Divisibility conditions in almost Moore digraphs with selfrepeats

Moore digraph is a digraph with maximum out-degree d, diameter k and order M d , k = 1 + d + … + d k . Moore digraphs exist only in trivial cases if d = 1 (i.e., directed cycle C k ) or k = 1 (i.e., complete symmetric digraph). Almost Moore digraphs are digraphs of order one less than Moore bound. We shall present new properties of almost Moore digraphs with selfrepeats from which we prove nonexistence of almost Moore digraphs for some k and d.