Edge-connectivity in regular multigraphs from eigenvalues

TL;DR

This paper establishes eigenvalue bounds that guarantee certain levels of edge-connectivity in regular multigraphs, linking spectral properties to structural robustness.

Contribution

It provides new spectral criteria for edge-connectivity in regular multigraphs, extending previous results with precise eigenvalue bounds.

Findings

If bbe2(G)c rac{d-1+\u221a{9d^2-10d+17}}{4}, then G is 2-edge-connected.

For t e 2, G is (t+1)-edge-connected when bbe2(G) < d - t.

G is (t+1)-edge-connected if bbe2(G) < d - t + 1 when t is odd.

Abstract

Let $G$ be a $d$-regular multigraph, and let $\lambda_2(G)$ be the second largest eigenvalue of $G$. In this paper, we prove that if $\lambda_2(G) < \frac{d-1+\sqrt{9d^2-10d+17}}4$, then $G$ is 2-edge-connected. Furthermore, for $t\ge2$ we show that $G$ is $(t+1)$-edge-connected when $\lambda_2(G)<d-t$, and in fact when $\lambda_2(G)<d-t+1$ if $t$ is odd.