A best bound for $\lambda_2(G)$ to guarantee $\kappa(G) \geq 2$

TL;DR

This paper establishes a tight upper bound on the second largest eigenvalue of a regular graph to ensure it has vertex connectivity at least 2, advancing understanding of spectral conditions for graph connectivity.

Contribution

The paper provides the best possible eigenvalue bound for guaranteeing vertex connectivity of at least 2 in regular graphs and characterizes extremal graph families.

Findings

Derived a sharp eigenvalue bound for connectivity ≥ 2

Characterized extremal graphs achieving the bound

Solved the problem for the case t=1

Abstract

Let $G$ be a connected $d$-regular graph with a given order and the second largest eigenvalue $\lambda_2(G)$. Mohar and O (private communication) asked a challenging problem: what is the best upper bound for $\lambda_2(G)$ which guarantees that $\kappa(G) \geq t+1$, where $1 \leq t \leq d-1$ and $\kappa(G)$ is the vertex-connectivity of $G$, which was also mentioned by Cioab\u{a}. As a starting point, we solve this problem in the case $t =1$, and characterize all families of extremal graphs.