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

Abstract

Let G=(V,E) be a simple connected graph with vertex set V={1,2,…,n}. For any real number α, the topological index sα(G) of G is defined assα(G)=∑i=1n−1μiα,where μ1≥μ2≥…μn−1>μn=0 are the Laplacian eigenvalues of G. In this paper, we first express sα(G) explicitly in terms of resistance distances Ωij,i,j∈V. Then we generalize the following well-known equalityns−1(G)=Kf(G)to any integer k≥−1, where Kf(G)=∑i<jΩij 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.