Nonexistence of Almost Moore Digraphs of Degrees 4 and 5 with Self-Repeats.

An almost Moore (d, k)-digraph is a regular digraph of degree d > 1, diameter k > 1 and order N(d, k) = d + d2 + ?? ?? ?? + dk. So far, their existence has only been shown for k = 2, whilst it is known that there are no such digraphs for k = 3, 4 and for d = 2, 3 when k 3. Furthermore, under certain assumptions, the nonexistence for the remaining cases has also been shown. In this paper, we prove that (4, k) and (5, k)-almost Moore digraphs with self-repeats do not exist for k 5.