On the non-existence of odd degree graphs of diameter 2 and defect 2

In 1960, Hoffman and Singleton investigated the existence of Moore graphs for diameter and found that Moore graphs only exist for d = 2, 3, 7 and possibly 57. In 1980, Erd˝os et al. asked the following more general question: Given nonnegative numbers d and ∆, is there a (d, 2, ∆)-graph, that is, a graph of diameter 2, maximum degree d and order d 2 + 1 − ∆?. Erd˝os et al. solved the case for ∆ = 1: the C4 is the only possible graph. In this paper, we consider the next case (∆=2). We prove the nonexistence of such graphs for infinitely many values of odd d and we conjecture that they do not exit for any odd d greater than 5.