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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 ask for the optimal value of one of them while holding the other two fixed. Here we focus in {\em 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 {\em 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 $q=13$.