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 = d2 - 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 A Jn = d Jn and A2 + 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 (x2 + x + 1 - d), where fi (x) denotes the minimal polynomial of the Gauss period ζi + over(ζi, -), being ζi a primitive ith 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. © 2008 Elsevier B.V. All rights reserved.
Conde, J., & Gimbert, J. (2009). On the existence of graphs of diameter two and defect two. Discrete Mathematics, 309(10), 3166–3172. https://doi.org/10.1016/j.disc.2008.09.017