Sum of powers of the Laplacian eigenvalues and the kirchhoff index of a graph

Let G = (V, E) be a simple connected graph with vertex set V = { 1 , 2 , . . . , n }. For any real number alpha, the topological index s(alpha)(G) of G is defined as s(alpha)(G) = Sigma(n-1) (i =1) mu(alpha) (i) , where mu(1) >= mu 2 >= . . . mu(n -1) >= mu(n) = 0 are the Laplacian eigenvalues of G . In this paper, we first express s alpha (G ) explicitly in terms of resistance distances Omega(ij), i, j is an element of V . Then we generalize the following well-known equality ns -1 (G ) = Kf(G) to any integer k >= -1 , where Kf(G) = Sigma(i<j) Omega(ij) is the Kirchhoff index of G . As by-products, we get the expressions for the first Zagreb index and the Laplacian Estrada index in terms of the resistance distances. (c) 2023 Elsevier Inc. All rights reserved.