Geodesic
Non-existence of bipartite graphs of diameter at least $4$ and defect $2$
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 2, pp. 163-182.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The Moore bipartite bound represents an upper bound on the order of a bipartite graph of maximum degree $\Delta $ and diameter $D$. Bipartite graphs of maximum degree $\Delta $, diameter $D$ and order equal to the Moore bipartite bound are called Moore bipartite graphs. Such bipartite graphs exist only if $D=2,3,4$ and 6, and for $D=3,4,6$, they have been constructed only for those values of $\Delta $ such that $\Delta - 1$ is a prime power. The scarcity of Moore bipartite graphs, together with the applications of such large topologies in the design of interconnection networks, prompted us to investigate what happens when the order of bipartite graphs misses the Moore bipartite bound by a small number of vertices. In this direction the first class of graphs to be studied is naturally the class of bipartite graphs of maximum degree $\Delta $, diameter $D$, and two vertices less than the Moore bipartite bound (defect 2), that is, bipartite ( $\Delta , D$, - 2)-graphs.
Keywords: keywords degree/diameter problem, Moore bipartite bound, Moore bipartite graphs, defect, dickson polynomials of the second kind 
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     title = {Non-existence of bipartite graphs of diameter at least $4$ and defect $2$},
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Pineda-Villavicencio, Guillermo. Non-existence of bipartite graphs of diameter at least $4$ and defect $2$. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 2, pp. 163-182. http://geodesic.mathdoc.fr/item/JAC_2011__34_2_a6/