Coulson-type integral formulas for the general energy of a vertex

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 Arizmendi and Juárez-Romero proposed the concept of vertex energy and Arizmendi et al. further obtained a Coulson-type integral formula for the vertex energy of a graph. In this paper, we will introduce the concept of the general energy of a vertex, and give some Coulson-type integral formulas for the general energy of a vertex.