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

Abstract

Let ${\lambda }_2(G)$ and ${\kappa }^{\prime }(G)$ 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 ${\kappa }^{\prime }(G)\ge t+1$. In this paper, we settle down for all $t\ge 3$.

16 pages, corrected the typos, revised the proof of main theorem, updated the references