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arXiv:1608.07939v1 [math.CO] 29 Aug 2016

Applications of a theorem by Ky Fan in the

theory of weighted Laplacian graph energy

Reza Sharafdini∗, Alireza Ataei, Habibeh Panahbar

Department of Mathematics, Persian Gulf University, Bushehr 75169-13817, Iran

Abstract

The energy of a graph Gis equal to the sum of the absolute values of

the eigenvalues of G, which in turn is equal to the sum of the singular

values of the adjacency matrix of G. Let X,Yand Zbe matrices, such

that X+Y=Z. The Ky Fan theorem establishes an inequality between

the sum of the singular values of Zand the sum of the sum of the singular

values of Xand Y. This theorem is applied in the theory of graph energy,

resulting in several new inequalities, as well as new proofs of some earlier

known inequalities.

Key words:Ky Fan theorem; Mean deviation; Vertex weight; Eigenvalue;

Energy (of graph); Singular value (of matrix);

AMS Subject Classiﬁcation: 05C50; 05C90; 15A18; 15A42; 92E10.

1 Introduction

In this paper, we are concerned with simple graphs. Let G= (V , E)be a simple

graph, with nonempty vertex set V={v1,...,vn}and edge set E={e1,...,em}.

That is to say, Gis a simple (n, m)-graph. Let ωbe a vertex weight of G, i.e., ω

is a function from the set of vertices of Gto the set of positive real numbers. Gis

called ω-regular if for any u, v ∈V(G),ω(u) = ω(v). Observe that a well-know

vertex weight of a graph is the vertex degree weight assigning to each vertex its

degree. Let us denote it by deg.

The diagonal matrix of order nwhose (i, i)-entry is ω(vi)is called the diag-

onal vertex weight matrix of Gwith respect to ωand is denoted by Dω(G), i.e.,

Dω(G) = diag(ω(vi),...,ω(vn)) . The adjacency matrix A(G) = (aij )of Gis a

1

(0,1)-matrix deﬁned by aij = 1 if and only if the vertices viand vjare adjacent.

Then the matrices Ldeg(G) = Ddeg(G)−A(G)and L†

deg(G) = A(G) + Ddeg(G)

are called Laplacian and signless Laplacian matrix of G, respectively (see [11],

[12], [22], [23], [24] and [25]). Let us generalize these matrices for arbitrary ver-

tex weighted graphs. Let Gbe a simple graph with the vertex weight ω. Then we

shall call the matrices Lω(G) = Dω(G)−A(G)and L†

ω(G) = A(G) + Dω(G)the

weighted Laplacian and the weighted signless Laplacian matrix of Gwith respect

to the vertex weight ω.

Let X={x1, x2, ..., xn}be a data set of real numbers. The mean absolute

deviation (often called the mean deviation) MD(X)and variance Var(X)of Xis

deﬁned as

MD(X) = 1

n

n

X

i=1

|xi−x|,Var(X) = 1

n

n

X

i=1

(xi−x)2

where x=Pn

i=1 xi

nis the arithmetic mean of the distribution. Note that an easy

application of the Cauchy-Schwarz inequality gives that the mean deviation is a

lower bound on the standard deviation (see [3]).

MD(X)≤pVar(X).(1)

The mean deviation and variance of Gwith respect to ω, denoted by MDω(G)and

Varω(G), respectively, is deﬁned as

MDω(G) = MD(ω(v1),...,ω(vn)),Varω(G) = Var(ω(v1),...,ω(vn)).

It follows from Eq. (1) that MDω(G)≤pVarω(G). It is worth mention-

ing that Vardeg (G)is well-investigated graph invariant (see [2] and [19]). Let

λ1, λ2,...,λnbe eigenvalues of the adjacency matrix A(G)of graph G. It is

known that Pn

i=1 λi= 0. The notion of the energy E(G)of an (n, m)-graph G

was introduced by Gutman in connection with the π-molecular energy (see [13],

[14], [17] and [21]). It is deﬁned as

E(G) =

n

X

i=1

|λi|=nMD(λ1, λ2,...,λn).

For details of the theory of graph energy see [14], [16] and [29].

Let M∈Cn×nbe Hermitian with singular values si(M), i = 1,2,...,n.

If λi(M), i = 1,2,...,n are eigenvalues of M, then si(M) = |λi(M)|,i=

2

1,2,...,n. Getting motivated from this fact, Nikiforov established the concept of

matrix energy by analogy with graph energy [26]. Let M∈Cn×nwith singular

values si(M), i = 1,2,...,n. Then the energy of M, denoted by E(M), is de-

ﬁned as s1(M) + s2(M) + ...+sn(M). Consequently, if M∈Cn×nis Hermitian

with eigenvalues λ1(M), λ2(M),...,λn(M), we have

E(M) =

n

X

i=1

|λi(M)|.

Let n≥µ1, µ2, . . . , µn= 0 be eigenvalues of Laplacian matrix L(G)of graph

G. It is known that Pn

i=1 µi= 2m. Gutman and Zhou deﬁned the Laplacian

energy of an (n, m)-graph Gfor the ﬁrst time (see [18] ) as

LE(G) =

n

X

i=1 µi−2m

n=nMD(µ1,...,µn).

Numerous results on the Laplacian energy have been obtained, see for instance

[1], [4], [7], [15], [27], [28] and [33]. Note that in the deﬁnition of Laplacian

energy 2m

nis the average vertex degree of G. This motivates us to extend their

deﬁnition to the graphs equipped with an arbitrary vertex weight. Let Gbe a graph

with the vertex set V={v1,...,vn}and with an arbitrary vertex weight ω. Let

µ1, µ2,...,µnbe eigenvalues of the weighted Laplacian matrix Lω(G)of graph

Gwith respect to the vertex weight ω. Then we propose the weighted Laplacian

energy LEω(G)of Gwith respect to the vertex weight ωas

LEω(G) =

n

X

i=1 µi−ω=nMD(µ1,...,µn),(2)

where

ω=Pn

i=1 ω(vi)

nand n

X

i=1

µi=nω.

Note that LEdeg (G) = LE(G).

Let Isbe the unit matrix of order s. For the considerations that follow it will

be necessary to note that instead via Eq. (2), the weighted Laplacian energy can

be expressed also as

LEω(G) = E(Lω(G)−ωIn).(3)

The following results are already known.

The next lemma is known for the vertex degree weight [5]; Its proof for an

arbitrary vertex weight is done in a similar fashion.

3

Lemma 1. Let Gbe a bipartite graphs with n vertices and with a vertex weight

ω. Then Lω(G)and L†

ω(G)are similar.

Lemma 2. [20, Section 7.1, Ex. 2] If A= (aij )n

i,j=1 is a positive semi-deﬁnite

matrix and aii = 0 for some i, then aij = 0 = aj i,j= 1,...,n.

Theorem 1, supporting the concept of matrix energy proposed by Nikiforov,

was ﬁrst obtained by Ky Fan [8] using a variational principle. It also appears

in Gohberg and Krein [10] and in Horn and Johnson [20]. No equality case is

discussed in these references. Thompson [31, 32] employs polar decomposition

theorem and inequalities due to Fan and Hoffman [9] to obtain its equality case.

Day and So [6] give the details of a proof for the inequality and the case of equal-

ity.

Theorem 1. Let Aand Bbe two complex square matrices of size n(A, B ∈Cn×n)

and let C=A+B. Then

E(C)≤E(A) + E(B).(4)

Moreover equality holds if and only if there exists an unitary matrix Psuch that

P A and P B are both positive semi-deﬁnite matrices .

Let Abe a complex matrix of size n(A∈Cn×n). Let us denote the Hermitian

adjoint of Aby A∗. Then both A∗Aand AA∗are Hermitian positive semi-deﬁnite

matrices with the same nonzero eigenvalues. In particular A∗Aand AA∗are di-

agonalizable with real non-negative eigenvalues. Then by spectral theorem for

complex matrices we may deﬁne |A|:= (A∗A)1/2. Here we present the following

version of the polar decomposition theorem [20].

Theorem 2. For A∈Cn×n, there exist positive semi-deﬁnite matrices X, Y ∈

Cn×nand unitary matrices P, F ∈Cn×nsuch that A=P X =Y F . Moreover,

the matrices X, Y are unique, X=|A|,Y=|A∗|. The matrices Pand Fare

uniquely determined if and only if Ais non-singular.

There is a great deal of analogy between the properties of E(G)and LEω(G),

but also some signiﬁcant differences. These similarities and dissimilarities has

been investigated [30]. In this paper we apply Theorem 1 in the theory of graph

energy, resulting in several new inequalities, as well as new proofs of some earlier

known inequalities. It is worth mentioning that the idea of this paper inspired

from [27] and [28]; Our proofs are based on those of these references.

4

2 Graphs Gfor which LEω(G) = E(G)+E(Dω(G)−

ωIn)

In the case of vertex degree weight, the inequality in the following theorem was

proved in [28], whereas the equality in Eq. (5) was investigated in [27]. Based

on their proof, we generalize their results for a connected graph with an arbitrary

vertex weight.

Theorem 3. Let Gbe a connected graph with nvertices and with a vertex weight

ω. Then

LEω(G)≤nMDω(G) + E(G).(5)

Moreover the equality in Eq. (5) holds if and if Gis ω-regular.

Proof. We Know that

Lω(G)−ωIn= (Dω(G)−ωIn) + (−A(G)).(6)

Note that Dω(G)−ωInis a diagonal matrix whose eigenvalues are ω(vi)−ω,

i= 1,...,n. It follows from Theorem 1 that

n

X

i=1

si(Lω(G)−ωIn)≤

n

X

i=1

si(Dω(G)−ωIn) +

n

X

i=1

si(−A(G)).

Therefor

LEω(G)≤

n

X

i=1

|ω(vi)−ω|+

n

X

i=1

|λi(−A(G))|.

Then, due to the similarity between A(G)and −A(G), we have

LEω(G)≤nMDω(G) + E(G).

Let Gbe a ω-regular graph with eigenvalues λ1,...,λn. Then ω=ω(vi)for

each 1≤i≤nand Lω(G) = ωIn−A(G). It follows that ω−λ1, . . . , ω −λnare

all the eigenvalues of Lω(G). Therefore, by Eq. (2) we have

LEω(G) = E(G).

Conversely, suppose that the equality in Eq. (5) holds. Without loss of gener-

ality, we may assume that ω(v1) = max{ω(vi)|1≤i≤n)}. Suppose on the

contrary that Gis not ω-regular. Therefore

ω(v1)>ω. (7)

5

Let ai:= ω(vi)−ωfor i= 1,...,n. We have a1>0, via Eq. (7). Due to the

equality in Eq. (5), we may apply Theorem 1 to Eq. (6). Therefore, there exists

a unitary matrix Psuch that X=P(Dω(G)−ωIn)and Y=P(−A(G)) are

both positive semi-deﬁnite. Hence P∗Xand P∗Yare polar decompositions of the

matrices Dω(G)−ωInand −A(G), respectively. It follows from Theorem 2 that

X=|Dω(G)−ωIn|and Y=|A(G)|. Therefore X= diag(|a1|,|a2|,...,|an|).

Setting

P∗=

q11 ··· q1n

.

.

.....

.

.

qn1··· qnn

, A(G) =

0a12 ··· a1n

a12 0··· a2n

.

.

........

.

.

a1na2n··· 0

,

P∗X=Dω(G)−ωIn, implies

q11 ··· q1n

.

.

.....

.

.

qn1··· qnn

|a1|...

|an|

=

a1...

an

.

Then,

|a1|q11 |a2|q12 · · · |an|q1n

|a1|q21 |a2|q22 · · · |an|q2n

.

.

..

.

.....

.

.

|a1|qn1|a2|qn2· · · |a1|qnn

=

a1

a2...

an

.

Equality at ﬁrst column imposes q11 = 1 and qi1= 0, i = 2,...,n. It follows that

P=

1 0 ··· 0

q12 ··· q1n

.

.

.....

.

.

q1n··· qnn

.

We must then have

Y=−

1 0 · · · 0

q12 · · · q1n

.

.

.....

.

.

q1n· · · qnn

0a12 ··· a1n

a12 0··· a2n

.

.

........

.

.

a1na2n··· 0

=−

0a12 ··· a1n

∗0· · · ∗

.

.

........

.

.

∗a2n· · · ∗

.

6

The previous matrix is positive semi-deﬁnite and by Lemma 2, we obtain a1j= 0,

j= 2,...,n. This contradicts our assumption that Gis a connected graph and the

result follows.

3 Graphs Gfor which LEω(G) = E(G)

In Theorem 3 we showed that if Gis a ω-regular graph, then

LEω(G) = E(G).(8)

In what follows we consider the converse argument.

In the case of vertex degree weight, the ﬁrst part of the following theorem was

proved in [28], whereas the second part was proved in [27]. Based on their proof,

we generalize their results for a connected graph with an arbitrary vertex weight.

Theorem 4. Let Gbe a bipartite graph with a vertex weight ω. Then

LEω(G)≥E(G).(9)

Moreover, the equality in Eq. (9) holds if and only if Gis a ω-regular graph.

Proof. From the deﬁnition of weighted Laplacian matrix and weighted signless

Laplacian matrix, it is clear that

L†

ω(G)−ωIn−Lω(G)−ωIn= 2A(G).(10)

If Gis bipartite, then it follows from Lemma 1 that Lω(G)and L†

ω(G)have

the same spectra and therefore

n

X

i=1

si(L†

ω(G)−ωIn) =

n

X

i=1

si(Lω−ωIn) =

n

X

i=1

si(−[Lω(G)−ωIn]) = LEω(G).

So by Theorem 1, LEω(G)≥E(G).

Let Gbe a ω-regular graph. Then by Theorem 3, the equality in Eq. (9) holds.

Conversely, suppose that the equality in Eq. (9) holds. Therefore,

E(L†

ω(G)−ωIn)−(Lω(G)−ωIn)= 2E(G) = E(G)+E(G) = LEω(G)+LEω(G).

Since Gis bipartite it follows from Lemma 1 that

E(L†

ω(G)−ωIn)−(Lω(G)−ωIn)=EL†

ω(G)−ωIn+E−Lω(G)−ωIn.

(11)

7

Therefore, Theorem 1 asserts that there exists a unitary matrix P, such that

X=PL†

ω(G)−ωInand Y=P−Lω(G)−ωIn,(12)

are both positive semi-deﬁnite matrices. Hence P∗Xand P∗Yare polar decom-

positions of

L†

ω(G)−ωInand −Lω(G)−ωIn,

respectively. By Theorem 1 we obtain

X=|L†

ω(G)−ωIn|and Y=| − Lω(G)−ωIn|.

In view of the fact that Gis bipartite, we conclude that X=Y. Therefore, it

follows from Eq. (12) that

L†

ω(G) + Lω(G) = 2ωIn,

implying the result.

In the case of vertex degree weight, the next theorem was proved in [28] and

based on their proof, we get also the following theorem.

Theorem 5. Let Gbe a bipartite graph with nvertices and with a vertex weight

ω. Then

max nnMDω(G), E(G)o≤LEω(G)≤nMDω(G) + E(G).(13)

Proof. The right side inequality is a direct consequent of Theorem 3. Let us prove

the left one. It is easy to see that

L†

ω(G) + Lω(G) = 2Dω(G),

from which

(L†

ω(G)−ωIn!+ Lω(G)−ωIn)!= 2 Dω(G)−ωIn!.

It follows from Theorem 1 that

E (L†

ω(G)−ωIn!+E Lω(G)−ωIn)!≥2E Dω(G)−ωIn!= 2nMDω(G).

8

In the other hand, since Gis bipartite, it follows from Lemma 1 that

LEω(G) = E (L†

ω(G)−ωIn!=E Lω(G)−ωIn)!.

Therefore

LEω(G)≥nMDω(G).(14)

Hence, the result follow from Eq. (14) and Theorem 4.

4 An upper bound on the Laplacian matrix energy

for the disjoint union of graphs

Here and throughout this section, Ldenotes the block matrix direct sum [20].

Let k∈N. Suppose that for each 1≤i≤k,Gi= (Vi, Ei)is an (ni, mi)-

graph with the vertex set Viand the edge set Ei. Let Vi’s are mutually disjoint. In

this case the disjoint union of Gi’s, denoted by Sk

i=1 Gi, is a non-connected graph

with the vertex set Sk

i=1 Viand the edge set Sk

i=1 Ei. It is easy to see that

A(

k

[

i=1

Gi) =

k

M

i=1

A(Gi).(15)

Moreover, if ωiis a vertex weight, assigned to Gi, then Sk

i=1 Giinherits naturally

a vertex degree weight from its components. This weight is nothing but ω:=

Sk

i=1 ωi, i.e., For each v∈Sk

i=1 Vi,ω(v) = ωi(v)if and only if v∈Vi. Note that

ωis a convex combination of ωi,i= 1,...,k, since

ω=1

Pk

j=1 nj k

X

i=1 X

v∈Vi

ωi(v)=

k

X

i=1 ni

Pk

j=1 njωi.(16)

Moreover

ω≥ωi, i = 1,...,k.

In the case of vertex degree weight, the next theorem was provedin [27] and based

on their proof, we get also the following result.

9

Theorem 6. Let k∈N. Suppose that for each 1≤i≤k,Giis a graph with ni

vertices and with a vertex weight ωi. Then

LEω(

k

[

i=1

Gi)≤

k

X

i=1

LEωi(Gi) +

k

X

i=1 ωi−ωni.(17)

Equality holds if and only if ωi=ωfor all i= 1,...,k.

Proof. In order to simplify the writing and omit some subscripts, for each 1≤

i≤k, we denote Iniand ωi−ωby Iiand bi, respectively. It is clear that

Lω(G)−ωIn=

k

M

i=1 Lωi(Gi)−ωIi=

k

M

i=1 Lωi(Gi)−ωiIi+

k

M

i=1

biIi(18)

Therefore, as a consequence of Eq. (3) and Theorem 1, the inequality in Eq. (17)

follows.

Now let us consider the the equality case in Eq. (17). Let ωi=ωfor all

i= 1,...,k. Therefore the matrix k

M

i=1

biIiis zero and consequently it follows

from Eq. (18) that the equality in Eq. (17) holds.

Conversely suppose on the contrary that there exists 1≤l≤ksuch that

ωl> ω. We may assume that l= 1. As a consequence of Theorem 1, Eq. (18)

and the equality in Eq. (17), there exists a unitary matrix Psuch that

X=P

k

M

i=1 Lωi(Gi)−ωiIiand Y=P

k

M

i=1

biIi,

are both positive semi-deﬁnite. Hence P∗Xand P∗Yare polar decompositions

of the matrices k

M

i=1 Lωi(Gi)−ωiIiand k

M

i=1

biIi,

respectively. By Theorem 2, we arrive at

Y=

k

M

i=1

|bi|Ii=P

k

M

i=1

biIi(19)

10

We can write the unitary matrix Pas

P=

P11 P12 ··· P1k

P21 P22 ··· P2k

.

.

.....

.

.

Pk1··· Pkk

,(20)

with the diagonal matrices Pj j,j= 1,...,k of order nj, respectively. From Eq.

(19) we have

|b1|I10··· 0

0|b2|I2··· 0

.

.

.....

.

.

0··· 0|bk|Ik

=

P11 P12 ··· P1k

P21 P22 ··· P2k

.

.

.....

.

.

Pk1··· Pkk

b1I10··· 0

0b2I2··· 0

.

.

.....

.

.

0··· 0bkIk

,

and then

|b1|I10··· 0

0|b2|I2··· 0

.

.

.....

.

.

0··· 0|bk|Ik

=

b1P11 P12 ··· P1k

b1P21 P22 ··· P2k

.

.

.....

.

.

b1Pk1··· Pkk

.(21)

As b1=ω1−ω > 0, via Eq. (21) we obtain P11 =I1and Pj1= 0,j=

2,...,k. Now it follows from X=PLk

i=1 Lωi(Gi)−ωiIithat Lω1(G1)−ω1I1

is positive semi-deﬁnite. Now we have the required contradiction, since by the

Rayleigh principle we ﬁnd that Lω1(G1)−ω1I1has a negative eigenvalue. Hence

the assertion follows.

References

[1] Aleksic, T. Upper bounds for Laplacian energy of graphs. MATCH Commun.

Math. Comput. Chem. 2008,60, 435–439.

[2] Bell, F.K. A Note on the Irregularity of Graphs. Linear Algebra Appl. 1992,

161,45–54.

[3] Cavers, M.S. The normalized laplacian matrix and general randic index of

graphs. Ph.D. Thesis, University of Regina, Regina, Saskatchewan, 2010.

11

[4] Das, C.K.; Mojallala, S.A.; Gutman, I. On energy and Laplacian energy of

bipartite graphs. Appl. Math. Comput. 2016,273, 759–766.

[5] Cvetkovic; Doob, M.; Sachs, H. Spectra of Graphs-Theory and Application.

third ed.; Johann Ambrosius Barth Verlag, Heidelberg: Leipzig, 1995.

[6] Day, J.; So, W. Singular value inequality and graph energy change. El. J.

Linear Algebra 2007,16, 291–297.

[7] Abreu, N.N.M. de ; Vinagre, C.M.; Bonifacio, A.S.; Gutman, I. The Lapla-

cian energy of some Laplacian integral graphs. MATCH Commun. Math.

Comput. Chem. 2008,60, 447–460.

[8] Fan, K. Maximum properties and inequalities for the eigenvalues of com-

pletely continuous operators. Proc. Nat. Acad. Sci.U.S.A. 1951,(37), 760–

766.

[9] Fan, K.; Hoffman, A.J. Some metric inequalities in the space of matrices.

Proc. Amer. Math. Soc. 1955 6, 111–116.

[10] Gohber, I.; Krein, M. Introduction to the Theory of Linear Nonselfadjoint

Operators. Amer. Math. Soc. Providence. 1969.

[11] Grone, R.; Merris, R. The Laplacian spectrum of a graph II. SIAM J. Discrete

Math. 1994,7, 221–229.

[12] Grone, R.; Merris, R.; Sunder, V.S. The Laplacian spectrum of a graph.

SIAM J. Matrix Anal. Appl. 1990,11, 218–238.

[13] Gutman, I. The energy of a Graph, Old and New Results. In Algebraic

Combinatorics and Applications; A., Betten.; A., Kohnert.; R., Laue.; A.,

Wassermann.;Eds.; Springer-Verlag: Berlin, 2001; pp. 196–211.

[14] Gutman, I. The energy of a graph. Ber. Math.-Statist. Sekt. Forschungsz.

Graz. 1978,103, 1–22.

[15] Gutman, I.; Abreu, N.M.M. de; Vinagre, C.T.M.; Bonifacio, A.S.;

Radenkovic, S. Relation between energy and Laplacian energy. MATCH

Commun. Math. Comput. Chem. 2008,59, 343–354.

[16] Gutman, I.; Polansky, O.E. Mathematical Concepts in Organic Chemistry.

Springer-Verlag: Berlin, 1986; Chapter 8.

12

[17] Gutman, I.; Zare Firoozabadi, S.; de la Pena, J.A.; Rada, J. On the energy of

regular graphs. MATCH Commun. Math. Comput. Chem. 2007,57, 435–442.

[18] Gutman, I.; Zhou, B. Laplacian energy of a graph. Linear Algebra Appl.

2006,414, 29–37.

[19] Gutman, I.; Paule, P. The variance of the vertex degrees of randomly gen-

erated graphs. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 2002,13,

30–35.

[20] Horn, R.; Johnson, C. Matrix Analysis. Cambridge Univ. Press: Cambridge,

1989.

[21] Indulal, G.; Vijayakumar, A. A note on energy of some graphs. MATCH

Commun. Math. Comput. Chem. 2008,59, 269–274.

[22] Merris, R. A survey of graph Laplacians. Linear Multilinear Algebra 1995,

39, 19–31.

[23] Merris, R. Laplacian matrices of graphs: a survey. Linear Algebra Appl.

1994,197–198, 143–176.

[24] Mohar, B., The Laplacian spectrum of graphs. In Graph Theory, Combi-

natorics, and Applications; Alavi, Y.; Chartrand, G.; Oellermann, O.R.;

Schwenk, A.J., Eds.; Wiley: New York, 1991; pp. 871–898.

[25] Mohar, B., Graph Laplacians. In Topics in Algebraic Graph Theory; Brualdi,

L.W.; Wilson, R.J., Eds.; Cambridge Univ. Press: Cambridge, 2004; pp.

113–136.

[26] Nikiforov, V. The energy of graphs and matrices. J. Math. Anal. Appl. 2007,

326, 1472–1475.

[27] Robbiano, M.; Jim´enez, R. Applications of a theorem by Ky Fan in the the-

ory of Laplacian energy of graphs. MATCH Commun. Math. Comput. Chem.

2009,62, 537–552.

[28] So, W.; Robbiano, M.; Abreu, N.M.M. de; Gutman, I. Applications of the

Ky Fan theorem in the theory of graph energy. Linear Algebra Appl. 2010,

432, 2163–2169.

[29] Li, X.; Shi, Y.; Gutman I. Graph Energy, Springer, New York, 2012.

13

[30] Sharafdini, R.; Panahbar, H. On the weighted version of Laplacian energy of

graphs. manuscript.

[31] Thompson, R.C. Convex and concave functions of singular values of matrix

sums. Paciﬁc J. Math. 1976,66, 285–290.

[32] Thompson, R.C. The case of equality in the matrix-valued triangle inequali-

tity. Paciﬁc J. Math. 1979,82, 279–280.

[33] Zhou, B.; Gutman, I.; Aleksic, T. A note on Laplacian energy of graphs.

MATCH Commun. Math. Comput. Chem. 2008,60, 441–446.

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