The generalized connectivity of alternating group graphs and (n, k)-star graphs

Let S subset of V(G) and kappa(G)(S) denote the maximum number r of edge-disjoint trees T-1, T-2, ..., T-r in G such that V(T-i) boolean AND V(T-j) = S for any i, j is an element of{1, 2, ..., r} and i not equal j. For an integer k with 2 <= k <= n, the generalized k-connectivity of a graph G is defined as kappa(k)(G) = min{kappa(G)(S)vertical bar S subset of V(G) and vertical bar S vertical bar = k}. The generalized k-connectivity is a generalization of traditional connectivity. In this paper, we focus on the alternating group graphs and (n, k)-star graphs, denoted by AG(n) and S-n,S-k, respectively. We study the generalized 3-connectivity of the two kinds of graphs and show that kappa(3)(AG(n)) = 2n - 5 for n >= 4 and kappa(3)(S-n,S-k) = n - 2 for n >= k + 1 and k >= 4, which generalize the known result about star graphs given by Li et al. (2016). In addition, as the alternating group network AN(n) is isomorphic to S-n,S-k for k = n - 2, the generalized 3-connectivity of AN(n) for n >= 6 can be obtained directly. (C) 2018 Elsevier B.V. All rights reserved.