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 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 of Fowler and Pisanski. The proof provides additional insight about eigenvalue distribution of large subcubic graphs.

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