Bounds for the Hückel Energy of a Graph

Let G be a graph on n vertices with r := bn/2c and let λ1 ≥ · · · ≥ λn be adjacency eigenvalues of G. Then the Hückel energy of G, HE(G), is defined as HE(G) = ( 2 Pr i=1 λi, if n = 2r; 2 Pr i=1 λi + λr+1, if n = 2r + 1. The concept of Hückel energy was introduced by Coulson as it gives a good approximation for the π-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, λ, μ) = (4t + 4t+ 2, 2t + 3t+ 1, t + 2t, t + 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. E-mail Addresses: e [email protected] (E. Ghorbani), [email protected] (J.H. Koolen), [email protected] (J.Y. Yang) ∗This work was done while the first author was visiting the department of mathematics of POSTECH. He would like to thank the department for its hospitality and support. †He was partially supported by a grant from the Korea Research Foundation funded by the Korean government (MOEHRD) under grant number KRF-2008-314-C00007. 1 ar X iv :0 90 8. 26 67 v2 [ m at h. C O ] 4 S ep 2 00 9