Linking rings structures and semisymmetric graphs: Cayley constructions

An LR structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR structures were introduced in Potočnik and Wilson (2014) as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we consider algebraic methods of constructing LR structures, using number theory, Cayley graphs, affine groups, abelian groups and fields.