Constructions of graphs with any possible two-fold automorphism and automorphism groups

TL;DR

This paper constructs graphs with specific automorphism groups, including any semisimple product with Z2, and explores properties of their canonical double covers, expanding understanding of graph symmetries.

Contribution

It introduces methods to construct connected nonbipartite graphs with canonical double covers having any desired automorphism group, including semisimple products with Z2.

Findings

Constructed graphs with automorphism groups as any semisimple product of Z2 and a given group

Showed that the canonical double cover of asymmetric graphs has an abelian automorphism group of odd order

Provided examples of asymmetric graphs with any specified automorphism group

Abstract

It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected nonbipartite and vertex determining graphs whose canonical double covers have auromorphisms group isomorphic to any semisimple product of $\mathbb{Z}_2$ with any abstract group $H$. Later we show, that the canonical double cover of any asymmetric graph have abelian automorphisms group of odd order. The above construction provides an example of asymmetric graph for any such group. By modifying the aforementioned construction we obtain graphs which have any possible number and type of graphs with isomorphic double covers.