Bounds on median eigenvalues of graphs of bounded degree

TL;DR

This paper establishes bounds on the median eigenvalues of graphs with bounded maximum degree, resolving an open problem for most degrees and extending previous energy bounds.

Contribution

It provides new upper and lower bounds on median eigenvalues for graphs of maximum degree d, addressing an open problem in spectral graph theory.

Findings

Median eigenvalues are bounded above by √(d-1) for all graphs with max degree d.

Lower bounds on median eigenvalues are established for specific cases, including triangle-free graphs.

An upper bound on the average energy of graphs with maximum degree d is derived.

Abstract

We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are bounded below by $-\sqrt{d-1}$: when the graph is triangle-free, when $d-1$ is a perfect square, or when $d \ge 75$. These results resolve, for all but finitely many values of $d$, an open problem of Mohar on median eigenvalues of graphs of maximum degree $d$. As a byproduct, we establish an upper bound on the average energy of graphs of maximum degree at most $d$, generalizing a previous result of van Dam, Haemers, and Koolen for $d$-regular graphs.