International Journal of Computer Engineering and ... · APPLICATION OF ALPHA CUT COLORING OF A...

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),

ISSN 0976 - 6375(Online), Volume 5, Issue 8, August (2014), pp. 01-12 © IAEME

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APPLICATION OF ALPHA CUT COLORING OF A FUZZY GRAPH

Dr. D. Savithiri1, A. Gayathrridevi

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Department of Mathematics, Nehru Institute of Technology

Department of Mathematics, Kalaignar Karunanidhi Institute of Technology

ABSTRACT

A Fuzzy graph G � �V,µ, σ� is a nonempty set V together with a pair of functions

µ V� 0,1 and σ VXV� 0,1 such that for all x, y in V, σ�x, y� � µ�x�^ µ�y� .We callµ the

fuzzy vertex set of G and σ the fuzzy edge set of G and σ is a fuzzy relation on µ. Chromatic

number of a graph G is the number of different colors assign to a vertex set of a graph such that

adjacent vertices receive different colors. In this paper, coloring function based α cut of graph G is

used to color all the vertices of a graph which gives a fuzzy number. In this paper, we have derived

properties of α cut sets using α cut coloring and algorithm is developed to generate fuzzy graph.

Index Terms: Fuzzy Sets, Fuzzy Graph, Fuzzy Number, � cut, � Cut Coloring, Fuzzy Number.

I. INTRODUCTION

Graph theory is a growing area in mathematical research, and has a large applications. Graphs

are represented as ubiquitous models of both natural and human-made structures. They pave us way

to model many types of relations and process dynamics In physical, biological[6] and social systems.

Problems related to practical interest can be symbolised by graphs. Graph Theory is considered to

began with the first paper Seven Bridges of Konigsberg which was written by Leonhard Euler and

published in 1736[1].

In recent years, there is an increased demand for the application of mathematics in which

Graph theory has proven to be particularly useful to a large number of rather diverse fields. The

exciting and rapidly growing area of graph theory is rich in theoretical results as well as applications

to real-world problems. With the increasing importance of the computer, there has been a significant

movement away from the traditional calculus courses and toward courses on discrete mathematics,

including graph theory.

In the field of computer science, graphs are used to represent the networks of communication,

data organization, computational devices, the flow of computation, etc., Graphs can be used to form

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many situations in the real world, for example, the users of a social network and their friendships, the

cities in a country and the streets that connect them, project management, to manage dependencies

between tasks, mathematical relationships like Fibonacci expansions using trees, electronic circuits,

Kirchhoff laws are deeply related with the graph structure of circuits.

A Graph coloring[7] is the assignment of a color to each vertex of the graph so that no two

adjacent vertices are assigned the same color. A vertex coloring is a way of coloring the vertices of a

graph such that no two adjacent vertices share the same color. Similarly, an edge coloring assigns a

color to each edge so that no two adjacent edges share the same color. The minimum number of

colors needed to color a graph G is called its chromatic numberand it is denoted byχ(G).

In 1852, the first graph coloring problem was first raised when August De Morgan, Professor

of Mathematics at University College, London wrote to Sir William RowanHamilton in Dublin about

a problem which was asked to him by a former student named Francis Guthrie. Guthrie noticed that

it was possible to color the countries of England using four colors so that no two adjacent countries

were assigned the same color. The question raised thereby was whether four colors would be

sufficient for all possible decompositions of the plane into regions.

A graph coloring is one of the most important concepts in graph theory and is used in many

real time applications like Job scheduling[2], Aircraft scheduling[2], Computer network security[10],

Map coloring and GSM mobile phone networks[3]. Automatic channel allocation for small wireless

local area networks.[8].

A graph is a expedient way of representing information involving relationship between

objects. The objects are represented by vertices and relations by edges. When there is vagueness in

the description of the objects or in its relationships or in both, it is natural that we need to design a

'Fuzzy Graph Model'.

The notion of fuzzy sets was introduced by L.A. Zadeh in 1965[12]. It involves the concept

of a membership function defined on a universal set. The value ofthe membership function lies in

[0,1].Using the concept of fuzzy subsets, the concept of fuzzy graph was introduced A. Rosenfeld in

1975[9]We know that a graph is a symmetric binary relation on a nonempty set V. Similarly, a fuzzy

graph is a symmetric binary fuzzy relation on a fuzzy subset. The first definition of a fuzzy graph

was by Kaufmann[5] in 1973, based on Zadeh's fuzzy relations [12]. But it was Azriel Rosenfeld [9]

who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975. The

Application of fuzzy relations are widespread and important; especially in the field of clustering

analysis, neural networks, computer networks, pattern recognition, decision making and expert

systems. In each of these fields, the basic mathematical structure is that of a fuzzy graph.

II. BASIC DEFINITIONS

A. Fuzzy Graph

A fuzzy graph G = (V, µ, � ) is a nonempty set V together with a pair of functions

µ : V� [0, 1] such that for all x, y in V , ρ(x, y) ≤ �(x) ^ �(y). We call µ the fuzzy vertex set of G

and � is a fuzzy relation on µ.

B. Degree of a Vertex

Let G: = (V, ρ)be a fuzzy graph. The degree of a vertex v in G to be d(v) = ∑ ρ�v, u�� .



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C. Alpha cut

The α cut of Fuzzy set A is defined as �α is made up of members whose membership is not

less than α. �α= { α � �/ !�"� # α }. α cut set of fuzzy set is crisp set. In this paper, α cut set

depends on vertex and edge membership value.

D. Union

Consider the union [4] $ � $% & $' of two graphs $% � �(%, �%� and $' � �(', �'� . Then

G = ((% & (', �% & �'� . Let * be a fuzzy subset of (*and �*a fuzzy subset of �*, i = 1, 2. Define

the fuzzy subsets % & ' of (% & (' and �% & �'of �% & �' as follows:

� % & '��+� � , %�+� -. + � (%\ (' '�+� -. + � ('\ (% %�+� ( '�+�, -. + � (% 0 ('1

And (�% & �'��+2� � , �%�+2� -. +2 � �%\ �'�'�+2� -. +2 � �'\ �%�%�+2� (�'�+2�, -. +2 � �% 0 �'1

E. Join

Consider the join[4] $ � $% 3 $' of two graphs $% � �(%, �%� and $' � �(', �'� . Then

G = ((% & (', �% & �' & �′� where X’ is the set of all edges joining the vertices of (%456 ('.and

where we assume (% 0 (' � 7. Let * be a fuzzy subset of (* and �* a fuzzy subset of �* , i = 1,

2. Define the fuzzy subsets % 3 ' of (% & (' and �% 3 �'of �% & �' & �′ as follows: � % 3 '��+� � � % & '��+� 8+ � (% & ('

And ��% 3 �'��+2� � 9�% & �'�+2�-. +2 � �% & �' %�+�^� '�2�-. + 2 � �′ 1

F. Complement

The Complement[11] of a fuzzy graph $: ��, �is a fuzzy graph $ ; : ��,; <� where �= � �and <�+, 2� � ��+�^��2� > �+, 2�for all u, v in V.

III. PROPERTIES OF ALPHA CUT SETS �AUB�B � ABUBB �A 3 B�B � AB 3 BB AB==== = A;B except for B � 0.5

IV. PROPERTY OF UNION�DEF�B � DBE FB

Let us take fuzzy graph � � �(, , ��where V has 5 vertices and membership values of those

vertices are � G0.9, 0.7,0.8, 0.7,1Kand graph A has 6 edges. Membership values of those edges are

in �



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u v w x y u v w x y

u 0 0.5 0 0 0.8 u 0 e6 0 0 e3

v 0.5 0 0.3 0 0 v e6 0 e3 0 0 �= w 0 0.3 0 0.4 0.6 w 0 e3 0 e5 e4

x 0 0 0.4 0 0.7 x 0 0 e5 0 e1

y 0.8 0 0.6 0.7 0 y e3 0 e4 e1 0

Adjacent matrix 1 Adjacent matrix 2

Let us take fuzzy graph B� �(, , �� where V has 4 vertices and membership values of those

vertices are � G0.4, 0.8 ,1, 0.9Kand graph A has 5 edges. Membership values of those edges are

in �

u v w x u v w x

u 0.0 0.6 0.0 0.7 u 0 e5 0 e3

v 0.6 0.0 0.5 0.9 v e5 0 e2 e1 � = w 0.0 0.5 0.0 0.3 w 0 e2 0 e2

x 0 .7 0.9 0.3 0.0 x e3 e1 e2 0


Adjacent matrix 1 represents the membership values of edges and Adjacent matrix 2

represents the name of the edges between the vertices.

Let A=(V, F) be a fuzzy graph where V = {(u, 0.9), (v, 0.7), (y, 1), (x, 0.7), (w, 0.8)} and

F = {(e1,0.7), (e3, 0.3), (e3, 0.8), (e4, 0.6),(e5, 0.4), (e6, 0.5)}

Let fuzzy graph B =(V, F) be a fuzzy graph where V = {(u, 0.4), (v, 0.8), (x, 0.9), (w, 1)} and

F = {(e1,0.9), (e2, 0.3), (e2, 0.5), (e3, 0.7),(e5, 06)}

Consider the union[4] $ � $% & $' of two graphs $% � �(%, �%� and $' � �(', �'� . Then

G = ((% & (', �% & �'�MNO * be a fuzzy subset of (* and �* a fuzzy subset of �* , i = 1, 2. Define

the fuzzy subsets % & ' of (% & (' and �% & �'of �% & �' as follows:

� % & '��+� � , %�+� -. + � (%\ (' '�+� -. + � ('\ (% %�+� ( '�+�, -. + � (% 0 ('1

And ( �% & �'��+2� � , �%�+2� -. +2 � �%\ �'�'�+2� -. +2 � �'\ �%�%�+2� (�'�+2�, -. +2 � �% 0 �'1

A. ALGORITHM FOR UNION

A UNION B

Vertices av[],bv[]

Edges am[],bm[]



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For each av[]&bv[]

Cv[]=max(av[],bv[])

For each am[]&bm[]

Cm[]=max(am[],bm[])

For each cm[]

If cm[]<cut then cm[]=0.0

Display cm[]

Drawgraph(cm[])

A CUT

For each am[]

If am[]<cut then am[]=0.0

Drawgraph(am[])

B CUT

For each bm[]

If bm[]<cut then bm[]=0.0

Drawgraph(bm[])

A CUT UNION B CUT

For each am[]&bm[]

Cm[]=max(am[],bm[])

Display cm[]

Drawgraph(cm[])

drawGraph(vertices[nRows][nCols])

{

for(i = 0; i <nRows; ++i)

{

for (j = i; j <nCols; ++j)

{

drawLine(i, j);

}

}

}

Fig.1: Graph A Fig.2: Graph B



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Fig.3: Graph A U B

For B � 0.6

Fig 4: Graph �P.Q Fig 5. Graph RP.Q

Fig 6: Graph �P.QS RP.QFig 7. Graph ��SR�P.Q

For B � 0.6 χP.Q � χ��SR�P.Q � �P.QS RP.Q�� 3

��SR�P.Q � �P.QS RP.Q



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V. PROPERTY OF JOIN �D 3 F� V.W � DV.W 3 FV.W

Let us take fuzzy graph � � �(, , �� where V has 3 vertices and membership values of

those vertices are � G0.9, 0.8, 0.7Kand graph A has 2 edges. Membership values of those edges are

in �

u1 u2 u3 u1 u2 u3

u1 0 0.5 0 u1 0 e3 0

� = u2 0.5 0 0.6 u2 e3 0 e2

u3 0 0.6 0 u3 0 e2 0


Let us take fuzzy graph B� �(, , �� where V has 3 vertices and membership values of those

vertices are � G0.5, 0.6 , 0.3K and graph A has 2 edges. Membership values of those edges are in �.

v1 v2 v3 v1 v2 v3

v1 0 0.5 0 v1 0 e2 0 �= v2 0.5 0 0.4 v2 e2 0 e1

v3 0 0.4 0 v3 0 e1 0


Adjacency matrix 1 represents the membership values of edges and Adjacent matrix 2

represents the name of the edges between the vertices.

Let A=(V, F) be a fuzzy graph whenV = {(u1, 0.9), (u2, 0.8), (u3, 0.7)} and F = { (e2 0.6), (e3, 0.5)}

Let B =(V, F) be a fuzzy graph whenV = {(v1, 0.5), (v2, 0.6), (v3, 0.3)} and F = {(e1,0.4), (e2, 0.5)}

Consider the join[4] $ � $% 3 $' of two graphs $% � �(%, �%� and $' � �(', �'� . Then

G = ((% & (', �% & �' & �′� where X’ is the set of all edges joining the vertices of (%456 (' .

and where we assume (% 0 (' � 7.. Let * be a fuzzy subset of (* and �* a fuzzy subset of �* , i = 1, 2. Define the fuzzy subsets % 3 ' of (% & (' and �% 3 �'of �% & �' & �′ as follows: � % 3 '��+� � � % & '�+� 8+ � (% & ('

And ��% 3 �'��+2� � 9�% & �'�+2�-. +2 � �% & �' %�+�^� '�2�-. + 2 � �′ 1

A. ALGORITHM FOR JOIN

A JOIN B

Vertices av[],bv[]

Edges ae[],be[]

For each av[]&bv[]

Ce[]=min(av[],bv[])

For each ce[]

If ce[]>cut then remove ce[]

Display ce[]

Drawgraph(ce[])



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A CUT

For each av[]

If av[]>cut then rav[]=av[]

B CUT

For each bv[]

If rbv[]<cut then rbv[]=bv[]

A CUT JOIN B CUT

For each rav[]&rbv[]

rce[]=min(rav[],rbv[])

Drawgraph(cm[])

drawGraph(vertices[nRows][nCols])

{

for(i = 0; i <nRows; ++i)

{

for (j = i; j <nCols; ++j)

{

drawLine(i, j);

}

}

}

Fig 8 : Graph A Fig 9 : Graph B

Fig 10: Graph A + B



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For

Fig 11: Graph Fig 12: Graph

Fig 13: Graph Fig 14: Graph

For

VI. PROPERTY OF COMPLEMENT = EXCEPT FOR

Let us take fuzzy graph where V has 4 vertices and membership values of

those vertices are and graph A has 4 edges. Membership values of those edges

are in

u v w x

u 0 0.5 0.1 0.3

v 0.5 0 0 0.2

w 0.1 0 0 0.2

x 0.3 0.2 0.2 0

The Complement of a fuzzy graph is a fuzzy graph where and

for all u, v in V.



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A. ALGORITHM FOR COMPLEMENT

Vertices av[]

Edges am[],cm[]

For each av[]&av[+1]

Cm[]=min(av[],av[+1])

Am[]=cm[]-am[]

For each am[]


Display cm[]

Drawgraph(cm[])

ii)

A CUT

For each am[]


CUT COMPLEMENT

For each am[]&am[+1]

Cm[]=min(am[],am[+1])

Am[]=cm[]-am[]

Display cm[]

Drawgraph(cm[])

Fig 15: Graph A Fig 16: Graph �<

For B � 0.5

Fig 17: Graph �P.X



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Fig 18: Graph �<P.X Fig 19: Graph �P.X=====

�P.X===== Y �<P.X

For B � 0.3

Fig 20: Graph �P.Z

Fig 21: Graph �P.Z===== Fig 22: Graph �<P.Z

�P.Z===== = �<P.Z

For B � 0.3, χP.Z � χ��P.Z===== � �<P.Z�� 4

VII. CONCLUSION

In this paper, we have derived fuzzy chromatic number for the properties of alpha cut sets

using alpha cut coloring of fuzzy graph whose edges and vertices both are fuzzy set. Here chromatic

number of fuzzy graph will be decrease when the value of alpha cut of the fuzzy graph will increase.

An interesting future work is to determine base station of mobile WDM communication networks.



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VIII. REFERENCES

[1] N.Biggs, E. Lloyd, R. Wilson, “Graph Theory”, Oxford University Press1736-1936.

[2] Daniel Marx, “Graph Coloring problems and their applications in scheduling”.

[3] A. Gamst, “Application of graph theoretical methods to GSM radio network”.

[4] F. Harrary, “Graph Theory”, Addison Wesley, Third printing, October 1972, “ Industrial and

applied Mathematics”, vol. 11, no.4, 2007.

[5] A. Kauffman, “Introduction a la Theorie des Sous-emsemblesFlous”, Vol.I, Masson et Cie,

1973.

[6] A. Mashaghi, A, et al, “Investigation of a protein complex network”, European Physical

Journal B 41 (1): 113–121, 2004.

[7] NarshingDeo,“Graph Theory with Applications to engineering and computer science”,

Prentice Hall of India private limited, New delhi, 1997.

[8] PerriMehonen, JanneRiihijarvi, Marina Petrova,“Automatic Channel allocation for small

wireless area networks using graph coloring algorithm approach”, IEEE, 2004.

[9] A. Rosenfeld,“ Fuzzy Graphs, In Fuzzy Sets and their Applications to Cognitive and Decision

Processes”, L.A. Zadeh, Fu, K.S. Shimura, M., Eds, Academic Press, New York 77-95,1975.

[10] ShariefuddinPirzada and AshayDharwadker, “Journal of the Korean Society for small

wireless area networks using graph coloring algorithm approach”, IEEE, 2004.

[11] M.S. Sunitha, A. Vijayakumar, “Complement of a fuzzy graph”, Indian Journal of Pure and

Applied Mathematics 33, 1451 – 1464, 2002.

[12] L. A.Zadeh,“Fuzzy Sets”, Inform, Control. 8 338-53, 8, 1965.

[13] L.A. Zadeh,“Similarity Relations and Fuzzy Ordering”, Inform. Sci. 3, 177-200, 1971.

[14] M.Manjuri and B.Maheswari, “Strong Dominating Sets of Strong Product Graph of Cayley

Graphs with Arithmetic Graphs”, International Journal of Computer Engineering &

Technology (IJCET), Volume 4, Issue 6, 2013, pp. 136 - 144, ISSN Print: 0976 – 6367,

ISSN Online: 0976 – 6375.

[15] K. Dhanalakshmi and B. Maheswari, “Accurate and Total Accurate Dominating Sets of

Interval Graphs”, International Journal of Computer Engineering & Technology (IJCET),

Volume 5, Issue 1, 2014, pp. 85 - 93, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.

[16] László Lengyel, “The Role of Graph Transformations in Validating Domain-Specific

Properties”, International Journal of Computer Engineering & Technology (IJCET),

Volume 3, Issue 3, 2012, pp. 406 - 425, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.

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