On edge-path eigenvalues of graphs

Let G be a graph with vertex set V(G) = {v(1), ... , v(n)} and EP(G) be an n x n matrix whose (i, j)-entry is the maximum number of internally edge-disjoint paths between v(i) and v(j), if i not equal j, and zero otherwise. Also, define (EP) over bar (G) = EP(G) + D, where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing v(i), whose EP(G) is a multiple of J-I. Among other results, we determine the spectrum and the energy of the matrix (EP) over bar (G) for an arbitrary bicyclic graph G.