Regular digraphs of diameter 2 and maximum order

It is known that Moore digraphs of degree d > 1 and diameter k > 1 do not exist (see [16] or [4]). For degree 2, it has been shown that for diameter k ~ 3 there are no digraphs of order 'close' to, i.e., one less than, the Moore bound (14). For diameter 2, it is known that digraphs close to Moore bound exist for any degree because the line digraphs of complete digraphs are an example of such digraphs. However, it is not known whether these are the only digraphs close to Moore digraphs. In this paper, we shall consider the general case of directed graphs of diameter 2, degree d ~ 3 and with the number of vertices n = d + d , that is, one less than the Moore bound. U sing the eigenvalues of the corresponding adjacency matrices we give a number of necessary conditions for the existence of such digraphs. Furthermore, for degree 3 we prove that there are no digraphs close to Moore bound other than the line digraph of K 4 • '