Median eigenvalues and the HOMO–LUMO index of graphs

Motivated by the problem about HOMO–LUMO separation that arises in mathematical chemistry, Fowler and Pisanski [2] and [3] introduced the notion of the HL-index which measures how large in absolute value may be the median eigenvalues of a graph. In this note we provide rather tight lower and upper bounds on the maximum value of the HL-index among all graphs with given average degree. In particular, we determine the exact value of this parameter when restricted to chemically relevant graphs, i.e. graphs of maximum degree 3, and thus answer a question from [2], [3] and [6]. The proof provides additional insight about eigenvalue distribution of large subcubic graphs.