The vertex connectivity and the third largest eigenvalue in regular (multi-)graphs

Abstract

Let $G$ be a simple graph or a multigraph. The vertex connectivity $\kappa(G)$ of $G$ is the minimum size of a vertex set $S$ such that $G-S$ is disconnected or has only one vertex. We denote by $\lambda_{3}(G)$ the third largest eigenvalue of the adjacency matrix of $G$. In this paper, we present an upper bound for $\lambda_{3}(G)$ in a $d$-regular (multi-)graph $G$ which guarantees that $\kappa(G)\geq t+1$, which is based on the result of Abiad et al. [Spectral bounds for the connectivity of regular graphs with given order. Electron. J. Linear Algebra 34:428-443, 2018]. Furthermore, we improve the upper bound for $\lambda_{3}(G)$ in a $d$-regular multigraph which assures that $\kappa(G)\geq 2$.