A Note on the Spectral Radius of Weighted Signless Laplacian Matrix ()
1. Introduction
A weighted graph is a graph that has a numeric label associated with each edge, called the weight of edge. In many applications, the edge weights are usually represented by nonnegative integers or square matrices. In this paper, we generally deal with simple connected weighted graphs where the edge weights are positive definite square matrices. Let
be a simple connected weighted graph with vertex set
. Let
be the positive definite weight matrix of order t of the edge ij and assume that
. The weight of a vertex
defined as
; where
denotes the vertex j is adjacent to i.
Unless otherwise specified, by a weighted graph we mean a graph with each edge weight is a positive definite square matrix.
The weighted signless Laplacian matrix
of weighted graph G is a block matrix and defined as
, where
Clearly,
is a square matrix of order nt. The eigenvalues may be denoted by
, where
. Also let
,
and
denote the spectral radius of G and the largest eigenvalues of
and
, respectively. If
is the weighted adjacency matrix of G, then note that
,
where
.
In literature, there are a lot of studies deal with upper and lower bounds for the spectral radius of signless Laplacian matrix of unweighted graphs. For a simple connected and unweighted graph G, there are some known upper bounds on the spectral radius of signless Laplacian matrix as follows such that
is
degree of vertex i and
.
, (1)
, (2)
, (3)
, (4)
. (5)
In this paper, some upper bounds for the spectral radius of signless Laplacian matrix of weighted graphs are given. Also some results on weighted and unweighted graphs are obtained by using these bounds. The following lemmas are convenient for the graphs we consider.
Lemma 1. [1] .
If A is a real symmetric
matrix with eigenvalues
, then for any
,
.
The equality holds if and only if
is an eigenvector of A corresponding to the least eigenvalue
.
Lemma 2. [2] .
If A is a real symmetric
matrix with eigenvalues
then for any
,
,
.
The equality holds if and only if
is an eigenvector of A corresponding to the largest eigenvalue
and
for some
.
Lemma 3. [3] .
Let G be a weighted graph and let
be the positive definite weight matrix of order t of the edge ij. Also let
be an eigenvector of
corresponding to the largest eigenvalue
for all i, j. Then
.
Lemma 4. [4] .
Let G be a weighted graph and let
be the positive definite weight matrix of order t of the edge ij. If
and
are not Hermitian matrices for all j, j∼i and for all k, k∼j, j∼i, then for any
,
,
2. Main Results
In this section, some upper bounds for the spectral radius of weighted signless Laplacian matrix are found.
Theorem 5.
Let
be a simple connected weighted graph. Then
. (6)
Proof.
Let
be an eigenvector corresponding to the eigenvalue
and
be the vector component of
such that
. (7)
Since
is nonzero, so is
. We have
. (8)
From the i-th Equation of (8), we get
,
i.e.,
.
From (7) and using Lemma 2, we get
.
From Lemma 1 and Lemma 3, we have
.
Thus
.
Hence the theorem follows.
Corollary 6.
Let G be a simple connected weighted graph where each edge weight
is a positive number. Then
.
Proof.
For weighted graphs where the edge weights
are positive number, we have
and
, for all
. Using Theorem 5 we get the required result.
Corollary 7. [5] .
Let G be a simple connected unweighted graph. Then
,
where
is the degree of vertex i.
Proof.
For an unweighted graph,
and
for all
and
. Using Corollary 6 we get the required result.
Theorem 8.
Let G be a simple connected weighted graph. Then
. (9)
Proof.
Let us consider the matrix
. The
-th element of
is
Let
be an eigenvector corresponding to the eigenvalue
of
and
be the vector component of
such that
. (10)
Since
is nonzero, so is
. We have
. (11)
From the i-th Equation of (11), we get
,
i.e.,
.
From (10) and using Lemma 1 and Lemma 2, we get
(12)
Thus
.
Hence the theorem follows.
Corollary 9.
Let G be a simple connected weighted graph where each edge weight
is a positive number. Then
.
Proof.
For weighted graphs where the edge weights
are positive number, we have
and
, for all
. Using Theorem 8 we get the required result.
Corollary 10. [6] .
Let G be a simple connected unweighted graph. Then
,
where
is the degree of vertex i.
Proof.
For an unweighted graph,
and
for all
and
. Using Corollary 9 we get the required result.
Theorem 11.
Let G be a simple connected weighted graph. Then,
, (13)
where
Proof.
From (12), we get
The proof is complete.
Corollary 12.
Let G be a simple connected weighted graph where each edge weight
is a positive number. Then
,
where
.
Proof.
For weighted graphs where the edge weights
are positive number, we have
and
, for all
. Using Theorem 11 we get the required result.
Corollary 13. [6] .
Let G be a simple connected unweighted graph. Then
,
where
is the degree of vertex i and
is the average of the degrees of the vertices adjacent to vertex i.
Proof.
For an unweighted graph,
and
for all
and
. Using Corollary 12 we get the required result.
Theorem 14.
Let G be a simple connected weighted graph. Then
, (14)
where 
Proof.
Let
be an eigenvector corresponding to the eigenvalue
. We assume that
be the vector component of
such that
. (15)
Since
is nonzero, so is
. We have
![]()
In order to prove the Inequality (14), we consider a simple quadratic function of
:
(16)
From the i-th Equation of (16), we get
![]()
i.e.,
![]()
Since
is the positive definite matrix of edge ij,
matrix is also positive definite. From Lemma 1, we have
(17)
Four cases arise;
1)
and
are real symmetric matrices for all j, j∼i and for all k, k∼j, j∼i.
2)
is a real symmetric matrix for all j, j∼i and
is not a real symmetric matrix for all k, k∼j, j∼i.
3)
is not a real symmetric matrix for all j, j∼i and
is a real symmetric matrix for all k, k∼j, j∼i.
4)
and
are not real symmetric matrices for all j, j∼i and for all k, k∼j, j∼i.
From (15), (17) and using Lemma 2 and Lemma 4, we get
(18)
for Case (1), Case (2), Case (3) and Case (4). Hence,
![]()
From Lemma 3, we have
![]()
i.e.,
![]()
Thus
![]()
From the inequality above, for every different value to b, we can get several distinct upper bounds. In particular, if
, we get
.
This completes the proof.
Corollary 15.
Let G be a simple connected weighted graph where each edge weight
is a positive number. Then
,
where
.
Proof.
For weighted graphs where the edge weights
are positive number, we have
and
, for all
. Using Theorem 14 we get the required result.
Corollary 16. [5] .
Let G be a simple connected unweighted graph. Then
,
where
is the degree of vertex i and
is the average of the degrees of the vertices adjacent to vertex i.
Proof.
For an unweighted graph,
and
for all
and
. Using Corollary 15 we get the required result.
Theorem 17.
Let G be a simple connected weighted graph. Then,
, (19)
where ![]()
Proof.
. The
-th element of
is
![]()
Let
be an eigenvector corresponding to the eigenvalue
of
,
and
are the vector components of
such that
, (20)
. (21)
Since
is nonzero, so is
. We have
. (22)
From the i-th Equation of (22), we get
.
From Lemma 2, we get
. (23)
From (21), (23) and using Lemma 1, we get
(24)
Similarly, from the j-th Equation of (22), we have
(25)
From (24) and (25), we get
![]()
Thus
.
Hence the theorem is proved.
Corollary 18.
Let G be a simple connected weighted graph where each edge weight
is a positive number. Then
,
where
.
Proof.
For weighted graphs where the edge weights
are positive number, we have
and
, for all
. Using Theorem 17 we get the required result.
Corollary 19. [7] .
Let G be a simple connected unweighted graph. Then
,
where
is the degree of vertex i and
is the average of the degrees of the vertices adjacent to vertex i.
Proof.
For an unweighted graph,
and
for all
and
. Using Corollary 18 we get the required result.
Conflict of Interests
The authors declare that there is no conlict of interests regarding the publication of this paper.