\(\ell\)-connectivity, integrity, tenacity, toughness and eigenvalues of graphs (Q2091163)

The \(\ell\)-edge connectivity \(\kappa_\ell(G)\) of a simple graph \(G\) is the minimum number of vertices of \(G\) whose removal produces either a disconnected graph with at least \(\ell\) components or a graph with fewer than \(\ell\) vertices. The first two main results of the paper under review provide sufficient conditions for the bound \(\kappa_\ell(G)\geq k\) to hold, where \(\ell\geq k\geq 2\), expressed in terms of bounds involving other invariants on \(G\) including the maximum degree \(\Delta\), the minimum degree \(\delta\), the edge connectivity \(\kappa'\), the girth \(g(G)\), the clique number \(\omega(G)\), and eigenvalues \(\lambda_i\) of \(G\). Lower bounds on the integrity \(I(G)\), tenacity \(T(G)\) and toughness \(t(G)\) are also provided, expressed in terms of \(\Delta\), \(\kappa'\), the number of vertices \(n\), the Laplacian eigenvalue \(\mu_{n-1}\) and the normalised Laplacian eigenvalues \(\rho_i\). Some of these results extend previous results in the literature.