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(2) - 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 AJ(n) = dJ(n) and A(2) + A + (1 - d)I-n = J(n) + B, where J(n) 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 F-i,F-d(x) =f(i)(x(2) + x + 1 - d), where f(i)(x) denotes the minimal polynomial of the Gauss period zeta(i) + (zeta) over bar (i), being zeta(i) a primitive ith root of unity. We formulate a conjecture on the irreducibility of F-i,F-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. (C) 2008 Elsevier B.V. All rights reserved.