Enumerations of vertex orders of almost Moore digraphs with selfrepeats

An almost Moore digraph G   of degree d>1d>1, diameter k>1k>1 is a diregular digraph with the number of vertices one less than the Moore bound. If G   is an almost Moore digraph, then for each vertex u∈V(G)u∈V(G) there exists a vertex v∈V(G)v∈V(G), called repeat of u   and denoted by r(u)=vr(u)=v, such that there are two walks of length ⩽k⩽k from u   to vv. The smallest positive integer p   such that the composition rp(u)=urp(u)=u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G   is an almost Moore digraph of diameter k⩾3k⩾3 then G contains exactly k selfrepeats or none. In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex.