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, . . ., λ_n 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≥⋯≥μ_n=0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted by LEL_α(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.