In this paper, we found the bounds of the extreme eigenvalues of normalized Laplacian matrices and signless Laplacian matrices by using their traces. In addition, we found the bounds for k-th eigenvalues of normalized Laplacian matrix and signless Laplacian matrix.

Normalized Laplacian Matrix Signless Laplacian Matrix Bounds of Eigenvalue
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  : For a simple connected graph G with n vertices and m edges,

where.

2. Another Oliveira and de Lima’s bound  :

where.

3. Li, Liu et al. bound’s   :

4. Rojo and Soto’s bound  : If is the largest eigenvalue of then

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  . 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

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

Proof. Clearly

and

Since was real symmetric matrix, we found the result from Theorem 1.

Example 2. Let with and

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