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Let G be a graph of order n and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of G . The energy of G is defined as E ( G ) = ∑ k = 1 n | λ k | . A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The general Laplacian energy-like invariant of G , denoted by L E L α ( G ) , is defined as ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 0 , where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α is an irrational number where 0 < | α | < 1 , whereas they do not hold when | α | > 1 . [ABSTRACT FROM AUTHOR]

Let G be a simple graph. Its energy is defined as E ( G ) = ∑ k = 1 n | λ k | , where λ 1 , λ 2 , … , 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 μ 1 ≥ μ 2 ≥ ⋯ ≥ 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 ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 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. [ABSTRACT FROM AUTHOR]