The energy of a graph

Abstract

The energy, E( G), of a simple graph G is defined to be the sum of the absolute values of the eigen values of G. If G is a k-regular graph on n vertices,then E(G)⩽k+ k(n−1)(n−k) =B 2 and this bound is sharp. It is shown that for each ϵ>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k< n−1 and E(G) B 2 <ϵ . Two graphs with the same number of vertices are equienergetic if they have the same energy. We show that for any positive integer n⩾3, there exist two equienergetic graphs of order 4 n that are not cospectral.