On k-geodetic digraphs with excess one

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 minimum out-degree d, then N (d, k) ≥ 1 + d + d2 + . . . + dk = M (d, k), where M (d, k) is the Moore bound obtained if and only if G is strongly geodetic. Thus, strongly geodetic digraphs only exist for d = 1 or k = 1, hence for d, k ≥ 2 we wish to determine if N (d, k) = M (d, k) + 1 is possible. A k-geodetic digraph with minimum out-degree d and order M (d, k) + 1 is denoted as a (d, k, 1)-digraph or said to have excess 1. 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.