Oriented incidence energy and threshold graphs

Abstract

Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n ≥ 9, there exists at least ([n/9/2]) + 1 distinct pairs of graphs on n vertices having equal oriented incidence energy.