On the difference of two generalized connectivities of a graph

The concept of k-connectivity. k(k)' (G) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity. k(k)(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if. k(k)' (G) not equal n -k + 1 where k >= 3, then 0 <= k(k)'(G) -k(k)(G) <= n-k-1; otherwise, -[k/2] + 1 <= k(k)'(G)-k(k)(G) <= n-k. Moreover, all of these bounds are sharp. Some specific study is focused for the case k = 3. As results, we characterize the graphs with. k(3)'(G) =k(3)(G) = t for t is an element of {1, n-3, n-2}, and give a necessary condition for k(3)'(G) =k(3)(G) by showing that for a connected graph G of order n and size m, if k(3)'(G) =k(3)(G) = t where 1 <= t <= n-3, then m <= ((n-2)(2)) + 2t. Moreover, the unique extremal graph is given for the equality to hold.