On the structure of almost Moore digraphs containing selfrepeats

An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices is one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r(u) = v, Such that there are two walks of lenght ≤ k from u to v. The smallest positive integer p such that the composition r(u) = u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G is an almost Moore digraph then G contains exactly k selfrepeats or none. In this paper, we present the possible vertex orders of an almost digraph containing selfrepeats for d ≥ 4, k ≥ 3.