Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2013, Article ID 520610, 8 pages
http://dx.doi.org/10.1155/2013/520610
Research Article
Bounds for the Largest Laplacian Eigenvalue of
Weighted Graphs
Sezer Sorgun
Department of Mathematics, Faculty of Sciences and Arts, Nevşehir University, 50300 Nevşehir, Turkey
Correspondence should be addressed to Sezer Sorgun; srgnrzs@gmail.com
Received 29 November 2012; Accepted 14 March 2013
Academic Editor: Jun-Ming Xu
Copyright © 2013 Sezer Sorgun. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let G be weighted graphs, as the graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph
are the eigenvalues of Laplacian matrix of a graph G. We obtain two upper bounds for the largest Laplacian eigenvalue of weighted
graphs and we compare these bounds with previously known bounds.
1. Introduction
Let 𝐺 = (𝑉, 𝐸) be simple graphs, as graphs which have no
loops or parallel edges such that 𝑉 is a finite set of vertices
and 𝐸 is a set of edges.
A weighted graph is a graph each edge of which has been
assigned to a square matrix called the weight of the edge. All
the weight matrices are assumed to be of same order and to be
positive matrix. In this paper, by “weighted graph” we mean
“a weighted graph with each of its edges bearing a positive
definite matrix as weight,” unless otherwise stated.
The notations to be used in paper are given in the
following.
Let 𝐺 be a weighted graph on 𝑛 vertices. Denote by 𝑤𝑖,𝑗
the positive definite weight matrix of order 𝑝 of the edge 𝑖𝑗,
and assume that 𝑤𝑖𝑗 = 𝑤𝑗𝑖 . We write 𝑖 ∼ 𝑗 if vertices 𝑖 and 𝑗
are adjacent. Let 𝑤𝑖 = ∑𝑗:𝑗∼𝑖 𝑤𝑖𝑗 . be the weight matrix of the
vertex 𝑖.
The Laplacian matrix of a graph 𝐺 is defined as 𝐿(𝐺) =
(𝑙𝑖𝑗 ), where
if 𝑖 = 𝑗,
𝑤;
{
{ 𝑖
𝑙𝑖,𝑗 = {−𝑤𝑖𝑗 ; if 𝑖 ∼ 𝑗,
{
otherwise.
{0;
(1)
The zero denotes the 𝑝 × 𝑝 zero matrix. Hence 𝐿(𝐺) is
square matrix of order 𝑛𝑝. Let 𝜆 1 denote the largest eigenvalue
of 𝐿(𝐺). In this paper we also use to avoid the confusion that
𝜌1 (𝑤𝑖𝑗 ) is the spectral radius of 𝑤𝑖𝑗 matrix. If 𝑉 is the disjoint
union of two nonempty sets 𝑉1 and 𝑉2 such that every vertex
𝑖 in 𝑉1 has the same 𝜌1 (𝑤𝑖 ) and every vertex 𝑗 in 𝑉2 has the
same 𝜌1 (𝑤𝑗 ), then 𝐺 is called a weight-semiregular graph. If
𝜌1 (𝑤𝑖 ) = 𝜌1 (𝑤𝑗 ) in weight semiregular graph, then 𝐺 is called
a weighted-regular graph.
Upper and lower bounds for the largest Laplacian eigenvalue for unweighted graphs have been investigated to a great
extent in the literature. Also there are some studies about
the bounds for the largest Laplacian eigenvalue of weighted
graphs [1–3]. The main result of this paper, contained in
Section 2, gives two upper bounds on the largest Laplacian
for weighted graphs, where the edge weights are positive
definite matrices. These upper bounds are attained by the
same methods in [1–3]. We also compare the upper bounds
with the known upper bounds in [1–3]. We also characterize
graphs which achieve the upper bound. The results clearly
generalize some known results for weighted and unweighted
graphs.
2. The Known Upper Bounds for the Largest
Laplacian Eigenvalue of Weighted Graphs
In this section, we present the upper bounds for the largest
Laplacian eigenvalue of weighted graphs and very useful
lemmas to prove theorems.
2
International Journal of Combinatorics
Theorem 1 (Horn and Johnson [4]). Let 𝐴 ∈ 𝑀𝑛 be Hermitian, and let the eigenvalues of 𝐴 be ordered such that 𝜆 𝑛 ≤
𝜆 𝑛−1 ≤ ⋅ ⋅ ⋅ ≤ 𝜆 1 . Then,
𝜆 𝑛 𝑥𝑇 𝑥 ≤ 𝑥𝑇𝐴𝑥 ≤ 𝜆 1 𝑥𝑇 𝑥
𝑥𝑇𝐴𝑥
= max 𝑥𝑇 𝐴𝑥
𝑥 = ̸ 0 𝑥𝑇 𝑥
𝑥𝑇 𝑥=1
(2)
𝜆 min
(i) 𝐺 is a bipartite semiregular graph;
(ii) 𝑤𝑖𝑗 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗.
𝜆 max = 𝜆 1 = max
𝑥𝑇 𝐴𝑥
= 𝜆 𝑛 = min 𝑇 = min
𝑥𝑇 𝐴𝑥
𝑥 ≠ 0 𝑥 𝑥
𝑥𝑇 𝑥=1
where 𝑤𝑖𝑗 is the positive definite weight matrix of order p of the
edge 𝑖𝑗. Moreover equality holds in (7) if and only if
(3)
Corollary 6 (see [2]). Let 𝐺 be a simple connected weighted
graph where each edge weight 𝑤𝑖𝑗 is a positive number. Then
for all 𝑥 ∈ C𝑛 .
Lemma 2 (Horn and Johnson [4]). Let 𝐵 be a Hermitian 𝑛×𝑛
matrix with eigenvalues 𝜆 1 ≥ 𝜆 2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆 𝑛 ; then for any
𝑥 ∈ 𝑅𝑛 (𝑥 ≠ 0), 𝑦 ∈ 𝑅𝑛 (𝑦 ≠ 0),
󵄨󵄨 𝑇 󵄨󵄨
(4)
󵄨󵄨𝑥 𝐵𝑦󵄨󵄨 ≤ 𝜆 1 √𝑥𝑇 𝑥√𝑦𝑇 𝑦.
󵄨
󵄨
Equality holds if and only if 𝑥 is an eigenvector of 𝐵 corresponding to 𝜆 1 and 𝑦 = 𝛼𝑥 for some 𝛼 ∈ 𝑅.
Lemma 3 (see [1]). Let 𝐺 be a (𝜌1 (𝑤𝑖 ), 𝜌1 (𝑤𝑗 ))-semiregular
bipartite graph of order 𝑛 such that the first 𝑙 vertices of the
same largest eigenvalue 𝜌1 (𝑤𝑖 ) and the remaining 𝑚 vertices
of the same largest eigenvalue 𝜌1 (𝑤𝑗 ). Also let 𝑥 be a common
eigenvector of 𝑤𝑖𝑗 corresponding to the largest eigenvalue
𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗, where 𝑤𝑖 = ∑𝑘∈𝑁𝑖 𝑤𝑖𝑘 for all 𝑖. Then 𝜌1 (𝑤𝑖 ) +
𝜌1 (𝑤𝑗 ) is the largest eigenvalue of 𝐿(𝐺) and the corresponding
eigenvector is
𝑇
𝑇
(𝜌
−𝜌1 (𝑤𝑗 )𝑥𝑇 , . . . , −𝜌1 (𝑤𝑗 )𝑥𝑇 ) .
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
1 (𝑤𝑖 )𝑥 , . . . , 𝜌1 (𝑤𝑖 )𝑥 , ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑙
𝑚
(5)
Theorem 4 (see [1]). Let G be a simple connected weighted
graph. Then
}
{
(6)
𝜆 1 ≤ max {𝜌1 ( ∑ 𝑤𝑖𝑘 ) + ∑ 𝜌1 (𝑤𝑗𝑘 )} ,
𝑖∼𝑗
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
}
{
where 𝑤𝑖𝑗 is the positive definite weight matrix of order p of the
edge 𝑖𝑗. Moreover equality holds in (6) if and only if
(i) 𝐺 is a weight-semiregular bipartite graph,
(ii) 𝑤𝑖𝑗 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗.
Theorem 5 (see [2]). Let G be a simple connected weighted
graph. Then
𝜆1
√
√
{
{
√ ∑ 𝜌 (𝑤 ) ( ∑ 𝜌 (𝑤 ) + ∑ 𝜌 (𝑤 ))
{
{
{√ 𝑘:𝑘∼𝑖 1 𝑖𝑘 𝑟:𝑟∼𝑖 1 𝑖𝑟 𝑠:𝑠∼𝑘 1 𝑘𝑠
≤ max {
{
𝑖∼𝑗 {
{
{ + ∑ 𝜌1 (𝑤𝑗𝑘 ) ( ∑ 𝜌1 (𝑤𝑗𝑟 ) + ∑ 𝜌1 (𝑤𝑘𝑠 ))
𝑟:𝑟∼𝑗
𝑘:𝑘∼𝑗
𝑠:𝑠∼𝑘
{√
}
}
}
}
}
,
}
}
}
}
}
}
(7)
𝜆 1 ≤ max {√2𝑤𝑖 (𝑤𝑖 + 𝑤𝑖 )} ,
𝑖
(8)
where 𝑤𝑖 = (∑𝑘:𝑘∼𝑖 𝑤𝑖𝑘 𝑤𝑘 )/𝑤𝑖 and 𝑤𝑖 is the weight of vertex 𝑖.
Moreover equality holds if and only if 𝐺 is a bipartite regular
graph.
Corollary 7 (see [2]). Let 𝐺 be a simple connected weighted
graph where each edge weight 𝑤𝑖𝑗 is a positive number. Then
𝜆 1 ≤ max {√𝑤𝑖 (𝑤𝑖 + 𝑤𝑖 ) + 𝑤𝑗 (𝑤𝑗 + 𝑤𝑗 )} ,
𝑖∼𝑗
(9)
where 𝑤𝑖 = (∑𝑘:𝑘∼𝑖 𝑤𝑖𝑘 𝑤𝑘 )/𝑤𝑖 and 𝑤𝑖 is the weight of vertex
𝑖. Moreover equality holds if and only if 𝐺 is a bipartite
semiregular graph.
Theorem 8 (see [2]). Let G be a simple connected weighted
graph. Then
𝜆1
2
{
}
{
{ 𝜌1 (𝑤𝑖 ) + 𝜌1 (𝑤𝑗 ) + √ (𝜌1 (𝑤𝑖 ) − 𝜌1 (𝑤𝑗 )) + 4𝛾𝑖 𝛾𝑗 }
}
≤ max {
,
}
}
𝑖∼𝑗 {
2
{
}
{
}
(10)
where 𝛾𝑖 = (∑𝑘:𝑘∼𝑖 𝜌1 (𝑤𝑖𝑘 )𝜌1 (𝑤𝑘 ))/𝜌1 (𝑤𝑖 ) and 𝑤𝑖𝑗 is the positive definite weight matrix of order p of the edge 𝑖𝑗. Moreover
equality holds in (10) if and only if
(i) 𝐺 is a weighted-regular graph or 𝐺 is a weight-semiregular bipartite graph;
(ii) 𝑤𝑖𝑗 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗.
Corollary 9 (see [2]). Let 𝐺 be a simple connected weighted
graph where each edge weight 𝑤𝑖𝑗 is a positive number. Then
𝜆 1 ≤ max {𝑤𝑖 + 𝑤𝑖 } ,
𝑖
(11)
where 𝑤𝑖 = (∑𝑘:𝑘∼𝑖 𝑤𝑖𝑘 𝑤𝑘 )/𝑤𝑖 and 𝑤𝑖 is the weight of vertex
𝑖. Moreover equality holds if and only if 𝐺 is a bipartite
semiregular graph or 𝐺 is a bipartite regular graph.
International Journal of Combinatorics
3
Theorem 10 (see [3]). Let G be a simple connected weighted
graph. Then
𝜆1
{
{
{
≤ max {√
𝑖 {
{
{
𝜌12
(𝑤𝑖 ) +
∑ 𝜌12
𝑘:𝑘∼𝑖
(𝑤𝑖𝑘 ) + ∑ 𝜌1 (𝑤𝑖 𝑤𝑖𝑘 + 𝑤𝑖𝑘 𝑤𝑘 ) }
}
}
𝑘:𝑘∼𝑖
,
}
∑ 𝜌1 (𝑤𝑖𝑠 𝑤𝑠𝑡 )
+ ∑
}
}
1≤𝑖,𝑡≤𝑛 𝑠∈𝑁𝑖 ∩𝑁𝑡
}
(12)
where 𝑤𝑖𝑘 is the positive definite weight matrix of order 𝑝 of the
edge 𝑖𝑘 and 𝑁𝑖 ∩ 𝑁𝑘 is the set of common neighbours of 𝑖 and
𝑘. Moreover equality holds in (12) if and only if
(i) 𝐺 is a weight-semiregular bipartite graph;
(ii) 𝑤𝑖𝑘 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑘 ) for all 𝑖, 𝑘.
Corollary 11 (see [3]). Let 𝐺 be a simple connected weighted
graph where each edge weight 𝑤𝑖𝑗 is a positive number. Then
2
2
{
∑ 𝑤𝑖𝑘
+ ∑ (𝑤𝑖 𝑤𝑖𝑘 + 𝑤𝑘 𝑤𝑖𝑘 )
{
{ 𝑤𝑖 + 𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑖
𝜆 1 ≤ max{√
+ ∑
∑ 𝑤𝑖𝑠 𝑤𝑠𝑡
{
𝑖 {
1≤𝑖,𝑡≤𝑛 𝑠∈𝑁𝑖 ∩𝑁𝑡
{
}
}
}
. (13)
}
}
}
}
Moreover equality holds if and only if 𝐺 is a bipartite semiregular graph.
3. Two Upper Bounds on the Largest Laplacian
Eigenvalue of Weighted Graphs
In this section we present two upper bounds for the largest
eigenvalue of weighted graphs and compare the bounds with
some examples.
Theorem 12. Let G be a simple connected weighted graph.
Then
{
}
2
{
{
}
𝜌1 (𝑤𝑖 ) + 𝜌1 (𝑤𝑗 ) + √(𝜌1 (𝑤𝑖 ) − 𝜌1 (𝑤𝑗 )) + 4 ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ( ∑ 𝜌1 (𝑤𝑗𝑘 )) }
{
}
{
}
{
}
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
{
}
𝜆 1 ≤ max {
,
}
{
}
𝑖∼𝑗 {
2
}
{
}
{
}
{
}
{
}
{
}
where 𝑤𝑖𝑗 is the positive definite weight matrix of order p of the
edge 𝑖𝑗. Moreover equality holds in (14) if and only if
From the 𝑖th equation of (18), we have
(𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑖 ) 𝑥𝑖 = − ∑ 𝑤𝑖𝑘 𝑥𝑘 ,
(i) 𝐺 is a weighted-regular graph or 𝐺 is a weightsemiregular bipartite graph;
(ii) 𝑤𝑖𝑗 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗.
Proof. Let 𝑋 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )𝑇 be an eigenvector corresponding to the largest eigenvalue 𝜆 1 of 𝐿(𝐺). We assume that
𝑥𝑖 is the vector component of 𝑋 such that
𝑥𝑇𝑖 𝑥𝑖
=
max {𝑥𝑇𝑘 𝑥𝑘 } .
𝑘∈𝑉
𝑘:𝑘∼𝑖
(15)
if 𝑖 = 𝑗
if 𝑖 ∼ 𝑗
otherwise.
𝑥𝑇𝑖 (𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑖 ) 𝑥𝑖 = − ∑ 𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘
(20)
󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
(21)
𝑘:𝑘∼𝑖
≤ ∑ 𝜌1 (𝑤𝑖𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑘 𝑥𝑘
by (4)
(22)
(16)
≤ ∑ 𝜌1 (𝑤𝑖𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑗 𝑥𝑗
by (16) .
𝑘:𝑘∼𝑖
(23)
(17)
From (23) we have
(𝜆 1 − 𝜌1 (𝑤𝑖 )) 𝑥𝑇𝑖 𝑥𝑖 ≤ 𝑥𝑇𝑖 (𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑖 ) 𝑥𝑖
We have
𝐿 (𝐺) 𝑋 = 𝜆 1 𝑋.
that is,
𝑘:𝑘∼𝑖
be. The (𝑖, 𝑗)th element of 𝐿(𝐺) is
𝑤;
{
{ 𝑖
{−𝑤𝑖,𝑗 ;
{
{0;
(19)
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑖
Since 𝑋 is nonzero, so is 𝑥𝑖 . Let
𝑥𝑇𝑗 𝑥𝑗 = max {𝑥𝑇𝑘 𝑥𝑘 }
(14)
(18)
by (2)
≤ ∑ 𝜌1 (𝑤𝑖𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑗 𝑥𝑗 ,
𝑘:𝑘∼𝑖
(24)
4
International Journal of Combinatorics
that is,
that is,
(𝜆 1 − 𝜌1 (𝑤𝑖 )) 𝑥𝑇𝑖 𝑥𝑖 ≤ ∑ 𝜌1 (𝑤𝑖𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑗 𝑥𝑗 .
𝑘:𝑘∼𝑖
(25)
𝜆1 ≤
𝜌1 (𝑤𝑖 ) + 𝜌1 (𝑤𝑗 )
2
From the 𝑗th equation of (18), we get
2
√(𝜌1 (𝑤𝑖 ) − 𝜌1 (𝑤𝑗 )) + 4 ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ( ∑ 𝜌1 (𝑤𝑗𝑘 ))
(𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑗 ) 𝑥𝑗 = − ∑ 𝑤𝑗𝑘 𝑥𝑘 ,
𝑘:𝑘∼𝑗
that is,
𝑥𝑇𝑗 (𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑖 ) 𝑥𝑗 = − ∑ 𝑥𝑇𝑗 𝑤𝑗𝑘 𝑥𝑘
(27)
󵄨
󵄨
≤ ∑ 󵄨󵄨󵄨󵄨𝑥𝑇𝑗 𝑤𝑗𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
(28)
𝑘:𝑘∼𝑗
𝑘:𝑘∼𝑗
≤ ∑ 𝜌1 (𝑤𝑗𝑘 ) √𝑥𝑇𝑗 𝑥𝑗 √𝑥𝑇𝑘 𝑥𝑘
}
2
}
√(𝜌1 (𝑤𝑖 )−𝜌1 (𝑤𝑗 )) +4 ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ( ∑ 𝜌1 (𝑤𝑗𝑘 )) }
}
}
}
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
}
.
+
}
}
2
}
}
}
}
}
}
by (4)
(29)
≤ ∑
𝑘:𝑘∼𝑗
by (15) .
(30)
Similarly, from (30) we get
(𝜆 1 − 𝜌1 (𝑤𝑗 )) 𝑥𝑇𝑗 𝑥𝑗 ≤ 𝑥𝑇𝑗 (𝜆 1 𝐼𝑝,𝑝 − 𝑤𝑗 ) 𝑥𝑗
(36)
This completes the proof of (14).
Now suppose that equality holds in (14). Then all inequalities in the previous argument must be equalities.
From equality in (23), we get
𝑥𝑇𝑖 𝑥𝑖 = 𝑥𝑇𝑘 𝑥𝑘
by (2)
≤ ∑ 𝜌1 (𝑤𝑗𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑗 𝑥𝑗 ,
(31)
𝑘:𝑘∼𝑗
that is,
(𝜆 1 − 𝜌1 (𝑤𝑗)) 𝑥𝑇𝑗 𝑥𝑗 ≤ ∑ 𝜌1 (𝑤𝑗𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑘 𝑥𝑘 .
𝑘:𝑘∼𝑗
(37)
𝑎2 𝑥𝑇𝑖 𝑥𝑖 = 𝑥𝑇𝑖 𝑥𝑖 ,
(38)
𝑎2 = 1 as 𝑥𝑇𝑖 𝑥𝑖 > 0.
(39)
(32)
that is,
𝜌1 (𝑤𝑖 )) 𝑥𝑇𝑗 𝑥𝑗 𝑥𝑇𝑖 𝑥𝑖
≤ ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ⋅ ( ∑ 𝜌1 (𝑤𝑗𝑘 )) 𝑥𝑇𝑗 𝑥𝑗 𝑥𝑇𝑖 𝑥𝑖 .
𝑘:𝑘∼𝑖
for all 𝑘, 𝑘 ∼ 𝑖.
Since 𝑥𝑖 ≠ 0, we get that 𝑥𝑘 ≠ 0 for all 𝑘, 𝑘 ∼ 𝑖. From equality in (22) and Lemma 2, we get that 𝑥𝑖 is an eigenvector of 𝑤𝑖𝑘
for the largest eigenvalue 𝜌1 (𝑤𝑖𝑘 ). Hence we say that 𝑥𝑘 = 𝑎𝑥𝑖
for some 𝑎, for any 𝑘, 𝑘 ∼ 𝑖.
On the other hand, from (37) we get
So, from (25) and (32) we have
(𝜆 1 − 𝜌1 (𝑤𝑗 )) (𝜆 1 −
𝑘:𝑘∼𝑗
2
{
{
{
{
{
{
{𝜌1 (𝑤𝑖 )+𝜌1 (𝑤𝑗 )
≤ max {
𝑖∼𝑗 {
2
{
{
{
{
{
{
𝑘:𝑘∼𝑗
𝜌1 (𝑤𝑗𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑗 𝑥𝑗
𝑘:𝑘∼𝑖
+
(26)
From equality in (21), we have
(33)
󵄨
󵄨
− ∑ 𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 = ∑ 󵄨󵄨󵄨󵄨𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨 .
𝑘:𝑘∼𝑗
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑖
(40)
Since 𝑥𝑘 = 𝑎𝑥𝑖 , from (40) we get
Hence we get
󵄨
󵄨
− ∑ 𝑎𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑖 = ∑ |𝑎| 󵄨󵄨󵄨󵄨𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
(𝜆 1 − 𝜌1 (𝑤𝑗 )) (𝜆 1 − 𝜌1 (𝑤𝑖 ))
≤ ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ⋅ ( ∑ 𝜌1 (𝑤𝑗𝑘 )) ,
𝑘:𝑘∼𝑖
(34)
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑖
= ∑ |𝑎| 𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘
𝑘:𝑘∼𝑗
𝑘:𝑘∼𝑖
as 𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 > 0.
(41)
Hence we get
that is,
𝑎 = −1
𝜆21 − 𝜆 1 (𝜌1 (𝑤𝑗 ) + 𝜌1 (𝑤𝑖 )) + 𝜌1 (𝑤𝑖 ) 𝜌1 (𝑤𝑗 )
− ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ⋅ ( ∑ 𝜌1 (𝑤𝑗𝑘 )) ≤ 0,
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
(35)
(42)
from equalities in (41). Therefore we have
𝑥𝑘 = −𝑥𝑖
for all 𝑘, 𝑘 ∼ 𝑖.
(43)
International Journal of Combinatorics
5
Similarly from equality in (29), we get that 𝑥𝑗 is an
eigenvector of 𝑤𝑗𝑘 for the largest eigenvalue 𝜌1 (𝑤𝑗𝑘 ). Hence
we say that 𝑥𝑘 = 𝑏𝑥𝑗 for some 𝑏, for any 𝑘, 𝑘 ∼ 𝑗. From
equality in (16) we have
𝑥𝑇𝑗 𝑥𝑗 = 𝑥𝑇𝑘 𝑥𝑘
for 𝑘 ∼ 𝑖,
Conversely, suppose that conditions (i)-(ii) of the theorem hold for the graph 𝐺. Let 𝐺 be (𝜌1 (𝑤𝑖 ), 𝜌1 (𝑤𝑗 ))semiregular bipartite graph. Let 𝑥 be a common eigenvector
of 𝑤𝑖𝑘 corresponding to the largest eigenvalue 𝜌1 (𝑤𝑖𝑘 ) for all
𝑖, 𝑘. Then we have
(44)
𝜌1 (𝑤𝑖 ) = ∑ 𝜌1 (𝑤𝑖𝑘 ) ,
that is,
𝑘:𝑘∼𝑖
𝑏2 𝑥𝑇𝑗 𝑥𝑗
=
𝑥𝑇𝑗 𝑥𝑗 ,
(45)
(55)
𝜌1 (𝑤𝑗 ) = ∑ 𝜌1 (𝑤𝑗𝑘 ) .
𝑘:𝑘∼𝑗
that is,
𝑏2 = 1
as 𝑥𝑇𝑗 𝑥𝑗 > 0.
(46)
By Lemma 3, we get
𝜆 1 = 𝜌1 (𝑤𝑖 ) + 𝜌1 (𝑤𝑗 ) ,
Applying the same methods as previously, we get
𝑏 = −1.
(47)
Therefore we have
𝑥𝑘 = −𝑥𝑗
for all 𝑘, 𝑘 ∼ 𝑗.
(48)
(56)
that is,
𝜆1 =
𝜌1 (𝑤𝑖 ) + 𝜌1 (𝑤𝑗 )
2
For 𝑖 ∼ 𝑗
2
√(𝜌1 (𝑤𝑖 ) − 𝜌1 (𝑤𝑗 )) + 4 ( ∑ 𝜌1 (𝑤𝑖𝑘 )) ( ∑ 𝜌1 (𝑤𝑗𝑘 ))
𝑥𝑖 = −𝑥𝑗 .
Hence we take that 𝑈 = {𝑘 : 𝑥𝑘 = 𝑥𝑖 } and 𝑊 = {𝑘 : 𝑥𝑘 =
−𝑥𝑖 } from (43), (48), and (49). So, 𝑁𝑗 ⊂ 𝑈 and 𝑁𝑖 ⊂ 𝑊. Also,
𝑈 ≠ 𝑊 ≠ 0 since 𝑥𝑖 ≠ 0. Further, for any vertex 𝑠 ∈ 𝑁𝑁𝑖 there
exists a vertex 𝑟 ∈ 𝑁𝑖 such that 𝑟 ∼ 𝑗ℓ𝑟 ∼ 𝑠, where 𝑁𝑁𝑖 is
the neighbor of neighbor set of vertex 𝑖. Therefore 𝑥𝑟 = −𝑥𝑖
and 𝑥𝑠 = 𝑥𝑖 . So 𝑁𝑁𝑖 ⊂ 𝑈. By similar argument we can present
that 𝑁𝑁𝑗 ⊂ 𝑊. Continuing the procedure, it is easy to see,
since 𝐺 is connected, that 𝑉 = 𝑈 ∪ 𝑊 and that the subgraphs
induced by 𝑈 and 𝑊, respectively, are empty graphs. Hence
𝐺 is bipartite. Moreover, 𝑥𝑖 is a common eigenvector of 𝑤𝑖𝑘
and 𝑤𝑖 for the largest eigenvalue 𝜌1 (𝑤𝑖𝑘 ) and 𝜌1 (𝑤𝑖 ).
For 𝑖, 𝑘 ∈ 𝑈
𝜆 1 𝑥𝑖 = 𝑤𝑖 𝑥𝑖 + ∑ 𝑤𝑖𝑘 𝑥𝑖 = 𝑤𝑘 𝑥𝑖 + ∑ 𝑤𝑖𝑘 𝑥𝑖 ,
(50)
𝑤𝑖 𝑥𝑖 = 𝑤𝑘 𝑥𝑖 .
(51)
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑖
that is,
Since 𝑥𝑖 is an eigenvector of 𝑤𝑖 corresponding to the
largest eigenvalue of 𝜌1 (𝑤𝑖 ) for all 𝑖, we get
𝜌1 (𝑤𝑖 ) 𝑥𝑖 = 𝜌1 (𝑤𝑘 ) 𝑥𝑖 ,
(52)
that is,
(𝜌1 (𝑤𝑖 ) − 𝜌1 (𝑤𝑘 )) 𝑥𝑖 = 0,
(53)
that is,
𝜌1 (𝑤𝑖 ) = 𝜌1 (𝑤𝑘 )
as 𝑥𝑖 ≠ 0.
(54)
Therefore we get that 𝜌1 (𝑤𝑖 ) is constant for all 𝑖 ∈ 𝑈.
Similarly we can show that 𝜌1 (𝑤𝑗 ) is constant for all 𝑗 ∈ 𝑊.
Hence 𝐺 is a bipartite semiregular graph.
𝑘:𝑘∼𝑖
+
(49)
𝑘:𝑘∼𝑗
2
.
(57)
Corollary 13 (see [1]). Let 𝐺 be a simple connected weighted
graph where each edge weight 𝑤𝑖,𝑗 is a positive number. Then
𝜆 1 ≤ max {𝑤𝑖 + 𝑤𝑗 } .
(58)
𝑖∼𝑗
Moreover equality holds in (58) if and only if 𝐺 is bipartite
semiregular graph.
Proof. We have 𝜌1 (𝑤𝑖 ) = 𝑤𝑖 and 𝜌1 (𝑤𝑖𝑗 ) = 𝑤𝑖𝑗 for all 𝑖, 𝑗. From
Theorem 12, we get the required result.
Corollary 14 (see [5]). Let 𝐺 be a simple connected unweighted graph. Then
𝜆 1 ≤ max {𝑑𝑖 + 𝑑𝑗 } ,
(59)
𝑖∼𝑗
where 𝑑𝑖 is the degree of vertex 𝑖. Moreover equality holds in
(59) if and only if 𝐺 is a bipartite regular graph or 𝐺 is a
bipartite semiregular graph.
Proof. For unweighted graph, 𝑤𝑖,𝑗 = 1 for 𝑖 ∼ 𝑗. Therefore
𝑤𝑖 = 𝑑𝑖 . Using Corollary 6, we get the required results.
Theorem 15. Let G be a simple connected weighted graph.
Then
𝜆1
≤ max {√(𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 )) (𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 ))} ,
𝑖∼𝑗
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
(60)
6
International Journal of Combinatorics
where 𝑤𝑖𝑗 is the positive definite weight matrix of order p of the
edge 𝑖𝑗. Moreover equality holds in (60) if and only if
≤ 𝜌1 (𝑤𝑗 ) 𝑥𝑇𝑗 𝑥𝑗 + ∑ 𝜌1 (𝑤𝑗𝑘 ) √𝑥𝑇𝑗 𝑥𝑗 √𝑥𝑇𝑘 𝑥𝑘
(i) 𝐺 is a weighted-regular bipartite graph;
≤ 𝜌1 (𝑤𝑗 ) 𝑥𝑇𝑗 𝑥𝑗 + ∑ 𝜌1 (𝑤𝑗𝑘 ) 𝑥𝑇𝑗 𝑥𝑗
(ii) 𝑤𝑖𝑗 have a common eigenvector corresponding to the
largest eigenvalue 𝜌1 (𝑤𝑖𝑗 ) for all 𝑖, 𝑗.
Proof. Let 𝑋 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 )𝑇 be an eigenvector corresponding to the largest eigenvalue 𝜆 1 of 𝐿(𝐺). We assume that
𝑥𝑖 is the vector component of 𝑋 such that
𝑥𝑇𝑖 𝑥𝑖 = max {𝑥𝑇𝑘 𝑥𝑘 } .
Since 𝑋 is nonzero, so is 𝑥𝑖 . Let
by (41) .
𝑘:𝑘∼𝑗
(68)
Hence we get
𝜆 1 ≤ 𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 ) .
(61)
𝑘∈𝑉
by (2)
𝑘:𝑘∼𝑗
𝑘:𝑘∼𝑗
(69)
From (49) and (58), we have
𝑥𝑇𝑗 𝑥𝑗 = max {𝑥𝑇𝑘 𝑥𝑘 }
(62)
𝑘:𝑘∼𝑗
𝜆21 ≤ (𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 )) (𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 )) ,
be. We have
𝑘:𝑘∼𝑖
𝐿 (𝐺) 𝑋 = 𝜆 1 𝑋.
𝑘:𝑘∼𝑗
(70)
(63)
From the 𝑖th equation of (43), we have
that is,
𝜆 1 𝑥𝑖 = 𝑤𝑖 𝑥𝑖 − ∑ 𝑤𝑖𝑘 𝑥𝑘 ,
(64)
𝑘:𝑘∼𝑖
𝜆1
≤ max {√(𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 )) (𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 ))} .
that is,
𝑖∼𝑗
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
𝜆 1 𝑥𝑇𝑖 𝑥𝑖 = 󵄨󵄨󵄨𝑥𝑇𝑖 𝑤𝑖 𝑥𝑖 − ∑ 𝑥𝑇𝑖 𝑤𝑖𝑘 𝑥𝑘 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑘:𝑘∼𝑖
󵄨
󵄨
󵄨󵄨 𝑇
󵄨󵄨
󵄨󵄨 𝑇
󵄨
≤ 󵄨󵄨󵄨𝑥𝑖 𝑤𝑖 𝑥𝑖 󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨𝑥𝑖 𝑤𝑖𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
𝑘:𝑘∼𝑗
(71)
𝑘:𝑘∼𝑖
≤ 𝜌1 (𝑤𝑖 ) 𝑥𝑇𝑖 𝑥𝑖 + ∑ 𝜌1 (𝑤𝑖𝑘 ) √𝑥𝑇𝑖 𝑥𝑖 √𝑥𝑇𝑘 𝑥𝑘
by (2)
𝑘:𝑘∼𝑖
≤ 𝜌1 (𝑤𝑖 ) 𝑥𝑇𝑖 𝑥𝑖 + ∑ 𝜌1 (𝑤𝑖𝑘 ) 𝑥𝑇𝑖 𝑥𝑖
𝑘:𝑘∼𝑖
by (40) .
𝑘:𝑘∼𝑖
(65)
This completes the proof of (60).
Now we show the case of equality in (60). By similar
method in Theorem 12. In the part of equalit, the necessary
condition can show easily. So we will show the sufficient
condition.
Suppose that conditions (i)-(ii) of Theorem hold for the
graph 𝐺. We must prove that
𝜆1
= max {√(𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 )) (𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 ))} .
Hence we get
𝑖∼𝑗
𝜆 1 ≤ 𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 ) .
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
that is,
󵄨󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨
𝜆 1 𝑥𝑇𝑗 𝑥𝑗 = 󵄨󵄨󵄨󵄨𝑥𝑇𝑗 𝑤𝑗 𝑥𝑗 − ∑ 𝑥𝑇𝑗 𝑤𝑗𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
𝑘:𝑘∼𝑗
󵄨
󵄨
󵄨
󵄨
󵄨
󵄨
≤ 󵄨󵄨󵄨󵄨𝑥𝑇𝑗 𝑤𝑗 𝑥𝑗 󵄨󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨𝑥𝑇𝑗 𝑤𝑗𝑘 𝑥𝑘 󵄨󵄨󵄨󵄨
𝑘:𝑘∼𝑗
𝑘:𝑘∼𝑗
(72)
(66)
By the same method, from the 𝑗th equation of (43), we have
𝜆 1 𝑥𝑗 = 𝑤𝑗 𝑥𝑗 − ∑ 𝑤𝑗𝑘 𝑥𝑘 ,
𝑘:𝑘∼𝑖
(67)
Let 𝐺 be regular bipartite graph. Therefore we have
𝜌1 (𝑤𝑖 ) = 𝛼 for 𝑖 ∈ 𝑈 and 𝜌1 (𝑤𝑗 ) = 𝛼 for 𝑗 ∈ 𝑊 such
that 𝑉 = 𝑈 ∪ 𝑊. Let 𝑥 be a common eigenvector of 𝑤𝑖𝑘
corresponding to the largest eigenvalue 𝜌1 (𝑤𝑖𝑘 ) for all 𝑖, 𝑘.
Hence we have
𝜌1 (𝑤𝑖 ) = ∑ 𝜌1 (𝑤𝑖𝑘 ) .
𝑘:𝑘∼𝑖
(73)
From (71) we get that
𝜆 1 ≤ 2𝛼.
(74)
International Journal of Combinatorics
7
On the other hand, the following equation can be easily
verified:
𝑥
𝑥
.
( .. )
( )
(𝑥)
)
(2𝛼) (
(−𝑥)
( )
(−𝑥)
..
.
So from (74) and (76) we obtain
𝜆1
}
{
= max {√(𝜌1 (𝑤𝑖 ) + ∑ 𝜌1 (𝑤𝑖𝑘 )) (𝜌1 (𝑤𝑗 ) + ∑ 𝜌1 (𝑤𝑗𝑘 ))} .
𝑖∼𝑗
𝑘:𝑘∼𝑖
𝑘:𝑘∼𝑗
}
{
(77)
Corollary 16. Let 𝐺 be a simple connected weighted graph
where each edge weight 𝑤𝑖,𝑗 is a positive number. Then
(−𝑥)
𝑤1
0
...
(
( 0
=(
(−𝑤𝑘+1,1
−𝑤𝑘+1,2
...
( −𝑤𝑛,1
𝜆 1 ≤ max {2√𝑤𝑖 𝑤𝑗 } .
𝑖∼𝑗
⋅
0
−𝑤1,𝑘+1
⋅
0
−𝑤2,𝑘+1
⋅
...
...
−𝑤𝑘,𝑘+1
⋅
𝑤𝑘
⋅ −𝑤𝑘+1,𝑘 𝑤𝑘+1
⋅ −𝑤𝑘+2,𝑘
0
⋅
...
...
⋅ −𝑤𝑛,𝑘
0
⋅ −𝑤1,𝑛
⋅ −𝑤2,𝑛
⋅ ...
)
⋅ −𝑤𝑘,𝑛 )
⋅
0 )
)
⋅
0
⋅ ...
⋅ 𝑤𝑛 )
𝑥
𝑥
.
( .. )
( )
(𝑥)
)
×(
(−𝑥) .
( )
(−𝑥)
..
.
(−𝑥)
(78)
Moreover equality holds in (78) if and only if 𝐺 is bipartite
semiregular graph.
Proof. We have 𝜌1 (𝑤𝑖 ) = 𝑤𝑖 and 𝜌1 (𝑤𝑖𝑗 ) = 𝑤𝑖𝑗 for all 𝑖, 𝑗. From
Theorem 15 we get the required result.
Corollary 17. Let 𝐺 be a simple connected unweighted graph.
Then
𝜆 1 ≤ max {2√𝑑𝑖 𝑑𝑗 } ,
𝑖∼𝑗
(79)
where 𝑑𝑖 is the degree of vertex 𝑖. Moreover equality holds in
(79) if and only if 𝐺 is a bipartite regular graph or 𝐺 is a
bipartite semiregular graph.
Proof. For unweighted graph, 𝑤𝑖,𝑗 = 1 for 𝑖 ∼ 𝑗. Therefore
𝑤𝑖 = 𝑑𝑖 . Using Corollary 16, we get the required results.
(75)
(76)
Example 18. Let 𝐺1 = (𝑉1 , 𝐸1 ) and 𝐺2 = (𝑉2 , 𝐸2 ) be a
weighted graph where 𝑉1 = {1, 2, 3, 4}, 𝐸1 = {{1, 3}, {2, 4},
{3, 4}} and each weight is the positive definite matrix of order
three. Let 𝑉2 = {1, 2, 3, 4, 5, 6, 7}, 𝐸2 = { {1,4},{2,4},{3,4},
{4,5},{5,6},{5,7} } such
that each weight is the positive definite matrix of order two.
Assume that the following Laplacian matrices of 𝐺1 and 𝐺2
are as follows:
Thus 2𝛼 is an eigenvalue of 𝐿(𝐺). Since 𝜆 1 is the largest
eigenvalue of 𝐿(𝐺), we get
2𝛼 ≤ 𝜆 1 .
[
[
[
[
[
[
[
[
[
[
[
[
[
[
𝐿 (𝐺1 ) = [
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1
0
0
0
0
0 −1 0
0
0
0
5
3
0
0
0
0 −5 −3 0
0
0
3
3
0
0
0
0 −3 −3 0
0
0
0
0
2 −1 0
0
0
0 −2 1
0
0
0 −1 2 −1 0
0
0
1 −2
0
0
0
0 −1 2
0
0
0
0
1
−1 0
0
0
0
0
7
2 −2 6
2
0 −5 −3 0
0
0
2 11 1
0 −3 −3 0
0
0 −2 1 13 −2 −2
0
0
0
2 −1 0
6
0
0
0 −1 2 −1 2
[ 0
0
0
0
2
2 −2 8
6
1
6 −2 −3 8
0 −1 2 −2 −2 10 −2 −2
0
]
0 ]
]
]
0 ]
]
0 ]
]
]
1 ]
]
]
−2 ]
]
],
−2 ]
]
]
]
−2 ]
]
]
10 ]
]
−2 ]
]
]
−3 ]
]
12 ]
8
International Journal of Combinatorics
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
𝐿 (𝐺2 ) = [
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
1
1
0
0
0
0 −1 −1 0
0
0
0
0
1
2
0
0
0
0 −1 −2 0
0
0
0
0
0
0
1
1
0
0 −1 −1 0
0
0
0
0
0
0
1
3
0
0 −1 −3 0
0
0
0
0
0
0
0
0
1
1 −1 −1 0
0
0
0
0
0
0
0
0
1
4 −1 −4 0
0
0
0
0
4 −1 −1 0
0
0
−1 −2 −1 −3 −1 −4 4 14 −1 −5 0
0
0
0
0
0
0
0
0
0 −1 −1 3
0
0
0
0
0
0 −1 −5 3 18 −1 −6 −1
0
0
0
0
0
0
0
0 −1 −1 1
1
0
0
0
0
0
0
0
0
0 −1 −6 1
6
0
0
0
0
0
0
0
0
0 −1 −1 0
0
1
]
0 ]
]
0 ]
]
]
0 ]
]
]
0 ]
]
]
0 ]
]
]
0 ]
]
].
0 ]
]
]
−1 ]
]
]
7 ]
]
]
0 ]
]
]
0 ]
]
1 ]
]
[ 0
0
0
0
0
0
0
0 −1 −7 0
0
1
7 ]
−1 −1 −1 −1 −1 −1 4
3 −1 −1 −1
(80)
The largest eigenvalues of 𝐿(𝐺1 ) and 𝐿(𝐺2 ) are 𝜆 1 = 25, 66,
𝜆 2 = 26.16 rounded two decimal places and the previously
mentioned bounds give the following results:
(6)
(7) (10) (12) (14) (60)
𝐺1 32.90 32.88 27.90 29.55 30.88 30.93 .
𝐺2 34.12 29.86 27.11 27.22 34.05 33.90
(81)
For 𝐺1 , we see that the upper bounds in (14) and (60)
are better than upper bounds in (6) and (7). But they are not
better than upper bounds in (10) and (12) from (81).
For also 𝐺2 , we see that upper bounds in (14) and (60) are
only better than the upper bound in (6).
Consequently, we cannot exactly compare all the bounds
for weighted graphs, where the weights are positive definite
matrices. Modifications according to each weight of edges,
especially for matrices can be shown.
References
[1] K. Ch. Das and R. B. Bapat, “A sharp upper bound on the largest
Laplacian eigenvalue of weighted graphs,” Linear Algebra and Its
Applications, vol. 409, pp. 153–165, 2005.
[2] K. Ch. Das, “Extremal graph characterization from the upper
bound of the Laplacian spectral radius of weighted graphs,”
Linear Algebra and Its Applications, vol. 427, no. 1, pp. 55–69,
2007.
[3] S. Sorgun and Ş. Büyükköse, “On the bounds for the largest Laplacian eigenvalues of weighted graphs,” Discrete Optimization,
vol. 9, no. 2, pp. 122–129, 2012.
[4] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge
University Press, Cambridge, UK, 1985.
[5] W. N. Anderson Jr. and T. D. Morley, “Eigenvalues of the Laplacian of a graph,” Linear and Multilinear Algebra, vol. 18, no. 2,
pp. 141–145, 1985.
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