On the Construction of Radially Moore Digraphs

Abstract

The Moore bound states that a digraph with maximum out-degree $d$
and radius $k$ has at most $1+d+\cdots +d^k$ vertices.
Regular digraphs attaining this bound and whose diameter is at most
$k+1$ are called radially Moore digraphs. Körner [4] proved
that these extremal digraphs exist for any value of $d\ge 1$ and $k\ge 1$.

In this paper, we introduce a digraph operator based on the line
digraph, which allows us to construct new radially Moore digraphs
and recover the known ones. Furthermore, we show that for $k=2$,
a radially Moore digraph with as many central vertices as the degree
$d$ does exist.