where푝(퐺, 푘) denotes the number of 푘-matchings in 퐺 and we define 푝(퐺, 0) = 1. In

thispaper the matching polynomial and Hosoya index of some graphsareobtained.

Keywords: matching polynomial, Hosoya index.

References

[1] J. Aihara, Matrix representation of an olefinic reference structure for monocyclic

conjugated compounds, Bull. Chem. Soc. Japan 52 (1979) 15291530.

[2] R.A. Beezer, E.J. Farrell, The matching polynomial of a distance-regular graph, Internet.

J. Math. Math.Sci. 23 (2) (2000) 8997.

[3] D. Cvetkovic', M. Doob, H. Sachs, Spectra of Graph-Theory and Applications, Academic

Press, New York, 1980.

[103]


THE MAXIMUM RANDIC INDEX OF PENTACYCLIC GRAPHS

M. Pourbabaee, H. R. Tabrizidooz and A. R. Ashrafi

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan,

Kashan 87317-51167, I. R. Iran

Abstract

Let G be a simple and connected graph with n vertices and m edges. Then the

cyclomatic number of G is c(G) = m n + 1. A graph with c = 5 is called

pentacyclic. Throughout this paper G is a pentacyclic connected graph. The

Randić index R = R(G) is defined as follows:

R = R(G) = ,v,u )v(d)u(d1

whered(u) denotes the degree of a vertex u and the summation runs over all

edges uv of G.his topological index was first proposed by Randić [1] in 1975,

suitable for measuring the extent of branching of the carbonatom skeleton of

saturated hydrocarbons. In this paper, the maximum value of the Randić index

in the class of all pentacyclic nvertex connected graphs is computed. It is

proved that, the maximum of this topological index for this class is

29n

3611

, n 9. The maximum will be occurred in the graph G depicted in

Figure 1. If 5 n 8, the maximum values of the Randić index are +√

,

+√

, +√

and √

, respectively. The graphs with these maximum values are

depicted in Figure 2.

[104]


Keywords: Randić index, pentacyclic graph.

References

1. M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97

(1975) 6609–6615.

[105]


SEARCHING NUMBER IN GRAPH

ZAHRA SADRI IRANI and ASEFEH KARBASIOUN

Department of Mathematics, Faculty of Science, Islamic Azad University, Falavarjan Branch, Falavarjan, I. R. Iran

[email protected]

ABSTRACT

Searching questions are often designed like police pursue thief on the game of graph. The

least police who is leading to arrest of thief named searching number in the graph and it is

showed by Cn(G) and if it be Cn(G) ≤ C we will name C-searchable graph. In this essay we

introduce limit of special searching number such as Cayley graphs, flat graph, product graph

and infinite graph.

INTRODUCTION

Searching questions are often designed like police pursue thief on the game of graph. In this

manner two players play game on without direction and bilateral edge connected graph to

begin with each of officers are placing on one of the vertex and thief are placing on the

another ( one ) of the vertex. Then games be formed by number of cycle and this game is

starting to begin with police moves and then the if moves too, movement of police and thief

has two shapes: they are staying on their location or they are moving to adjacent vertex.

When they are placing on the same vertex therefore game is over .The least police

who is leading to arrest of thief named searching number in the graph and it is showed by

Cn(G) and if it be Cn(G) ≤ C we will name C-searchable graph. This definition has scientific

applications in control topic, robotic, sensor and electronics security.

[106]


REFERENCES

[1]. Alspach, D. Dyer, D. Hanson and B. Yang, Time constrained graph searching, Theoret. Comput.

Sci. 399 (2008), 158-168.

[2].M. Aigner and M. Frome, A game of cops and robbers, Discrete Appl. Math. 8 (1984), 112.

[3] R. P. Anstee and M. Farber, On bridge graphs and cop-win graphs, J. Combin. Theory Ser. B 44.

[107]


A GRAPH RELATED TO THE ELEMENTS OF FINITE GROUPS

Amin Saeidi and Seiran Zandi*

Kharazmi University

Abstract

Let G be a finite group. We define the graph г(G) with V(г) = G A and E(г) = {(x, y) V(г) :

(|x|,|y|) = 1}, where A is the set of all elements x of G such that either x = 1 or π(|x|)= π(G).

Then we call г the OPgraph of G. In this note, we obtain many results concerning these

graphs.

Keywords: Finite groups, order prime graphs.

[108]


FIXING NUMBER OF NONCOMMUTING GRAPHS

Hossein Shabani

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I R Iran

Abstract

The fixing number of a graph G is the minimum cardinality of a set S ⊂ V(G) such that every nonidentity automorphism of G moves at least one member of S, i.e., the automorphism group of the graph obtained from G by fixing every node in S is trivial [1,2]. Let H be a non-abelian group and let Z(H) be

the center of H. Associate a graph (H) with H as follows: Take H \ Z(H) as the

vertices of (H) and the join of two distinct vertices x and y, whenever xyyx.

The graph (H) is called the non-commuting graph of H [3]. In this paper, the

fixing number of noncommuting graph and some of its induced subgraphs are computed.

References

1. Frank Harary, Methods of Destroying the Symmetries of a Graph, Bull.

Malaysian Math. Sc. Soc. (Second Series) 24 (2001) 183 191. 2. David Erwin and Frank Harary, Destroying automorphisms by fixing nodes,

Discrete Mathematics 306 (2006) 3244 – 3252. 3. B. H. Neumann, A problem of Paul Erdös on groups, J. Aust. Math. Soc.

Ser. A 21 (1976) 467 472. 4. A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a

group, J. Algebra 298 (2006) 468 492.

Corresponding author (Email: [email protected]).

[109]


AN APPROACH FOR STATE PREDICTION BY HIDDEN MARKOV

MODEL GRAPH

Sanaz Sheikhi and S. Morteza Babamir

Department of Computer Engineering, University of Kashan, Iran

Abstract

High cost of system failure, which is a result of complexity and uncertainty clarify the important role

of prediction mechanisms to avoid involving erroneous states. For this means, probabilistic model

checking approaches are considered as suitable alternatives.

In this paper we present an initiative for prediction about system's states in consideration with

an incomplete sequence of behavioral observations. Since most systems operate like a black box, so

their internal states are not visible and only dependent outputs to the states are visible thus we

consider Hidden Markov Model (HMM) as a appropriate basis for our approach as in HMM states

cannot be observed; rather each state has a probability distribution for possible observations.

There are many algorithm working on HMM graph, and among them Forward algorithm

calculates the probability of being in a specific state. It's inputs are a complete sequence of system

behavioral observations (O = O1,O2,...,OT) and graph of HMM (H = <S, A, V, B, >). HMM is graph

containing a set S of system states, a transition probability matrix A, a set V of observation symbols,

observation probability matrix B and an initial state distribution .The algorithm calculates Pr(O,

qt=si | H) , the probability of the firs T observations (O) and state of the system when observation Ot is

made being si having the model H.

Due to many environmental or structural reasons, the monitoring system may be unable of

capturing all the observations and some of them may be loosed. Our extension to this algorithm is

managing the lack of a complete observation sequence and it's effect on the probability calculation of

being in the desired state at time T. So we suppose that between each two observation Oi, Oj there

may exist missed observations chain with any length and it may contain any of the observation

symbols in set V. Hence the system would enter middle states and the probability of being in the

desired state eventually would change. To deal with this assumption we modify main formula of the

Forward algorithm. We recursively calculate the probability of going into all the possible middle

[110]


states, that are set S states, by any of the missed observations in the chain and summing over all of

these probabilities, that are occurring just before the last visible observation in the sequence. Finally

join it with the probability of transition from the state of the last observation of the chain to the

desired state by OT. Result is calculation of the probability of being in a specific state of a system after

an incomplete observation sequence is received. To evaluate our approach we used a graph of client-

server example as case study.

[111]


THE CHROMATIC POLYNOMIAL OF LINEAR PHENILENES

Z. Shiri-Barzoki and G. H. Fath-Tabar


Abstract

The chromatic polynomial of a graph counts the number of ways the graph can be colored using no more than a given number of colors. In this paper, we compute The chromatic polynomial of linear phenilenes.

Keywords: Chromatic polynomial, linear phenilenes.

Introduction and Results

Let G be a graph with vertex and edge sets V(G) and E(G), respectively. The chromatic

polynomial P(G,k) of a graph counts the number of ways the graph can be colored using no

more than a given number of colors(k color). The chromatic polynomial includes at least as

much information about the colorability of G as does the chromatic number. Indeed, χ= χ(G)

is the smallest positive integer that is not a root of the chromatic polynomial. In this paper,

we compute the chromatic polynomial of linear phenylenes Lp(h) depicted in Figure 1.

Figure 1. Linear Phenilenes.

Theorem. The chromatic polynomial of linear phenylenes Lp(h) is as follows:

푃(Lp(h), k) = (k − 5k + 10k − 10k + 5)(k − 3k + 3) ((k− 1) − k + 1).

[112]


References

1. N. Biggs, Algebraic Graph Theory, Cambridge University Press, February 1994.

2. K. Appel and W. Haken. Every planar graph is four colorable. Part I. Discharging.

[113]


THE EDGE WIENER TYPE TOPOLOGICAL INDICES

Abolghasem Soltani and Ali Iranmanesh

Department of Pure Mathematics, Tarbiat Modares University, Tehran, Iran

Abstract

Let G be a simple connected graph. The distance between the edges f, g E(G) is defined as

the distance between the vertices f and g in the line graph of G. The edge Wiener index

We(G) is the sum of all distances between edges of G. In this paper, we generalize the edge

Wiener index and compute explicit formula of some edge Wiener type indices for join of two

graphs.

Keywords: Edge Wiener type index, Reciprocal edge Wiener index, Join of graphs.

[114]


ORDERING GENERALIZED PETERSEN GRAPHS WITH

RESPECT TO SPECTRAL MOMENTS

F. TAGHVAEE AND A. R. ASHRAFI

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan,

Kashan 8731751167 , I. R.Iran

Abstract

Suppose G is a graph, A(G) its adjacency matrix, and µ1(G) ≤…≤ µn(G) are eigenvalues of

A(G). The number Sk(G) = ∑µik(G), 0 ≤ k ≤ n-1 are said to be the k-th spectral moment of G.

The sequence S(G) = (S0(G),…,Sn-1(G)) is called the spectral moments sequence of G.

Suppose G1 and G2 are graphs. If there exists an integer k, 1 ≤ k ≤ n − 1, such that for each i,

0 ≤ i ≤ k − 1, Si (G1) = Si (G2) and Sk (G1) < Sk (G2) then we write G1 <S G2< [1,2,3].

In this paper, the extremal graphs with respect to the S-order in the classes of all

generalized Petersen graphs are determined.

Keywords: Spectral moments, generalized Petersen graphs.

2010 AMS classification Number: 05C50, 15A18.

References

[1] X. –F. Pan, X. L. Hu, X. G. Liu and H. Q. Liu, The spectral moments of trees with given

maximum degree, Appl. Math. Lett. 24 (2011) 1265-1268.

[2] Y. P. Wu and Q. Fan, On the lexicographical ordering by spectral moments of bicyclic

graphs, Ars Combin., in press.

[3] Y. P. Wu and H. Q. Liu, Lexicographical ordering by spectral moments of trees with a

prescribed diameter, Linear Algebra Appl. 433 (2010) 1707-1713.

[115]


SOME PROPERTIES OF PRINCIPAL IDEAL GRAPH OF THE RING ℤ퐧

Ali Asghar Talebi1 and Shabnam Safarizade2

1Departman of Mathematics, University of Mazandaran, Babolsar, Iran

2Departman of Mathematics, University of Payame Noor, Mashhad, Iran

Abstract

Let R be an associative (not necessarily commutative) ring. In the present paper, we consider

'principal ideal graph of a ring' denoted by PIG(R). We determine the value of n for which

PIG(ℤ ) is bipartite, tree and planar. Next, we obtain the number of component and

domination number of PIG(ℤ ). We also obtain the relation between chromatic number and

clique number of PIG(R).

Keyworld: Ring, Bipartite graph, Tree, Planar graph, Chromatic number, Clique number.

1. Introduction

Let R be a ring. A graph G = (V, E) where the vertex set V = R\{0}, and the edge set

E = {uv ; ⟨u⟩ = ⟨v⟩, u ≠ v}, is called the 'principal ideal graph' of R and it is denoted by

PIG(R).

2. Some Properties

Theorem 2. 1. The vertex u is isolate in PIG(ℤ ), if and only if, n be an even number and u = .

Theorem 2. 2. The number of connected component of PIG(ℤ ), is equal to the number of m, such that m divide n , 1 ≤ m < 푛 .

Theorem 2. 3. The domination number of PIG(ℤ ) is the number of connected component of PIG(ℤ ).

Theorem 2. 4. The graph PIG(ℤ ) is bipartite if and only if n = 3, 4, 6.

[116]


Theorem 2.5. Let G = PIG(ℤ ). Then, (i) χ(퐺) = 휑(n), (ii) ω(G) = χ(퐺) = 휑(n).

Theorem 2. 6. PIG(ℤ ) is planar graph, if only if, 휑(n) ≤ 4.

Theorem 2. 7. If 푅 ⊆ 푅′ then, PIG(푅′) is a sub-graph of PIG(푅).

Theorem 2. 8. PIG(ℤ ) ≅ K ∪ K ∪ …∪ K ∪ K .

Corollary 2. 9. PIG(ℤ ) ≅ K ∪ PIG(ℤ ).

References

[1] S. Bhavanari, G. Lungisile and N. Dasari, 2011. Some results on principal ideal graph of a ring. African Journal of Mathematics Computer Science Research Vol. 4 (6) 235241, 2011.

[2] F. Harary, 1969. "Graph Theory", Adison Wisley Publication, Reading, Massachusetts.

[117]


CLASSIFYING CUBIC S-REGULAR GRAPHS OF ORDERS 36P AND 36P2

A. A. Talebi1 and N. Mehdipoor2

1Department of Mathematics, University of Mazandaran, Babolsar, Iran [email protected]

2Department of Mathematics, University of Mazandaran, Babolsar, Iran [email protected]

Abstract

A graph is s-regular if its automorphism group acts regularly on the set of s-arcs. In this

study, we classify the connected cubic s-regular graphs of order 36p and 36p2 for each s ≥1,

and each prime p.

2000 Mathematics Subject Classification: 05C25, 20b25.

Keywords: s-regular graphs, s-arc-transitive graphs.

[118]


The Top Ten Minimum and Maximum Values of Narumi-Katayama Index in Unicyclic Graphs

A. Zolfi and A. R. Ashrafi


University of Kashan, Kashan 87317 – 51167, I. R. Iran

Abstract

Let G be an nvertex graph with degree sequence d1, d2, …, dn. The NarumiKatayama index

is defined as NK(G) = .n1i id In this paper the first top ten minimum and maximum values

of this topological index were computed in the class of unicyclic graphs.

Keywords: NarumiKatayama index, chemical tree.

Persian Papers

of the

5th Conference on Algebraic Combinatorics and Graph

Theory

July 34, 2012, University of Kashan

[120]


:مراجع

[1] W. Ren, J. Zhao and Y. Ren, “Network coding based dependable and efficient data data survival in unattended wireless sensor networks”, the journal of communications, V. 4, N. 11, Dec 2011.

[121]


هاي هادامارد تعمیمی از ماتریس

اصغر بهمنی دانشکده ریاضی، دانشگاه صنعتی شریف، تهران

Email:[email protected]

کار مشترك با دکتر سعید اکبري چکیده

اي متناهی از اعداد مختلط ناصفر بیابیم کوشیم که زیرمجموعه ي تعمیمی جدید از ماتریس هادامارد، می با ارائه در این مقاله

∗퐴퐴وجود داشته باشد که 푛، ماتریسی از مرتبه 푛که براي هر عدد طبیعی = 퐷푖푎푔(휆 , … , 휆 ، براي اعدادي مانند (

휆 , … , 휆 .که مجموعه زنیم همچنین حدس می푆 = {±1, ±2, براي این منظور کافیست و روشهایی براي ساخت {±3

هایی ارائه از ساخت چنین ماتریس 50هاي زیر در انتها نیز جدولی براي مرتبه. کنیم هاي خاصی ارائه می هاي از مرتبه ماتریس

.دهیم می

.یافته ماتریس هادامارد تعمیم:ماتریس هادامارد، :هاي کلیدي واژه

[122]


مینیمم مجموع رنگ آمیزي گراف ها

زهرا تندپور، تهرانانشکده ریاضی، دانشگاه الزهراد

[email protected]

∑یک گراف باشد، Gاگر (G) کوچکترین مجموع ممکن بین ھمھk رنگ آمیزی ھای راسی سره ازG کھ رنگ ھا در

است کھ در واقع برابر Gمجموع رنگی راسی(G)∑ھمچنین. آن ھا اعداد طبیعی ھستند را تعیین می کند

min ( ) ∑ (G) شدت راسی .می باشدG کھ باs(G) نمایش می دھیم کوچکترین مقدارs است بھ طوری کھ

∑ (G) = ∑(G) شود.

:cرنگ آمیزی راسی سره V(G) → N رای گراف یک رنگ آمیزی مینیمال بG است ھرگاه

∑ c(v)∈ ( ) = ∑(G) باشد و ھمچنین اگردر چنین رنگ آمیزی بھ ازای ھر راسc(v) ≤ s(G), v ∈ V(G)

.استGیک رنگ آمیزی بھینھ برای گراف cباشد، آنگاه رنگ آمیزی مینیمال

در این تحقیق رنگ آمیزی مینیمال وھمچنین رنگ آمیزی بھینھ و کاربردھای آنھا را روی درخت ھا،

.گرافھای بازه ای، شکاف، دو بخشی، دوبخشی زنجیری، جدولی و ابر گراف ھا مورد بررسی قرار داده ایم

[123]


همبنددودیی -4قضیه شکافنده براي مترویدهاي

سمیھ غنی یارلو

گروه ریاضی، دانشگاه ارومیھ، ارومیھ[email protected]

چكیده

تا حد جداسازي از اندازه ھمبند - 4مینور – Nھمبند داخلي دودویي با - 4در این مقالھ قضیھ شكافنده براي مترویدھاي

.مي شود مشخص 5

[124]


ضلعی رشدیافته هاي شش شاخص وینر دستگاه

فرزانھ کتابی و علیرضا اشرفی

کاشانضی، دانشگاه ریاعلوم دانشکده

[email protected]

چکیده

هاي یک گراف همبند ي زوج رأس هاي بین همه صورت مجموع فاصله ها، شاخص وینر است که به ترین شاخص یکی از قدیمی

هاي ضلعی را تعریف نموده ، سپس براي حالت ضلعی و رشد یک دستگاه شش در این مقاله ابتدا دستگاه شش. شود تعریف می

ضلعی را براي هاي رشد یک دستگاه شش چنین حالت هم ایم؛ دست آورده را به رشد دستگاه، شاخص وینرمختلفی از

ایم؛ در پایان رشد تصادفی زنجیرهاي ها را محاسبه کرده هاي کاتاکوندنس نیز در نظر گرفته، سپس شاخص وینر آن دستگاه

هاي مسطح اي از گراف ضلعی یک نوع ویژه هاي شش دستگاه. ایم دست آورده ضلعی را بیان کرده ، شاخص وینر آن را به شش

ضلعی ، فرایند افزایش منظور از رشد یک دستگاه شش. ها هستند ضلعی ها محدود به شش ي وجوه آن اي که همه گونه هستند به

:ضلعی به صورت زیر است هاي مختلفی از رشد یک دستگاه شش حالت. ضلعی هاست پی تعداد شش در پی

ي ضلعی به دستگاه ساخته شده توان به صورت یک روش بازگشتی از اتصال یک شش ضلعی را می دستگاه شش) الف

.دست آورد ضلعی را به هاي شش ي دستگاه توان همه ضلعی می بنابراین با شروع از یک شش . قبلی در نظر گرفت

. ی دلخواه در نظر گرفتضلع صورت اتصال دو دستگاه شش توان به ضلعی را می دستگاه شش) ب

.ضلعی، دستگاه کاتاکوندنس، زنجیر خطی شاخص وینر، دستگاه شش :ها کلید واژه

[125]


نظریه ي شمارشی پولیا

فاطمه کوره پزان مفتخر[email protected]

چکیده

با استفاده از این فرمول و رابطھ آن با ریاضیات شمارشی و . پردازیم در این مقالھ بھ بازنگری فرمول شمارشی پولیا می

این . ، ارائھ می دھیم ھای متفاوت جھت برچسب گذاری یک گراف ، یک تابع مولد برای یافتن تعداد راه ترکیبیات جبری

سپس بھ کاربرد قضیھ پولیا در . القا می کند nی غیر یکریخت از مرتبھ ھا روش، راه حلی را برای شمارش تعداد گراف

.یابیم می را ای از ترکیبات شیمیایی پردازیم و با استفاده از آن تعداد ایزومرھای خانواده می شیمی

[126]

Conference Participants

1 * A. Abdolghaforian 43 S. Ghalandar 2 A. R. Abdollahi 44 ** L. Ghanbari Mamani 3 S. Akbari 45 * S. Ghaniyarlou 4 ** Z. Akbarzadeh-Ghanaee 46 * M. Ghasemi 5 ** Z. Alem 47 ** Z.GholamRezaee 6 **,* S. Alikhani 48 ** F.Gholami 7 A. Amini 49 Kh. Gholami-Mehmandoust 8 R. Amirjan 50 ** F.GholamiNezhad 9 * M. Ariannejad 51 A. Ghorbani

10 J. Asgari 52 * M. Ghorbani 11 A. R. Ashrafi 53 ** F. Gilasi 12 ** Z. Assadi Golzar 54 ** Z. Golmohamadi 13 N. Azimi 55 ** E. Haghi 14 * F. S. Babamir 56 ** M. Hakimi-Nejad 15 * A. A. Bahmani 57 ** A. Hamzeh 16 M. Bahramian 58 ** A. Heideri 17 * S. Bahramian 59 H. Hemadi 18 E. Baniasadi 60 M. Hemmasi 19 R. B. Bapat 61 ** S. HossienZadeh 20 Gh. Baradaran Khosroshahi 62 ** M. A. Hosseinzadeh 21 B. Bazigaran 63 ** S. Irandoust 22 Z. BeygMohammadi 64 M. A. Iranmanesh 23 ** A. Chatrazar 65 R. Jahaninezhad 24 H. Daghigh 66 R. Jahanipoor 25 * H. Damadi 67 ** M. Jahanbakhat 26 B. DaneshvarZadeh 68 ** M. Kadkhoda 27 B. Davvaz 69 A. Karbasiun 28 * M. DavoudiMonfared 70 G. Y. Katona 29 ** S. Davoudpour 71 * F. Ketabi 30 ** T. Dehghan-Zadeh 72 K. Khashayarmanesh 31 A. Dehnokhalaji 73 ** M. Khanzadeh 32 ** S. Djafari 74 H. Khodashenas 33 M. Emami 75 * S. M. Khoramizadeh 34 M. Eslami 76 D. Kiani 35 ** Z. Falahzadeh Abarghui 77 S. Kiani 36 M. Farhadi Jalalvand 78 ** F. KoorepazanMoftakhar 37 ** A. Farokh 79 ** A. Loghman 38 M.Farshi 80 * A. Madadi 39 ** Kh. Fathalikhani 81 A. D. Maden 40 G. H. FathTabar 82 ** E. Mahfuz 41 * Z. Foruzanfar 83 ** Sh. Malekpoor 42 Z. Ghadiri Herevani 84 * R. Manaviyat

[127]

85 H. R. Meimani 118 ** A. Saedi 86 M. Mirsadeghi 119 Gh. Safakish 87 S. Mirvakili 120 ** Sh. Safarizadeh 88 * M. Mizakhah 121 * S. M.Seyedi 89 ** Z. Mirzaykichi 122 ** H. Shabani 90 * M. Mirzargar 123 * R. Sharofdini 91 * M. Mogharab 124 ** S. Sheikhi 92 A. Mohebi 125 ** Z. Shiri 93 ** M. R. Molaee 126 L. Soleymani 94 * S. Moradi 127 ** A. Soltani 95 ** S. A. Moravegi 128 S. Soltani 96 Z. Mostaghim 129 ** M. Songhori 97 ** A. R. Naghipour 130 ** S. M. Shaker 98 * M. J. Nadjafi-Arani 131 Kh. Shamsiporshokoh 99 * O. Naseriyan 132 A. Shariatnia 100 F. Nasiri 133 V. Shirvani 101 R. NasiriGharaghani 134 H. R. Tabrizidouz 102 * M. R. Nezadi 135 B. Taeri 103 * S. Nikandish 136 ** F. TagvaeeArani 104 N. Nourouzi 137 F. Talebi Meydan Ghale 105 Gh. Omidi 138 A. A. Talebi Rostami 106 * Y. Pakravesh 139 * M. Tavakoli 107 ** M. Pourbabaee 140 * Z. Tondpour 108 ** A. Rafiepour 141 F. Yakhchibiglo 109 Y. Rahi 142 * Z. Yarahmadi 110 F. Rahimi 143 * A. A. Yazdanpour 111 Gh. Rahmani 144 H. YosefiAzari 112 F. Rahmati 145 * R. Zafarani 113 M. Rahmati 146 ** M. Zakeri 114 ** B. Razeghimanesh 147 S. Zandi 115 A. A. Rezaee 148 ** A. Zolfi 116 M. Romena 149 117 A. Saadatmandi * Accepted as oral presentation. ** Accepted as poster presentation. Invited Speakers.

5th Conference on Algebraic Combinatorics and Graph …conferences2012.kashanu.ac.ir/11.pdf · 5th...

Documents

Transcript of 5th Conference on Algebraic Combinatorics and Graph …conferences2012.kashanu.ac.ir/11.pdf · 5th...