On large bipartite graphs of diameter 3

We consider the bipartite version of the degree/diameter problem  , namely, given natural numbers d≥2d≥2 and D≥2D≥2, find the maximum number Nb(d,D) of vertices in a bipartite graph of maximum degree dd and diameter DD. In this context, the bipartite Moore bound Mb(d,D) represents a general upper bound for Nb(d,D). Bipartite graphs of order Mb(d,D) are very rare, and determining Nb(d,D) still remains an open problem for most (d,D)(d,D) pairs.This paper is a follow-up of our earlier paper (Feria-Purón and Pineda-Villavicencio, 2012 [5]), where a study on bipartite (d,D,−4)(d,D,−4)-graphs (that is, bipartite graphs of order Mb(d,D)−4) was carried out. Here we first present some structural properties of bipartite (d,3,−4)(d,3,−4)-graphs, and later prove that there are no bipartite (7,3,−4)(7,3,−4)-graphs. This result implies that the known bipartite (7,3,−6)(7,3,−6)-graph is optimal, and therefore Nb(7,3)=80. We dub this graph the Hafner–Loz graph after its first discoverers Paul Hafner and Eyal Loz.The approach here presented also provides a proof of the uniqueness of the known bipartite (5,3,−4)(5,3,−4)-graph, and the non-existence of bipartite (6,3,−4)(6,3,−4)-graphs.In addition, we discover at least one new largest known bipartite–and also vertex-transitive–graph of degree 11, diameter 3 and order 190, a result which improves by four vertices the previous lower bound for Nb(11,3).