Normal Cayley Graphs

Edward Dobson (Mississippi State University, USA)
     
A Cayley graph $\Gamma$ of a group $G$ is a normal Cayley graph of $G$  if $G_L$, the left-regular representation of $G$, is normal in $\mathrm{Aut}(\Gamma)$,  the full automorphism group of $\Gamma$.  Such graphs were first defined and  studied by Ming-Yao Xu in 1998, and are natural generalizations of the notion  of a GRR or graphical regular representation of a group $G$, which is a  graph whose automorphism group is permutation isomorphic to $G_L$.  Indeed,  normal Cayley graphs that are not GRR's can be thought of as graphs that are  not GRR's but whose automorphism group has as simple a structure as possible,  (in fact such graphs have automorphism group a subgroup of $\mathrm{Aut}(G)\cdot G_L$).  In this talk we will, at least from one point of view, discuss the main problems  regarding normal Cayley graphs, survey known results on normal Cayley graphs,  discuss possible future directions for additional research, and discuss related  problems not only in graph theory, but in permutation group theory as well.