چکیده (انگلیسی):

Le G be a graph of order n and λ1 ≥ λ2 ≥ ••• ≥ λn the eigenvalues of G. The energy of G is defined as E(G) = Ʃn k=1 |λk|. 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 = 0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted by LELα(G), is defined as Ʃ μk≠0 μαk when μ1 ≠ 0, and 0 when μ1 = 0, where α is a real number. In this paper we give some Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs in the case that α is a rational number. Based on this result, we further give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case that α is a rational number. We also show that our formulas hold when α is an irrational number with 0 < |α| < 1 and do not hold with |α| > 1 .