Sufficient Conditions for Graphs to Be k-Connected, Maximally Connected, and Super-Connected

Let G be a connected graph with minimum degree delta(G) and vertex-connectivity kappa (G). The graph G is k-connected if kappa (G) >= k, maximally connected if kappa (G) = delta (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.