Strongly regular semi-Cayley graphs
This paper studies strongly regular graphs \(G\) on \(2n\) vertices which admit a group of automorphisms \(\Gamma\) of order \(n\) with two orbits of length \(n\) on the vertices of \(G\), which are called semi-Cayley graphs. The Petersen and Hoffman-Singleton graphs provide examples. The authors use the group ring \(Z\Gamma\) to obtain non-existence results when \(\Gamma\) is Abelian. If \(G\) is the Cayley graph for a group which contains an Abelian subgroup \(\Gamma\) of index two then \(G\) is a semi-Cayley graph. This observation allows the authors to apply their theory to prove that the only strongly regular Cayley graphs for dihedral and generalised quaternion groups are the complete bipartite graphs \(K_{n,n}\). They conjecture that strongly regular Cayley graphs for groups with a cyclic normal subgroup of index two are trivial (i.e., complete bipartite). This conjecture is verified for 2-groups with order not equal to 16 or 64.
scientific article; zbMATH DE number 130431
scientific article