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

Abstract

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