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Paper IPM / M / 16991  


Abstract:  
We study bipartite distanceregular Cayley graphs with diameter three or four. We give sufficient conditions under which a bipartite Cayley graph can be constructed on the semidirect product of a group  the part of this bipartite Cayley graph which contains the identity element  and Z2. We apply this to the case of bipartite distanceregular Cayley graphs with diameter three, and consider cases where the sufficient conditions are not satisfied for some specific groups such as the dihedral group.
We also extend a result by Miklavic and Potocnik that relates difference sets to bipartite distanceregular Cayley graphs with diameter three to the case of diameter four. This new case involves certain partial geometric difference sets and  in the antipodal case  relative difference sets.
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