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

Let G be a simple graph or a multigraph. The vertex connectivity κ(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 λ₃(G) the third largest eigenvalue of the adjacency matrix of G. In this paper, we present an upper bound for λ₃(G) in a d-regular (multi-)graph G which guarantees that κ(G)≥ 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 λ₃(G) in a d-regular multigraph which assures that κ(G)≥ 2.