Remark on sum of powers of normalized signless Laplacian eigenvalues of graphs

Abstract

Let $G=(V,E)$, $V=\{v_{1},v_{2},\ldots ,v_{n}\}$, be a simple connected
graph of order $n$ and size $m$. Denote by $\gamma _{1}^{+}\geq \gamma
_{2}^{+}\geq \cdots \geq \gamma _{n}^{+}\geq 0$ the normalized signless
Laplacian eigenvalues of $G$, and by $\sigma _{\alpha }(G)$ the sum of $%
\alpha $-th powers of the normalized signless Laplacian eigenvalues of a
connected graph. The paper deals with bounds of $\sigma _{\alpha }$. Some
special cases, when $\alpha =\frac{1}{2}$ and $\alpha =-1$, are also
considered