Coulson-type integral formulas for the general energy of polynomials with real roots

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In 1940 Coulson obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. Recently several Coulson-type integral formulas have been obtained for various energies and some other invariants of graphs based on eigenvalues. For a complex polynomial ϕ(z)=∑_k=0ⁿa_kz^n−k=a₀∏_k=1ⁿ(z−z_k) of degree n and a real number α, the general energy of ϕ(z), denoted by E_α(ϕ), is defined as ∑_zk≠0|z_k|^α when there exists k₀∈{1,2,…,n} such that z_k0≠0, and 0 when z₁=⋯=z_n=0. In this paper we give Coulson-type integral formulas for the general energy of polynomials whose roots are all real numbers in the case that α∈Q. As a consequence of this result, we obtain an integral formula for the 2l-th spectral moment of a graph. Furthermore, we show that our formulas hold when α is an irrational number with 0 < |α| < 2 and do not hold with |α| > 2.