Bounds for the Hückel energy of a graph (Q2380300)

Summary: Let \(G\) be a graph on \(n\) vertices with \(r:=\lfloor n/2\rfloor\) and let \(\lambda_1\geq\cdots\geq\lambda_n\) be adjacency eigenvalues of \(G\). Then the Hückel energy of \(G\), \(\text{HE}(G)\), is defined as \[ \text{HE}(G)= \begin{cases} 2\sum\limits_{i=1}^r \lambda_i, &\text{if }n=2r;\\ 2\sum\limits_{i=1}^r \lambda_i+ \lambda_{r+1}, &\text{if }n=2r+1. \end{cases} \] 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 \(\text{HE}(G)\). 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 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.