Laplacian-like energy of unicyclic graphs

8/12/2019 Laplacian-like energy of unicyclic graphs

1/14

ON THE LAPLACIAN COEFFICIENTS OF

UNICYCLIC GRAPHS 1

Dragan Stevanovic a,b,, Aleksandar Ilic aaFaculty of Sciences and Mathematics, University of Nis, Serbia

bUniversity of PrimorskaFAMNIT, Glagoljaska 8, 6000 Koper, Slovenia

Abstract

Let G be a graph of order n and let P(G, ) =n

k=0(1)k

ck

nk

be the charac-teristic polynomial of its Laplacian matrix. Generalizing an approach of Mohar ongraph transformations, we show that among all connected unicyclic graphs of ordern, the kth coefficientck is largest when the graph is a cycle Cn and smallest whenthe graph is the a Sn with an additional edge between two of its pendent vertices.A relation to the recently established Laplacian-like energy of a graph is discussed.

MSC Classification: 05C50

Key words: Laplacian coefficients; Laplacian matrix; Laplacian-like energy;Unicyclic graph.

1 Introduction

Let G = (V, E) be a simple undirected graph with n =|V| vertices andm =|E| edges. The Laplacian polynomial P(G, ) ofG is the characteristicpolynomial of its Laplacian matrix L(G) =D(G) A(G),

P(G, ) = det(In

L(G)) =n

k=0

(

1)kcknk.

Corresponding author. Full postal address is: Department of Mathematics andInformatics, Faculty of Sciences and Mathematics, Visegradska 33, 18000 Nis, Ser-bia.

Email addresses: [email protected] (Dragan Stevanovic),[email protected] (Aleksandar Ilic).1 This work has been supported by the grant 144007 and 144015G of the SerbianMinistry of Science and Technological Development and the research program P1-0285 of the Slovenian Agency for Research.

Preprint submitted to Elsevier 3 December 2008


2/14

Laplacian coefficients ck can be expressed in terms of subtree structures ofGby the following result of Kelmans [1]. Let Fbe a spanning forest ofG withcomponents Ti,i = 1, 2, . . . , k, havingnivertices each, and let (F) =

ki=1 ni.

Theorem 1.1 The Laplacian coefficientcnk of a graphG is given by

cnk=FFk

(F),

whereFk is the set of all spanning forests ofG with exactlyk components.The Laplacian matrix L(G) has non-negative eigenvalues 1 2 . . . n1 n = 0. From Viettes formulas, ck = k(1, 2, . . . , n1) is a sym-metric polynomial of order n 1. In particular, we have c0 = 1, cn = 0,c1= 2m, cn1=n(G), where (G) denotes the number of spanning trees ofG (see [2]). For unicyclic graphs, c1 = 2n and cn1 is n times the size of acycle in G.

LetCnbe the cycle of order n. LetSnbe the graph obtained from then-vertexstar Sn by joining two of its pendent vertices with an edge. Our main resultsare the following

Theorem 1.2 Let G be a connected unicyclic graph on n vertices and let k

be an integer, 2 k n 1. Thenck(G)< ck(Cn) ifG=Cn.Theorem 1.3 Let G be a connected unicyclic graph on n vertices and let k

be an integer, 2 k n 2. Thenck(Sn)< ck(G) ifG=Sn.The plan of the paper is as follows: In the next section we generalize resultof Mohar [10] on and transformation of trees to general graphs using

Kelmans theorem. Then in Section 3 we introduce and transformation ofunicyclic graphs. Finally, in Section 4 we prove that the cycle Cn has maxi-mum Laplacian coefficients among connected unicyclic graphs and in Section5 we prove that Sn has minimum Laplacian coefficients among connected uni-cyclic graphs. In the Section 6 we prove that Cn has maximum, while S

n has

minimum Laplacian-like energy among unicyclic graphs. In addition, we cal-culate the Laplacian coefficients and Laplacian-like energy of extremal graphsCn and S

n.

2 and transformations

Let w be a vertex of a connected graph G. We say that P =wv1v2 . . . vp is apendant path of lengthp attached at vertex wif internal vertices v1, v2, . . . , vp1have degree two and vp is a leaf. Graph G \ P is a graph obtained from G byremoving vertices v1, v2, . . . , vp of path P.

Definition 2.1 Suppose thatdeg(w) 3 and thatP =wv1v2 . . . vp andQ=wu1u2 . . . uq are distinct pendant paths of lengthsp 1andq 1, respectively,attached atw. Form a graphG = (G,w,P,Q) by removing pathsP and Q

2


3/14

and replacing them with another path R = wv1v2 . . . vpu1u2 . . . uq of lengthp+qby deleting edge{w, u1} and inserting edge{vp, u1}. We say that G isa-transform ofG.

Fig. 1. -transformation applied to G at vertexw.Theorem 2.1 LetG =(G,w,P,Q). Then for everyk= 0, 1, . . . , n

ck(G) ck(G), (1)

with equality if and only ifk {0, 1, n 1, n}.Proof: Laplacian coefficients c0 = 1 and cn = 0 are constant, while thenumber of edges and c1= 2mremain the same after -transformation. Everyspanning tree ofGcontains both pathsPandQ, and similarly, every spanningtree ofG contains pendant pathR. This means thatG andG have the same

number of spanning trees and thus we have equalitycn1= n(G) =n(G

) =cn1(G

).

Now assume that 2 k n2. Consider coefficient cnk and arbitraryspanning forest F ofG with k components. Assume that w belongs to a treeT from F which contains a 1 vertices in G \ (P Q) (counting w as well),firstb, 0 b p, vertices on pathPand firstc, 0 c q, vertices on path Q.

Ifb= p, let T be the tree obtained from Tby deleting edge wu1 and addingedge vpu1. The forest F

ofG which corresponds to F is given by

F = (F

\T)

{T

},

and, obviously, (F) =(F).

On the other hand, ifb p1, let T be that part ofT with a+b verticesnot containing edge wu1, and letT

be the remaining part ofT. Furthermore,a tree S from F that contains the last vertex on path P is a path with d,1 d p bvertices. Now, let S be union ofSand T. The forest F ofGwhich corresponds to F is given by

F = (F\ {S, T}) {S, T}.

3


4/14

This correspondence is obviously an injection.

Next, consider the subsetF of those spanning forests F ofGwith k compo-nents which coincide on G\(T S) (the part of F on P between T and Smay be translated), and for which b + d= Mis fixed. For such forests,

(F) =|T| |S| k2i=1

ni= (a + b + c)d N,

for some constant valueN. Sincea 1 andd 1, the corresponding spanningforest F ofG also contains k trees, and we have

(F) =|T| |S| k2i=1

ni = (a + b)(c + d) N.

The sum of numbers (F) for forests inF, divided by N, is

S=M1b=0

(a + b + c)d=M1b=0

(a + b + c)(M b)

=M1b=0

(a + b)(M b) + cM1b=0

(M b)

=M1b=0

(a + b)(M b) + cMb=1

b.

On the other hand, the sum for corresponding forests ofG is

S = M1b=0

(a + b)(c + d) =M1b=0

(a + b)(c + M b)

=M1b=0

(a + b)(M b) + cM1b=0

(a + b)

=M1b=0

(a + b)(M b) + cM a + cM1b=0

b.

Thus,S S = cM(a1) 0. Since deg(w) 3, and hence G\(P Q)has other vertices than w, there exists a subsetF of spanning forests of Gfor which a > 1 (and cM > 0). Now, the inequality cnk(G) cnk(G

),follows from Kelmans formula by summing over all possible subsetsF ofk-component spanning forests of G. By setting k nk, the inequality (1)follows.

Definition 2.2 Letw be a vertex of degreep + 1 in a graphG, which is not astar, such that wv1, wv2, . . . , w vp are pendent edges incident with w andu is

the neighbor of w distinct fromv1, v2, . . . , vp. We form a graphG =(G, w)

by removing edges wv1, wv2, . . . , w vp and adding new edges uv1, uv2, . . . , u vp.

We say thatG is-transform ofG.

4


5/14

Theorem 2.2 Let G = (G, w) be a -transform of a graph G. Then forevery0 k n holds

ck(G) ck(G), (2)

with equality if and only ifk {0, 1, n 1, n}.

Fig. 2. -transformation applied to G at vertexw.

Proof: Laplacian coefficients c0 = 1 and cn = 0 are constant, while the

number of edges and c1 = 2m remain the same after -transformation. Also,Gand G have the same number of spanning trees and thus we have equalitycn1=n(G) =n(G

) =cn1(G).

Now assume that 2 k n2. Consider coefficient cnk and arbitraryspanning forest F in G with k components. Let T be a tree from F thatcontains vertex u with a 1 vertices in G\ {w, v1, v2, . . . , vp}. Let b, 0 b p+ 1 be the number of isolated vertices in F among pendent verticesw, v1, v2, . . . vp. If b = 0, corresponding tree T has same vertices as T

andtherefore, for F = (F \ {T}) {T}, we have (F) = (F). In special caseb = p+ 1, vertices w, v1, . . . , vp are isolated in both forests, and so F = F

,

and obviously (F) =(F

).Suppose now that 1 b p. Consider the subsetF of those spanning forestsofG with kcomponents which coincide on G without vertices u,w,v1, v2, . . . , vpand for which numbers a and bare fixed. The sum of all such (F) equals

S =

p + 1

b

(a +p + 1 b)

kb1i=1

ni,

because we can choose b isolated vertices from the set{w, v1, v2, . . . , vp} inp+1b ways and tree T has a+ p + 1

b vertices. On the other hand, in G

we can construct b isolated vertices in two ways, depending on whether weinclude the edge{u, w}in a forest. If we include the edge{u, w}in some treeof F we have to choose b isolated vertices in order to preserve the numberof trees in forest F; in other case, we choose b1 isolated vertices and theremaining tree has p+ 1(b1) vertices. Therefore, corresponding sum inthe graph G equals

S=

p

b

(a +p + 1 b) +

p

b 1

a(p + 1 (b 1))

kb1i=1

ni.

5


6/14

Fromp+1

b

=p

b

+ p

b1

, it follows

S S =

p

b 1

(a (p b + 2) (a +p + 1 b))

kb1i=1

ni=

=

pb 1

(p b + 1)(a 1)

kb1i=1

ni 0.

Since graph G= Sn, there exists a subsetF of spanning forests of G forwhicha >1 and the inequality (2) follows from Kelmans formula by summingover all possible subsetsF ofk-component spanning forests ofG.

3 and transformations

Definition 3.1 Letw be a vertex in a cycleCof a connected unicyclic graphG. Assume thatG has pendant pathP =wv1v2 . . . vp attached tow. Ifu is oneof two neighbors ofw inC, letG =(G,w,P) be a graph obtained fromG byremoving edge{u, w}and adding edge{u, vp}. We say thatG is a-transformofG.

Theorem 3.1 LetG =(G,w,P) be a-transform of a connected unicyclicgraphG. For every0 k n holds

ck(G) ck(G), (3)

with equality if and only ifk {0, 1, n}.

Fig. 3. Example of-transformation fora= 3, b= 2, c= 1 andd= 3.

Proof: Equality for c0 = 1, c1 = 2m and cn = 0 is obvious. The lengthof a cycle in G is greater than the length of a cycle in G and therefore,cn1(G)< cn1(G

).

Consider coefficient cnk and arbitrary spanning forest F ofG with k compo-nents. If vertices w and vp belong to the same tree in F, then correspondingtrees inF have the same number of vertices and therefore(F) =(F). Oth-erwise, vertex w is in a tree Tthat contains a 0 vertices in a graph G \ Pon the same side as u, b 1 vertices on the other side of cycle (including w)

6


7/14

and firstc, 0 c < p, vertices on path P. Furthermore, a tree Sthat containsthe last vertex on path P is a path with d 1 vertices.

After removing edge{u, w}, tree T is decomposed in two disjoint partstree T that contains a vertices and tree T with b+c vertices. Thus, forestF = (F

\ {T, S

}) {

T, S

}, where S is union of trees T and S. Therefore,

(F) (F) = ((a + d)(b + c) (a + b + c)d) k2i=1

ni=a(b + c d) N,

for N=k2

i=1 ni.

Consider a familyF of spanning forests ofGwhich coincide on G \ (T S),having fixed values a, b and c+ d = M. If we sum all differences for suchforests, we get

FF

(F) (F) = FF

a(b + c d) N

=aNM1c=0

(b + 2c M)

=aN M(b 1) 0.

SinceG contains a cycle, there exists at least one forest Fsuch that a >0 andb >1, and then from Kelmans formula it follows that cnk(G)< cnk(G

).

Definition 3.2 Let v and w be two neighboring vertices on a cycle in anunicyclic graphG, such thatv has degreep+ 2 andp pendent edges incidentwith v and w has degree q+ 2 and q pendent edges incident with w. We getgraph G = (G,v,w) by contracting edge{v, w} and adding a new pendentedge to vertexv. We say thatG is a-transform ofG.

Theorem 3.2 LetG =(G,v,w) be a-transform of a connected unicyclicgraphG. For every0 k n holds

ck(G) ck(G),

with equality if and only ifk {

0, 1, n}

.

Proof: This transformation obviously decreases the size of a cycle inG andpreserves coefficientsc0,c1,cn. For coefficient cn1strict inequality cn1(G)>cn1(G

) holds. In order to prove inequality for 2 k n2, we considercoefficient cnk and arbitrary spanning forest F

with k components of graphG. Assume that pq >0. Let L be the set ofp+q+ 1 pendent vertices thatare neighbors ofv.

Suppose that cycles in G and G are oriented clockwise, such that w comesafter v in G. Assume that v is in tree T, which contains a 1 vertices

7


8/14

Fig. 4. Example of-transformation forp= 3, q= 4, a= 2, b= 1 andc= 4.

(including v) different from p+ q+ 1 pendent vertices in counterclockwisedirection from v, and b 0 vertices (excluding v) in clockwise direction from

v. Consider the subsetF

of those spanning forests F

which coincide onG \ (v L) and contain exactly 0 c p + q+ 1 isolated vertices in L.Assume that 0< c < p + q+ 1. For F F we have following summation inKelmans formula:

S =

p + q+ 1

c

(a + b +p + q c + 1)

kc1i=1

ni.

InG there are two types of corresponding trees: trees that contain edge{v, w}and those that do not contain edge{v, w}. For trees of the first type, we haveF = (F \ T) {T}, and sum

S1=

p + q

c

(a + b +p + q c + 1)

kc1i=1

ni.

In the second type of forests we have two trees T and S in forest F insteadofT, and we can choose 0 x p isolated vertices within p neighbors ofvand 0 c1x qvertices among qneighbors ofw in order to preservethe number of components. Therefore, for this kind of forests Fwe have:

S2=x

p

x

q

c 1 x

(a +p x)(b + 1 + q (c 1 x)) kc1i=1

ni.

Using inequality xy x + y 1 for numbers that are greater than or equal toone, we estimate product:

(a +p x)(b + 1 + q (c 1 x)) a + b +p + q c + 1. (4)

8


9/14

We also exploit well-known combinatorial summation identity (see [11]):

k

a

k

b

n k

=

a + b

n

.

Therefore, in Kelmans formula for graph G, we have following sum:

S=S1+ S2

p + q

c

+

p + q

c 1

(a + b +p + q c + 1) kc1i=1

ni.

For p = q= 0 there is exactly one pendent vertex attached at v . If this vertexrepresents a tree in forest F, then tree T has a+b vertices. In graph G wehave two corresponding forests that coincide on F \{T, w}, but instead treesT and w have trees witha and b +1 anda +1 andb vertices. From inequalitya + b (a + 1)b + a(b + 1), in this case we have cnk(G) cnk(G

).

If c = 0 the corresponding trees in forests of G and G have equal numberof vertices. If c = p + q + 1, in the forest F we have p + q+ 1 isolatedvertices and one tree with a+ b vertices. For b = 0 we have (F) = (F),so we can assume that b 1. If we plug c = 1 and x = 0 in the product(a+p x)(b+ 1 + q (c 1 x)), we will get (a+p)(1 +b+q) and bothfactors are strictly greater than 1. This saving in estimation (4) is at least oneand is enough to assure inequality in this special case.

This means thatS S and finally cnk(G) cnk(G). Lemma 3.3 Letv,w andu be vertices of a triangle in unicyclic graphG(a,b,c),

such that vertexv hasa + 2 neighbors, vertexw hasb + 2 neighbors and vertexu has c+ 2 neighbors and a+ b + c + 3 = n. If we delete all pendent edges

from vertexu and attach them to vertexv we will get graphG(a + c,b, 0). Fork= 0, 1, . . . , n we have inequality:

ck(G(a,b,c)) ck(G(a + c,b, 0)).

Fig. 5. Example of graphsG(a,b,c) andG(a+c,b, 0)

9


10/14

Proof: Equality holds for coefficientsc0,c1,cn1andcn. Consider coefficientcnk and spanning forests of these graphs with k components such that thereare exactly x isolated vertices between leaves ofv , y isolated vertices betweenleaves of w and z isolated vertices between leaves of u. Let A = a+ 1x,B =b + 1

y andC=c + 1

z. If we include the edge

{v, u

}in a forest, than

we have trees of equal sizes in both graphs. If we include both edges{v, w}and{w, u}, the corresponding trees also have equal number of vertices, andthus(F) =(F) in these cases.

if we include the edge{w, u} and not the edge{v, w}(F) (F) =A(B+ C) (A + C 1)(B+ 1)

if we include the edge{v, w} and not the edge{w, u}(F) (F) = (A + B)C (A + B+ C 1)

if we exclude both edges{w, u} and{v, w}(F) (F) =ABC B(A + C 1)

After summing we have to prove inequality:

A(B+C)+(A+B)C+ABC (A+C1)(B+1)+(A+B+C1)+B(A+C1)(A 1)(C 1)(B+ 2) 0.

The last inequality is obviously true for positive integer numbers A,B and C.Since numbers a and c are greater than or equal to 1, for the case x= z= 0

we have strict inequality.

4 Cn has maximum Laplacian coefficients

Definition 4.1 A unicyclic graph G is said to be a sun graph if cycle vertices

have degree at most three and remaining vertices have degree at most two.

Using and transformations, we can now give a simultaneous proof toTheorem 1.2.

Proof: LetC=u1u2 . . . ukbe a cycle of connected unicyclic graphG, and letTi be a tree attached at ui, i = 1, 2, . . . , k(Tidoes not contain any other cyclevertices). For i= 1, 2, . . . , k, we can apply -transformation to that vertex ofTi of degree at least three which is at largest distance from the root ui, aslong as Ti is not a path. This way we increase Laplacian coefficients ofG byTheorem 2.1.

Thus, an unicyclic graph with maximum Laplacian coefficients has to be a sungraph. Now, we can apply -transformation to a sun graph as long as it is nota cycle Cn, thus, further increasing its Laplacian coefficients.

10


11/14

Thus, we see thatCnuniquely maximizes Laplacian coefficientsckfor 2 k n 1 among connected unicyclic graphs of order n.

5 Sn has minimum Laplacian coefficients

Definition 5.1 A caterpillar graph is a tree such that if all leaf vertices and

their incident edges are removed, the remainder of the graph forms a path.

Using and transformations together with Lemma 3.3, we can now give asimultaneous proof to Theorems 1.3 and 6.3.

Proof: LetC=u1u2 . . . ukbe a cycle of connected unicyclic graphG, and letTi be a tree attached at ui, i = 1, 2, . . . , k(Tidoes not contain any other cyclevertices). In every tree we can find a vertex v that is the parent of the leaf onthe deepest level and apply -transformation at v to decrease the Laplacian

coefficients. We perform this algorithm until we get a graph G, such that toevery vertex on a cycle only pendent edges are attachedjust like a caterpillarwith an additional edge that connects the first and the last vertex.

We further use the -transformation until the size of the cycle becomes equal tothree, which further reduces Laplacian coefficients. After applying Lemma 3.3we get that the Laplacian coefficients are minimal when all pendent vertices areattached to one triangle vertex. The uniqueness follows from strict inequalitieson Laplacian coefficients when performing and transformations.

6 Laplacian-like energy of unicyclic graphs

The Laplacian-like energy of graph G, LEL for short, is defined as follows:

LEL(G) =n1k=1

k.

This concept was introduced in [3] where it was shown that it has similarfeatures as molecular graph energy (for more details see [4]). One of presentauthors showed a connection between LEL and Laplacian coefficients in [5].

Theorem 6.1 Let G and H be two n-vertex graphs. If ck(G) ck(H) fork= 1, 2, . . . , n1thenLEL(G) LEL(H). Furthermore, if a strict inequalityck(G)< ck(H) holds for some1 k n 1, thenLEL(G)< LEL(H).Using this result, we can conclude following:

Corollary 6.2 Let G be a connected unicyclic graph on n vertices. Then

LEL(G)< LEL(Cn) ifG=Cn.Corollary 6.3 Let G be a connected unicyclic graph on n vertices. Then

LEL(Sn)< LEL(G) ifG=Sn.

11


12/14

Corollary 6.2 has been conjectured in [5]. Recall that the energy E(G) ofa graph G is the sum of absolute values of the eigenvalues of its adjacencymatrix A(G). Now we can point out a connection between Theorem 6.2 andthe following conjecture of Caporossi, Cvetkovic, Gutman and Hansen from[6], obtained by using system AutoGraphiX [7], which served as our initial

inspiration:

Conjecture 6.4 Among unicyclic graphs onn vertices, the cycleCn has max-

imum energy ifn 7 andn= 9, 10, 11, 13 and15. For all other values ofn,the unicyclic graph with maximum energy is P6n , where P

ln is the unicyclic

graph obtained by connecting a vertex of a cycleCl with a pendant vertex of a

pathPnl.

This conjecture is still open and best result so far is by Hou, Gutman andWoo [8], who showed that among connected unicyclic bipartite n-vertex graphseither Cn or P

6n have maximum energy. Zhou and Gutman [9] recently es-

tablished a connection between ordinary and Laplacian spectra of bipartitegraphs, and from Corollary 2 of their paper it follows that

2 LEL(G) =E(S(G)),

where S(G) is the subdivision graph of a bipartite graph G. IfG is unicyclic,bipartite and has n vertices, then the subdivision graph S(G) is unicyclic,bipartite, and has 2n vertices. Thus,

LEL(G) max

1

2E(C2n),

1

2E(P62n)

= max

LEL(Cn),LEL(P

3n)

,

and sinceP3n is not bipartite, we see that indeed Cnhas maximum LEL amongbipartite unicyclic graphs. However, this approach may not be used when Gis not bipartite.

Laplacian eigenvalues of a cycleCnare k= 22cos 2kn , wherek = 1, 2, . . . , n(see, e.g., [12]). Therefore, the Laplacian-like energy ofCn is

LEL(Cn) =n

k=1

2

1 cos2k

n

= 2

n

k=1

sink

n

= 2sinn+12

n

sinn

2

n

sin

2n

= 2 cot

2n.

Corollary 6.5 IfG is a connected unicyclic graph onn vertices, then

LEL(G) 2cot

2n.

Equality holds if and only ifG=Cn.

12


13/14

From Taylor expansion of cot x, we get that, for sufficiently large n,

LEL(Cn)4n

6n.

Consider the Laplacian matrix for graph S

n.

P(Sn, ) =

(n 1) 1 1 1 . . . 1 11 2 1 0 . . . 0 01 1 2 0 . . . 0 01 0 0 1 . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

1 0 0 0 . . . 1 01 0 0 0 . . . 0 1

=

= ( 1) P(Sn1, ) ( 1)n3( 3).

After simple determinant manipulations we can prove by mathematical induc-tion that P(Sn, ) =( n)( 3)( 1)n3. For n = 3, a triangle S3 =C3has eigenvalues 3, 3, 0. If we assume that formula holds for n1, from therecurent formula on P(Sn, ) we get:

P(Sn, ) = (

1)

(

n + 1)(

3)(

1)n4

(

1)n3(

3)

=( n)( 3)( 1)n3.

This means that Laplacian-like energy ofSn equals LEL(Sn) =

n+

3 +

n3. Laplacian coefficients can be easily calculated using symmetric sums,because only eigenvalues that are different from 1 are n and 3.

ck= (n + 3)

n 3k 1

+ 3n

n 3k 2

+

n 3

k

.

Corollary 6.6 IfG is a connected unicyclic graph onn vertices, then

LEL(G) n + n 3 + 3.

Equality holds if and only ifG is isomorphic to Sn.

References

[1] A.K. Kelmans, V.M. Chelnokov, A certain polynomial of a graph and graphswith extremal number of trees, J. Comb. Theory, Ser. B 16(1974), 197214.

13


14/14

[2] R. Merris, A survey of graph Laplacians, Lin. and Multilin. Algebra 39(1995),1931.

[3] J. Liu, B. Liu, A Laplacian-Energy-Like Invariant of a graph, MATCHCommun. Math. Comput. Chem.59(2008), 397419.

[4] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz103(1978), 122.

[5] D. Stevanovic, Laplacian-like energy of trees, MATCH Commun. Math.Comput. Chem. 61 (2009), 407417.

[6] G. Caporossi, D. Cvetkovic, I. Gutman, P. Hansen,Variable neighborhood searchfor extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf.Comput. Sci. 39(1999), 984996.

[7] G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 1.The AutoGraphiX system, Discrete Math. 212(2000), 2944.

[8] Y. Hou, I. Gutman, C. W. Woo,Unicyclic graphs with maximal energy, LinearAlgebra Appl. 356(2002), 2736.

[9] B. Zhou, I. Gutman, A connection between ordinary and Laplacian spectra ofbipartite graphs, Linear Multilin. Algebra 56(2008), 305310.

[10] B. Mohar,On the Laplacian coefficients of acyclic graphs, Linear Algebra Appl.722(2007), 736741.

[11] R. Stanley, Enumerative Combinatorics, Volumes 1 and 2, CambridgeUniversity Press, New York/Cambridge, 1999

[12] D. Cvetkovic, M. Doob, H. Sachs, Spectra of graphs - Theory and Application,3rd edition, Johann Ambrosius Barth Verlag, 1995.

14

Laplacian-like energy of unicyclic graphs

Documents

Transcript of Laplacian-like energy of unicyclic graphs