On digraphs with unique walks of closed lengths between vertices

It is known that regular digraphs of degree d, diameter k and unique walks of length not smaller than h and not greater than k between all pairs of vertices ([ h, k ]-digraphs), exist only for h k and h = k 1, if d ;::: 2. This paper deals with the problem of the enumeration of [k 1, kJ-digraphs in the case of diameter k = 2 or degree d = 2. It is shown, using algebraic techniques, that the line digraph L K d+1 of the complete digraph Kd+J is the only [1, 2]-digraph of degree d, that is to say the only digraph -up to isomorphismswhose adjacency matrix A fulfills the equation A + A 2 = J, where J denotes the all-one matrix. As a consequence, we deduce that there does not exist any other almostMoore digraph of diameter k = 2 with all selfrepeat vertices apart from Kautz digraph. In addition, the cycle structure of a [k 1, k]-digraph is studied. Thus, a formula that provides the number of short cycles (cycles of length :::; k) of such a digraph is obtained. From this formula, using graphical arguments, the enumeration of [k 1, k]-digraphs of degree 2 and diameter not greater than 4 is concluded.