Bounds on median eigenvalues of graphs of bounded degree

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.

Figures (1)

• Figure 1: The graph of the magic polynomial.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark
• Theorem 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Remark