Twins and Co-Twins in Circulant graphs

TL;DR

This paper explores the automorphism groups of circulant graphs, focusing on how twins and co-twins influence symmetry, providing new insights into their structure and classification.

Contribution

It introduces methods to analyze automorphism groups of circulant graphs by leveraging twin and co-twin structures, advancing understanding of their symmetry properties.

Findings

Simplified the analysis of automorphism groups using twin and co-twin structures.

Provided insights into the symmetry parameters of vertex-transitive and circulant graphs.

Enhanced classification approaches for automorphism groups of 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.