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