On the existence of graphs of diameter two and defect two

Abstract

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 A J n = d J 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 , d ( x ) = f i ( x 2 + x + 1 − d ) , where f i ( x ) denotes the minimal polynomial of the Gauss period ζ i + ζ i ¯ , being ζ i a primitive i th root of unity. We formulate a conjecture on the irreducibility of F i , 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 .