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