Bounds for the Huckel energy of a graph

Abstract

Let $G$ be a graph on $n$ vertices with $r:=\lfloor n/2\rfloor$ and let ${\lambda }_1\ge ...\ge {\lambda }_n$ be adjacency eigenvalues of $G$. Then the Hückel energy of $G$, HE($G$), is defined as $$\he(G) = {ll} 2\sum_{i=1}^{r} λ_i, & \hbox{if $n= 2r$;} 2\sum_{i=1}^{r} λ_i + λ_{r+1}, & \hbox{if $n= 2r+1$.} $$ The concept of Hückel 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.

13 pages; historical points and some references added