Bounds for the Hückel energy of a graph

Abstract

Let G be a graph on n vertices with r := left perpendicularn/2right perpendicular and let lambda(1) >= center dot center dot center dot >= lambda(n) be adjacency eigenvalues of G. Then the Huckel energy of G, HE(G), is defined as HE(G) = {2 Sigma(r)(i=1)lambda(i), if n = 2r; 2 Sigma(r)(i=1)lambda(i) + lambda(r+1), if n = 2r + 1. The concept of Huckel energy was introduced by Coulson as it gives a good approximation for the pi-electron energy of molecular graphs. We obtain two upper bounds and a lower bound for HE(G). When n is even, it is shown that equality holds in both upper bounds if and only if G is a strongly regular graph with parameters (n, k, lambda, mu) = (4t(2) + 4t + 2, 2t(2) + 3t + 1, t(2) + 2t, t(2) + 2t + 1), for positive integer t. Furthermore, we will give an infinite family of these strongly regular graph whose construction was communicated by Willem Haemers to us. He attributes the construction to J.J. Seidel.