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