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

Let G be a graph of order n and λ₁≥λ₂≥⋯≥λ_n be the eigenvalues of G. The energy of G is defined as E(G)=∑_k=1ⁿ|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 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≠0μ_k^α when μ₁≠0, and 0 when μ₁=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.