On incidence energy of a graph

The Laplacian-energy like invariant LEL(G) and the incidence energy IE(G) of a graph are recently proposed quantities, equal, respectively, to the sum of the square roots of the Laplacian eigenvalues, and the sum of the singular values of the incidence matrix of the graph G. However, IE(G) is closely related with the eigenvalues of the Laplacian and signless Laplacian matrices of G. For bipartite graphs, IE=LEL. We now point out some further relations for IE and LEL: IE can be expressed in terms of eigenvalues of the line graph, whereas LEL in terms of singular values of the incidence matrix of a directed graph. Several lower and upper bounds for IE are obtained, including those that pertain to the line graph of G. In addition, Nordhaus–Gaddum-type results for IE are established.