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Let G be a connected graph with minimum degree δG and vertex-connectivity κG. The graph G is k-connected if κG≥k, maximally connected if κG=δG, 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.