On the maximum and minimum sizes of a graph with given k-connectivity
Yuefang Sun · Discussiones Mathematicae Graph Theory · 2016
The concept of k-connectivity k (G), introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. For an integer k 2, the k-connectivity of a connected graph G with order n k is the smallest number of vertices whose removal from G produces a graph with at least k components or a graph with fewer than k vertices. In this paper, we get a sharp upper bound for the size of G with k (G) = t, where 1 t nk and k 3; moreover, the unique extremal graph is given. Based on this result, we get the exact values for the maximum size, denoted by g(n, k, t), of a connected graph G with order n and k (G) = t. We also compute the exact values and bounds for another parameter f (n, k, t) which is defined as the minimum size of a connected graph G with order n and k (G) = t, where 1 t nk and k 3.