On the Sum of Powers of Normalized Laplacian Eigenvalues of Graphs

For a graph G without isolated vertices and a real alpha not equal 0, we introduce a new graph invariant s(alpha)(*) (G)- the sum of the alpha th power of the non-zero normalized Laplacian eigenvalues of G. Recently, the cases alpha = 2 and -1 have appeared in various problems in the literature. Here, we present some lower and upper bounds of s(alpha)(*)(G) for a connected graph G, where alpha not equal 0, 1. We also discuss the case alpha = -1.