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arXiv · 2606.12948

Maximum spectral gap of regular graphs with bounded essential edge-connectivity

Yu Wang·Sanming Zhou
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Abstract

An edge-cut of a graph is said to be essential if its removal results in a graph with at least two non-trivial components. The essential edge-connectivity of a graph $G$ is the minimum cardinality among all essential edge-cuts of $G$. The spectral gap of $G$ is the difference between its largest and second largest eigenvalues. In this paper, we prove that for any integers $t$ and $r$ with $6\leq r\leq t\leq 2r-3$, the maximum spectral gap among all connected $r$-regular graphs with essential edge-connectivity at most $t$ is equal to $\frac{1}{2}(r+7-\sqrt{(r+7)^2-8t-32})$ when $t-r$ is odd and $\frac{1}{2}(r+6-\sqrt{(r+6)^2-8t-32})$ when $t-r$ is even. We construct a family of connected $r$-regular graphs achieving these bounds.

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Yu Wang, Sanming Zhou. 2026-06-11. Maximum spectral gap of regular graphs with bounded essential edge-connectivity. https://arxiv.org/abs/2606.12948

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