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

Let G be a simple graph or a multigraph. The vertex connectivity (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 ₃ (G) the third largest eigenvalue of the adjacency matrix of G. In this paper, we present an upper bound for ₃ (G) in a d-regular (multi-) graph G which guarantees that (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 ₃ (G) in a d-regular multigraph which assures that (G) 2.