Twins and Co-Twins in Circulant graphs

Abstract

Circulant graphs are a widely studied family of graphs whose members possess varying amounts of symmetry. Although considerable progress has been made in finding the automorphism groups of circulant graphs under certain restrictions, a complete classification is elusive. In general, the structure of the automorphism group of a graph with twins can be simplified by separating the effect of automorphisms that permute mutually twin vertices and those that operate on the twin quotient graph. Further simplification can be achieved in twin-free, vertex-transitive graphs that have co-twins, which we define to be vertices whose neighborhoods are complementary. In this paper, we demonstrate how the these simplifications can be used provide insight into the automorphism groups and symmetry parameters of vertex-transitive graphs in general and circulant graphs in particular.