Codes and Expansions (CodEx) Seminar

Hricha Acharya (Arizona State)
Median eigenvalues of graphs

The median eigenvalues of a graph are the eigenvalues of its adjacency matrix sitting at the center of the spectrum. In mathematical chemistry, they correspond to the HOMO and LUMO orbitals, which govern molecular reactivity. In this talk, we explore how the maximum degree of a graph sharply controls its median eigenvalues. In the first part, we resolve a conjecture of Fowler and Pisanski by proving that the median eigenvalues of every chemical graph lie in \([-1, 1]\), with the Heawood graph as the unique exception. In the second part, we turn to graphs of general maximum degree \(d\). We confirm a conjecture of Mohar that the median eigenvalues are bounded in absolute value by \(\sqrt{d-1}\), for all but finitely many values of \(d\). Throughout, the extremal graphs are not random: incidence graphs of projective planes emerge as the key extremal family, turning what begins as a mysterious exception into an organizing pattern. This talk is based on joint work with Zilin Jiang, Benjamin Jeter and Shengtong Zhang.