The number of h-strongly connected digraphs with small diameter

Let D s (n; h, d = k) denote the number of h-strongly connected digraphs of order n and diameter equal to k. In this paper it is shown that: i) Ds(n; 11" d = 3) = 4(~) (3/4 + o(I))n for every fixed h ;::: 1; ii) Ds(n; h, d = 4) = 4(~) (2h 2 + 2-2 + o(I))n for every fixed h ;::: 2; iii) Ds(n; 11" d = k) = 4(~) ((2h+l I)2-kh+:3h-2 + o(I))n for every fixed h ;::: 1 and k ;::: 5. Similar asymptotic formulas hold for the number of h-connected digraphs of order n and diameter equal to k when n -+ 00. This extends the corresponding results for h-connected graphs given in a recent paper by the author. 1 Notation and preliminary results All digraphs in this paper are finite, labeled, without loops or parallel directed edges. By [{~ we denote the complete digraph of order n such that any two distinct vertices .r and yare joined by two directed edges (x, y) and (y, x). For a digraph G the out degree d+(x) of a vertex x is the number of vertices of G that are adjacent from x and the indegree d(x) is the number of vertices of G adjacent to x. For h 2:: 2, we say that a digraph G is h-connected (resp. h-strongly connected) if either G is a complete digraph [{':+1 or else it has at least h + 2 vertices and for any set of vertices X c V(G), IXI h 1, the digraph G X is connected (resp. strongly connected). A connected (resp. strongly connected) digraph is also said to be I-connected (resp. I-strongly connected). For a strongly connected digraph G the distance d( x, y) from vertex x to vertex y is the length of a shortest path of the form (1;, ... , y). The eccentricity of a vertex .r is ecc(x) = maxYEv(G)d(x, y). The diameter of G, denoted Australasian Journal of Combinatorics 24(2001), pp.305-311 d( G) is equal to max:n,YEV(G)d(x, y) if G is strongly connected and 00 otherwise. By Ds(n; h, d = k) and Ds(n; h, d 2 k) (resp. D(n; h, d = k) and D(n; h, d 2 k)) we denote the number of h-strongly connected (resp. h-connected) digraphs G of order n and diameter d( G) k and d( G) 2 k, respectively. It is well known [1, p. 131] that almost all digraphs have diameter two and for every fixed integer h 2 1 almost all graphs are h-connected. Also in [2] it was proved that for every fixed integer h 2 1 almost all digraphs are h-strongly connected. Hence for every h 2 1 we have: Ds(n; h, d 2) = 4(~) (1 + 0(1)) and D(n; h, d = 2) = 4(~) (1 + 0(1)). If limn-too~ = 1 we denote this by fen) rv g(n), or fen) = g(n)(l + 0(1)). The following results will be useful in the proofs of the theorems given in the next section. Lemma 1.1 ([4]). The number of bipartite digmphs G whose partite sets are A, B (A n B = 0, 1 A 1= p, 1 B 1= q) such that d-(x) 2 1 for every :1; E B and all edges are directed from A towards B is equal to (2P I)Q. Lemma 1.2 ([4]). We have Ds(n;l,d= 3) 4(~)(3/4+0(1))n. Also we need an asymptotic evaluation of the maximum of an arithmetical function. Let where nl + ... + nk = n, ni 2 h for every 1 ~ i ~ k 1 and nk 2 1. Let us denote fen, k) = maxDf(n, h; nl, ... , nk), where D is defined by: nl + ... + nk = n; ni 2 h for every 1 ::; i ::; k 1 and nk 2 1. Theorem 1.3 ([5]). The following equalities hold: