On the existence of graphs of diameter two and defect two

In the context of the degree/diameter problem, the ‘defect’ of a graph represents the difference between the corresponding Moore bound and its order. Thus, a graph with maximum degree d and diameter two has defect two if its order is n = d − 1. Only four extremal graphs of this type, referred to as (d, 2, 2)-graphs, are known at present: two of degree d = 3 and one of degree d = 4 and 5, respectively. In this paper we prove, by using algebraic and spectral techniques, that for all values of the degree d within a certain range, (d, 2, 2)-graphs do not exist. The enumeration of (d, 2, 2)-graphs is equivalent to the search of binary symmetric matrices A fulfilling that AJn = dJn and A +A+(1− d)In = Jn +B, where Jn denotes the all-one matrix and B is the adjacency matrix of a union of graph cycles. In order to get the factorization of the characteristic polynomial of A in Q[x], we consider the polynomials Fi,d(x) = fi(x + x + 1 − d), where fi(x) denotes the minimal polynomial of the Gauss period ζi + ζi, being ζi a primitive i-th root of unity. We formulate a conjecture on the irreducibility of Fi,d(x) in Q[x] and we show that its proof would imply the nonexistence of (d, 2, 2)-graphs for any degree d > 5.