Extremality and Sharp Bounds for the $k$-edge-connectivity of Graphs

Boesch and Chen (SIAM J. Appl. Math., 1978) introduced the cut-version of the generalized edge-connectivity, named $k$-edge-connectivity. For any integer $k$ with $2\leq k\leq n$, the {\em $k$-edge-connectivity} of a graph $G$, denoted by $\lambda_k(G)$, is defined as the smallest number of edges whose removal from $G$ produces a graph with at least $k$ components. In this paper, we first compute some exact values and sharp bounds for $\lambda_k(G)$ in terms of $n$ and $k$. We then discuss the relationships between $\lambda_k(G)$ and other generalized connectivities. An algorithm in $\mathcal{O}(n^2)$ time will be provided such that we can get a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function $f(n,k,t)$ which is defined as the minimum size of a connected graph $G$ with order $n$ and $\lambda_k(G)=t$.