On New Record Graphs Close to Bipartite Moore Graphs

The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and optimising one of the parameters given restrictions on some of the others. Here we focus on bipartite Moore graphs, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with local bipartite Moore graphs. We find in some cases those local bipartite Moore graphs with local girths as close as possible to the local girths given by a bipartite Moore graph. Second, we construct a family of (q+2) -bipartite graphs of order 2(q^2+q+5) and diameter 3, for q a power of prime. These graphs attain the record value for q=9 and improve the values for q=11 and q=13 .