The energy of a graph

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) less than or equal to k + rootk(n - 1)(n - k) = B-2 and this bound is sharp. It is shown that for each epsilon > 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 < epsilon. Two B2 graphs with the same number of vertices are equienergetic if they have the same energy. We show that for any positive integer n greater than or equal to 3, there exist two equienergetic graphs of order 4n that are not cospectral. (C) 2004 Elsevier Inc. All rights reserved.