On Incidence Energy of Tree

Abstract

Let \(G = (V, E)\) be a simple graph, \(I(G)\) its incidence matrix. The incidence energy of \(G\), denoted by \(IE(G)\), is the sum of the singular values of \(I(G)\). The incidence energy \(IE(G)\) of a graph is a recently proposed quantity. However, \(IE(G)\) is closely related with the eigenvalues of the Laplacian and signless Laplacian matrices of \(G\). The trees with the maximal, the second maximal, the third maximal, the smallest, the second smallest, and the third smallest incidence energy were characterized. In this paper, the trees with the fourth and fifth smallest incidence energy are characterized by the quasi-order method and Coulson integral formula, respectively. In addition, the fourth maximal incidence energy among all trees on \(n\) vertices is characterized.