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.