Bounds for the Huckel energy of a graph

Let $G$ be a graph on $n$ vertices with $r:=\lfloor n\mathrm{/}2\rfloor$ and let ${\lambda }_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge {\lambda }_n$ be adjacency eigenvalues of $G$. Then the H"uckel energy of $G$, HE($G$), is defined as \he(G) = {ll} 2\sum_{i=1}^{r} \lambda_i, & \hbox{if $n= 2r$;} 2\sum_{i=1}^{r} \lambda_i + \lambda_{r+1}, & \hbox{if $n= 2r+1$.} The concept of H"uckel 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 $(n, k, \lambda, \mu) = (4t^2 +4t +2, 2t^2 +3t +1, t^2 +2t, t^2 + 2t +1),$forpositiveinteger$t$. 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.