Further results on almost Moore digraphs

The nonexistence of digraphs with order equal to the Moore bound Md k d d k for d k has lead to the study of the problem of the existence of almost Moore digraphs namely digraphs with order close to the Moore bound In it was shown that almost Moore digraphs of order Md k degree d diameter k d k contain either no cycle of length k or exactly one such cycle In this paper we shall derive some further necessary conditions for the existence of almost Moore digraphs for degree and diameter greater than As a consequence for diameter and degree d d we show that there are no almost Moore digraphs of order Md with one vertex in a C except the digraphs with every vertex in C