On bipartite graphs of diameter 3 and defect 2

We consider bipartite graphs of degree ∆ ≥ 2, diameter D = 3, and defect 2 (having 2 vertices less than the bipartite Moore bound). Such graphs are called bipartite (∆, 3,−2)-graphs. We prove the uniqueness of the known bipartite (3, 3,−2)-graph and bipartite (4, 3,−2)graph. We also prove several necessary conditions for the existence of bipartite (∆, 3,−2)graphs. The most general of these conditions is that either ∆ or ∆ − 2 must be a perfect square. Furthermore, in some cases for which the condition holds, in particular, when ∆ = 6 and ∆ = 9, we prove the non-existence of the corresponding bipartite (∆, 3,−2)-graphs, thus establishing that there are no bipartite (∆, 3,−2)-graphs, for 5 ≤ ∆ ≤ 10.