Abstract
Digraphs of maximum out-degree at most d>1, diameter at most k>1 and order N(d,k)=d+⋯+dk 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 Φn(1+x+⋯+xk), where Φn(x) denotes the nth 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.
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