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

Let G   be a simple graph. Its energy is defined as E(G)=∑k=1n|λk|, where λ1,λ2,…,λnλ1,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=0μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G  , denoted by LELα(G)LELα(G), is defined as ∑μk≠0μkα when μ1≠0μ1≠0, and 0 when μ1=0μ1=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/pα=1/p with p∈Z+\{1}p∈Z+\{1}. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.