On the nonexistence of almost Moore digraphs

Digraphs of maximum out-degree at most d > 1 , diameter at most k > 1 and order N ( d , k ) = d + ⋯ + d k are called almost Moore or ( d , k ) -digraphs. So far, the problem of their existence has been solved only when d = 2 , 3 or k = 2 , 3 , 4 . In this paper we derive the nonexistence of ( d , k ) -digraphs, with k > 4 and d > 3 , under the assumption of a conjecture related to the factorization of the polynomials Φ n ( 1 + x + ⋯ + x k ) , where Φ n ( x ) denotes the n th cyclotomic polynomial and 1 < n ≤ N ( d , k ) . Moreover, we prove that almost Moore digraphs do not exist for the particular cases when k = 5 and d = 4 , 5 or 6.