The minimal size of a graph with generalized connectivity κ3 = 2

Let G be a nontrivial connected graph of order n and k an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number of edge-disjoint trees T1, T2, . . . , T in G such that V (Ti) ∩ V (Tj) = S for every pair i, j of distinct integers with 1 ≤ i, j ≤ . Chartrand et al. generalized the concept of connectivity as follows. The k-connectivity, denoted by κk(G), of G is defined by κk(G) = min{κ(S)}, where the minimum is taken over all k-subsets S of V (G). Thus κ2(G) = κ(G), where κ(G) is the connectivity of G. This paper mainly determines the minimal number of edges of a graph of order n with κ3 = 2; that is, for a graph G of order n and size e(G) with κ3(G) = 2, it is proved that e(G) ≥ 65n , and the lower bound is sharp for all n ≥ 4 apart from n = 9, 10, whereas for n = 9, 10 examples are given to show that 6 5 n + 1 is the best possible lower bound. This gives a clear picture on the minimal size of a graph of order n with generalized connectivity κ3 = 2.