Median eigenvalues of subcubic graphs

TL;DR

This paper proves that for all connected graphs with maximum degree three, except the Heawood graph, the median eigenvalues are at most 1 in absolute value, resolving previously open questions.

Contribution

It establishes a universal bound on median eigenvalues for subcubic graphs, except for a specific known exception, advancing spectral graph theory.

Findings

Median eigenvalues of connected subcubic graphs are at most 1 in absolute value.

The Heawood graph is the unique exception.

Resolved open problems in spectral graph theory.

Abstract

We show that the median eigenvalues of every connected graph of maximum degree at most three, except for the Heawood graph, are at most $1$ in absolute value, resolving open problems posed by Fowler and Pisanski, and by Mohar.