On diregular digraphs with degree two and excess three

Moore digraphs, that is digraphs with out-degree $d$, diameter $k$ and order equal to the Moore bound $M(d,k)=1+d+d^2+\cdots +d^k$, arise in the study of optimal network topologies. In an attempt to find digraphs with a `Moore-like’ structure, attention has recently been devoted to the study of small digraphs with minimum out-degree $d$ such that between any pair of vertices $u,v$ there is at most one directed path of length $\le k$ from $u$ to $v$; such a digraph has order $M(d,k)+ϵ$ for some small excess $ϵ$. Sillasen et al. have shown that there are no digraphs with out-degree two and excess one. The present author has classified all digraphs with out-degree two and excess two. In this paper it is proven that there are no diregular digraphs with out-degree two and excess three for $k\ge 3$, thereby providing the first classification of digraphs with order three away from the Moore bound for a fixed out-degree.