Spectral radius and \(k\)-connectedness of a graph (Q1744085)

Let \(\delta\geq k\geq 3\) and \(n\geq (\delta-k+2)(k^2-2k+4)+3\). The authors show that if a connected graph \(G\) with \(n\) vertices and minimum degree at least \(\delta\) satisfies \(\lambda_1(G)\geq n-\delta+k-3\), where \(\lambda_1(G)\) is the spectral radius of adjacency matrix of \(G\), then \(G\) is \(k\)-connected unless \(G\) is isomorphic to the join of \(K_{k-1}\) with \(K_{\delta-k+2}\cup K_{n-\delta-1}\). The same condition for \(\lambda_1(G)\) implies that \(G\) is always \(k\)-edge-connected.