On the nonexistence of almost Moore digraphs

Digraphs of maximum out-degree at most d>1d>1, diameter at most k>1k>1 and order N(d,k)=d+⋯+dkN(d,k)=d+⋯+dk are called almost Moore   or (d,k)(d,k)-digraphs  . So far, the problem of their existence has been solved only when d=2,3d=2,3 or k=2,3,4k=2,3,4. In this paper we derive the nonexistence of (d,k)(d,k)-digraphs, with k>4k>4 and d>3d>3, under the assumption of a conjecture related to the factorization of the polynomials Φn(1+x+⋯+xk)Φn(1+x+⋯+xk), where Φn(x)Φn(x) denotes the nnth cyclotomic polynomial and 1