^{1}

^{*}

^{2}

^{*}

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.

Let

The adjacency matrix, Laplacian matrix and diagonal matrix of vertex degree of a

The normalized Laplacian matrix of G is defined as

The signless Laplacian matrix of G is defined as

Since

and

respectively.

Now we give some bounds for normalized Laplacian matrix and signless Laplacian matrix.

1. Oliveira and de Lima’s bound [

where

2. Another Oliveira and de Lima’s bound [

where

3. Li, Liu et al. bound’s [

4. Rojo and Soto’s bound [

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

Lemma 1. Let W and

Theorem 1 [

and

where

Theorem 2. Let G be a simple graph and

Proof. Clearly

and

Since

Example 1. Let

Theorem 3. Let G be a simple graph and

Proof. Clearly

and

Since

Example 2. Let