A note on sum of powers of the Laplacian eigenvalues of graphs

Abstract

For a graph $G$ and a real number $\alpha \ne 0$, the graph invariant $s_{\alpha }\left(G\right)$ is the sum of the $\alpha$th power of the non-zero Laplacian eigenvalues of $G$. This note presents some bounds for $s_{\alpha }\left(G\right)$ in terms of the vertex degrees of $G$, and a relation between $s_{\alpha }\left(G\right)$ and the first general Zagreb index, which is a useful topological index and has important applications in chemistry.