- Associate Professor Michael Giudici
- Adjunct Professor CaiHeng Li

Some problems on the automorphism group of Cayley graphs

The concept of a Cayley graph was first considered for finite groups by A. Cayley in 1878. They are graphs whose vertex set is a group and adjacency relation is defined by a subset of the group.

Cayley graphs play a significant role in many mathematical research fields. In algebraic graph theory they have been used to construct extremal graphs, such as expanders and graphs without short cycles. In computer science they are used in the design of interconnection networks for large interacting arrays of processors. Also they can be used to study combinatorial and geometrical objects.

In 1958, Dr. Sabidussi gave a characterization of Cayley graphs, that a graph is a Cayley graph if and only if its automorphism group contains a subgroup acting regularly on the vertex set of the Cayley graph. In my Ph.D project, we will study Cayley graphs by analysing the regular subgroups of their automorphism groups, and classify some important classes of Cayley graphs with high symmetry degree.

There has been a great deal of work on investigating isomorphic Cayley graphs over non-isomorphic groups. The aim of my Ph.D project will advance the discipline since it will investigate normal and non-normal Cayley graphs over isomorphic groups, and develop a clearer picture of the structure of the automorphism groups of Cayley graphs, in particular it will classify the regular subgroups of the automorphism groups.

- IPRS+APA