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arXiv · 2003.02393

Spectral Threshold for Extremal Cyclic Edge-Connectivity

Sinan G. Aksoy·Mark Kempton·Stephen J. Young
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Abstract

The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth $g$ at least 4, the cyclic edge-connectivity is bounded above by $(Δ-2)g$ where $Δ$ is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a $d$-regular graph of girth $g\geq4$ is sufficiently small, then the cyclic edge-connectivity is $(d-2)g$, providing a spectral condition for when this upper bound on cyclic edge-connectivity is tight.

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Sinan G. Aksoy, Mark Kempton, Stephen J. Young. 2021-04-05. Spectral Threshold for Extremal Cyclic Edge-Connectivity. https://arxiv.org/abs/2003.02393

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