Nonexistence of Almost Moore Digraphs of Diameter Three

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 + · · · + dk, 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. In this paper, we prove that almost Moore digraphs of diameter k = 3 do not exist for any degree d. The enumeration of almost Moore digraphs of degree d and diameter k = 3 turns out to be equivalent to the search of binary matrices A fulfilling that AJ = dJ and I +A+A2+A3 = J +P , where J denotes the all-one matrix and P is a permutation matrix . We use spectral techniques in order to show that such equation has no ∗Partially supported by the Ministry of Science and Technology, Spain, under the projects TIC200309188, MTM2006-15038-C02-02 and MTM2007-66842-C02-02. the electronic journal of combinatorics 15 (2008), #R87 1 (0, 1)-matrix solutions. More precisely, we obtain the factorization in Q[x] of the characteristic polynomial of A, in terms of the cycle structure of P , we compute the trace of A and we derive a contradiction on some algebraic multiplicities of the eigenvalues of A. In order to get the factorization of det(xI−A) we determine when the polynomials Fn(x) = Φn(1 + x + x 2 + x3) are irreducible in Q[x], where Φn(x) denotes the n-th cyclotomic polynomial, since in such case they become ‘big pieces’ of det(xI −A). By using concepts and techniques from algebraic number theory, we prove that Fn(x) is always irreducible in Q[x], unless n = 1, 10. So, by combining tools from matrix and number theory we have been able to solve a problem of graph theory.