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.