On the nonexistence of almost Moore digraphs of diameter four

Almost Moore digraphs appear in the context of the degree/diameter problem as a class of extremal directed graphs, in the sense that their order is one less than the unattainable Moore bound M(d, k) = 1 + d + · · · + d, where d > 1 and k > 1 denote the maximum out-degree and diameter, respectively. So far, the problem of their existence has only been solved when d = 2, 3 or k = 2, 3. In this paper we deal with the case of almost Moore digraphs of diameter k = 4. Their construction turns out to be equivalent to the search of binary matrices A fulfilling that AJ = dJ and I + A + A + A + A = J + P , where J denotes the all-one matrix and P is a permutation matrix . Since the eigenvalues of P are roots of unity, the factorization in Q[x] of the characteristic polynomial of A involves the polynomials Fn(x) = Φn(1+x+x +x+x), where Φn(x) denotes the nth cyclotomic polynomial. More precisely, if Fn(x) is irreducible in Q[x] then it is a factor of det(xI − A) and its multiplicity only depends on the cycle structure of P . We conjecture that Fn(x) is always irreducible in Q[x], unless n = 1, 3, 6. Under this assumption, we show how to derive the nonexistence of almost Moore digraphs of diameter four. Right now, by using tools from algebraic number theory, we have been able to prove that Fn(x) for n 6= 1, 3, 6 is either irreducible or factorizes into two irreducible factors of degree 2φ(n).