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 $N(d,k)=d+d^2+\cdots +d^k$. 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 $k\geq 3$. 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 $k\geq 5$.