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

TL;DR

This paper establishes sharp spectral bounds for the edge-connectivity of regular graphs for all cases where the edge-connectivity exceeds two, extending previous results that covered only lower cases.

Contribution

The paper provides comprehensive spectral bounds for the edge-connectivity of regular graphs for all values of edge-connectivity greater than two, filling a gap in existing spectral graph theory.

Findings

Sharp spectral bounds for $ abla$-regular graphs with $ abla extgreater 2$

Extension of previous bounds for $t=1,2$ to all $t extgreater 2$

Improved understanding of the relationship between eigenvalues and edge-connectivity

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 $t \ge 3$.