On diregular digraphs with degree two and excess two

An important topic in the design of efficient networks is the construction of (d,k,+ϵ)-digraphs, i.e. k-geodetic digraphs with minimum out-degree ≥d and order M(d,k)+ϵ, where M(d,k) represents the Moore bound for degree d and diameter k and ϵ>0 is the (small) excess of the digraph. Previous work has shown that there are no (2,k,+1)-digraphs for k≥2. In a separate paper, the present author has shown that any (2,k,+2)-digraph must be diregular for k≥2. In the present work, this analysis is completed by proving the nonexistence of diregular (2,k,+2)-digraphs for k≥3 and classifying diregular (2,2,+2)-digraphs up to isomorphism.