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arXiv · 2603.27434

Bounds on median eigenvalues of graphs of bounded degree

Hricha Acharya·Zilin Jiang·Shengtong Zhang
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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.

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Hricha Acharya, Zilin Jiang, Shengtong Zhang. 2026-03-28. Bounds on median eigenvalues of graphs of bounded degree. https://arxiv.org/abs/2603.27434

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