On mixed Moore graphs

The Moore bound for a directed graph of maximum out-degree d and diameter k   is Md,k=1+d+d2+⋯+dkMd,k=1+d+d2+⋯+dk. It is known that digraphs of order Md,kMd,k (Moore digraphs) do not exist for d>1d>1 and k>1k>1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k   is Md,k*=1+d+d(d-1)+⋯+d(d-1)k-1. Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore   graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter k=2k=2 are unique and that mixed Moore graphs of diameter k⩾3k⩾3 do not exist.