Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I

Abstract

Let G be a simple graph. Its energy is defined as E(G)=∑k=1n|λk|, where λ1,λ2,…,λn are the eigenvalues of G. A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted by LELα(G), is defined as ∑μk≠0μkα when μ1≠0, and 0 when μ1=0, where α is a real number. In this paper we give a Coulson-type integral formula for the general Laplacian-energy-like invariant for α=1/p with p∈Z+\\{1}. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.