Abstract

Let G be a connected graph with minimum degree δ G and vertex-connectivity κ G. The graph G is k -connected if κ G ≥ k, maximally connected if κ G = δ G, and super-connected if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we present sufficient conditions for a graph with given minimum degree to be k -connected, maximally connected, or super-connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. Analogous results for triangle-free graphs with given minimum degree to be k -connected, maximally connected, or super-connected are also presented.