Sharp spectral bounds for the edge-connectivity of a regular graph

Abstract

Let $\lambda_2(G)$ and $\kappa'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $\lambda_2(G)$ in a $d$-regular simple graph $G$ to guarantee that $\kappa'(G) \ge t+1$. In this paper, we settle down for all $t \ge 3$.