Enumerations of vertex orders of almost Moore digraphs with selfrepeats

An almost Moore digraph G of degree image, diameter image 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 image there exists a vertex image, called repeat of u and denoted by image, such that there are two walks of length image from u to image. The smallest positive integer p such that the composition image 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 image 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.