The Bounds for Eigenvalues of Normalized Laplacian Matrices and Signless Laplacian Matrices ()
1. Introduction
Let
be a simple graph with the vertex set
and edge set of E. For
, the degree of
, the set of neighbours of
are denoted by
and
, respectively. If
and
are adjacent, we denote
of short use
.
The adjacency matrix, Laplacian matrix and diagonal matrix of vertex degree of a
graph are denoted by
,
,
, respectively. Clearly

The normalized Laplacian matrix of G is defined as
i.e.,
where
![]()
The signless Laplacian matrix of G is defined as
i.e.,
where
![]()
Since
normalized Laplacian matrix and
signless Laplacian matrix are real symetric matrices, their eigenvalues are real. We denote the eigenvalues of
and
by
![]()
and
![]()
respectively.
Now we give some bounds for normalized Laplacian matrix and signless Laplacian matrix.
1. Oliveira and de Lima’s bound [1] : For a simple connected graph G with n vertices and m edges, ![]()
(1)
where
.
2. Another Oliveira and de Lima’s bound [1] :
(2)
where
.
3. Li, Liu et al. bound’s [2] [3] :
(3)
4. Rojo and Soto’s bound [4] : If
is the largest eigenvalue of
then
(4)
where the minimum is taken over all pairs
,
.
In this paper, we found extreme eigenvalues of normalized Laplacian matrix and signless Laplacian matrix of a G graph with using theirs traces.
To obtain bounds for eigenvalues of
and
we need the followings lemmas and theorems.
Lemma 1. Let W and
be nonzero column vectors,
, ![]()
,
and
is an identity matrix. Let
. Then,
![]()
![]()
![]()
![]()
Theorem 1 [5] . Let A be a
complex matrix. Conjugate transpose of A denoted by
. Let
whose eigenvalues are
Then
![]()
and
![]()
where
and ![]()
2. Main Results for Normalized Laplacian Matrix
Theorem 2. Let G be a simple graph and
be a normalized Laplacian matrix of G. If the eigenvalues of
are
, then
(5)
(6)
(7)
Proof. Clearly
![]()
and
![]()
Since
real symmetric matrix, we found the result from Theorem 1.
Example 1. Let
with
and
![]()
3. Main Results for Signless Laplacian Matrix
Theorem 3. Let G be a simple graph and
be a signless Laplacian matrix of G. If the eigenvalues of
are
, then
(8)
(9)
(10)
Proof. Clearly
![]()
and
![]()
Since
was real symmetric matrix, we found the result from Theorem 1.
Example 2. Let
with
and
![]()