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A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length ≤ k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N (d, k) be the smallest possible order for a k-geodetic digraph of In this paper, we will prove that a (d, k, 1)-digraph is always out-regular and that if it is not in-regular, then it must have 2 vertices of in-degree less than d, d vertices of in-degree d + 1 and the remaining vertices will have in-degree d. Furthermore, we will prove there exist no (2, 2, 1)-digraphs and no diregular (2, k, 1)-digraphs for k ≥ 3.

A 3-uniform friendship hypergraph is a 3-uniform hypergraph in which, for all triples of vertices x, y, z there exists a unique vertex w, such that xyw, xzw and yzw are edges in the hypergraph. Sós showed that such 3-uniform friendship hypergraphs on n vertices exist with a so called universal friend if and only if a Steiner triple system, S(2, 3, n − 1) exists. Hartke and Vandenbussche used integer programming to search for 3-uniform friendship hypergraphs without a universal friend and found one on 8, three non-isomorphic on 16 and one on 32 vertices. So far, these five hypergraphs are the only known 3-uniform friendship hypergraphs. In this paper we construct an infinite family of 3-uniform friendship hypergraphs on 2 k vertices and 2 k−1 (3 k−1 − 1) edges. We also construct 3-uniform friendship hypergraphs on 20 and 28 vertices using a computer. Furthermore, we define r-uniform friendship hypergraphs and state that the existence of those with a universal friend is dependent on the existence of a Steiner system, S(r − 1, r, n − 1). As a result hereof, we know infinitely many 4-uniform friendship hypergraphs with a universal friend. Finally we show how to construct a 4-uniform friendship hypergraph on 9 vertices and with no universal friend.

The degree/diameter problem for directed graphs is the problem of determining the largest possible order for a digraph with given maximum out-degree d and diameter k. An upper bound is given by the Moore bound M (d, k) = In this paper we will look at the structure of subdigraphs of almost Moore digraphs, which are induced by the vertices fixed by some automorphism ϕ. If the automorphism fixes at least three vertices, we prove that the induced subdigraph is either an almost Moore digraph or a diregularAs it is known that almost Moore digraphs have an automorphism r, these results can help us determine structural properties of almost Moore digraphs, such as how many vertices of each order there are with respect to r. We determine this for d = 4 and d = 5, where we prove that except in some special cases, all vertices will have the same order.