Let \(G\) be a simple graph of order \(n\) with independence number \(\alpha\). We prove in this paper that if, for any pair of nonadjacent vertices \(u\) and \(v\), \(d(u)+d(v) \geq n+1\) or \(|N(u) \cap N(v)| \geq \alpha\), then \(G\) is \((4, n-1)\)-connected unless \(G\) is some special graphs. As a corollary, we investigate edge-pancyclicity of graphs.