Motivated by the problem about HOMO-LUMO separation that arises inmathematical chemistry, Fowler and Pisanski introduced the notion of theHL-index which measures how large in absolute value may be the medianeigenvalues of a graph. In this note we provide rather tight lower and upperbounds on the maximum value of the HL-index among all graphs with given averagedegree. In particular, we determine the exact value of this parameter whenrestricted to chemically relevant graphs, i.e.\ graphs of maximum degree $3$,and thus answer a question of Fowler and Pisanski. The proof providesadditional insight about eigenvalue distribution of large subcubic graphs.
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