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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.
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