On minimal graphs for hamiltonian groups and their fixing set

TL;DR

This paper investigates the minimal graph orders with a given hamiltonian group as automorphisms and characterizes the fixing sets for these groups, advancing understanding of symmetry and automorphism properties in graph theory.

Contribution

It computes the minimal orders of graphs with hamiltonian automorphism groups and determines the fixing sets for these groups, providing new insights into their automorphism structures.

Findings

Identified minimal graph orders for hamiltonian groups as automorphism groups

Characterized fixing sets for finite hamiltonian groups

Enhanced understanding of automorphism group structures in graphs

Abstract

A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $\Gamma$ is the minimum cardinality of a subset $S$ of $V(\Gamma)$ such that the stabilizer of $S$ is trivial. For a given finite group $G$, the fixing set is defined as the set comprising all possible fixing numbers of graphs having group $G$ as their automorphism groups. We determine the fixing sets corresponding to finite hamiltonian groups.