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Abstract
A Cayley graph Γ of a group G is a graphical doubly regular representation (GDRR) of the group G if Aut Γ is generated by the left and the right regular representations L( G) and R( G) of G, and by the involution g↦ g −1 on G. Examples and properties of GDRRs and their automorphism groups are studied. The problem of determining groups having a GDRR is considered, and some obstructions for a group to have a GDRR are found. Necessary and sufficient conditions for a graph to be a GDRR of two nonisomorphic groups are given. Further, disconnected GDRRs are determined, and imprimitive block systems of GDRRs are characterized.
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