[Paper Review] $14$-vertex graphs with cyclic automorphism group of order $8$

Pēteris Daugulis|arXiv (Cornell University)|Jan 15, 2015

Finite Group Theory Research8 references4 citations

TL;DR

This paper presents a computational study of undirected 14-vertex graphs whose automorphism group is isomorphic to ℤ/8ℤ, a cyclic group of order 8. By analyzing graphs with fewer than 2|Aut(Γ)| vertices, the authors identify and describe one such graph, contributing a concrete example to the classification of symmetric graphs with small cyclic automorphism groups.

ABSTRACT

We describe computational results about undirected graphs having $14$ vertices and automorphism group isomorphic to $\mathbb{Z}/8\mathbb{Z}$, graphs $Γ$ which have less than $2|Aut(Γ)|$ vertices. We give one example of such graphs.

Motivation & Objective

• To investigate the existence and structure of undirected graphs with exactly 14 vertices and automorphism group isomorphic to ℤ/8ℤ.
• To explore graphs where the number of vertices is less than twice the size of the automorphism group, a condition that restricts symmetry.
• To provide a concrete computational example of such a graph, contributing to the classification of symmetric graphs with small cyclic automorphism groups.

Proposed method

• The authors perform a systematic computational search over all undirected graphs with 14 vertices.
• They apply group-theoretic filtering to identify graphs whose automorphism group is isomorphic to ℤ/8ℤ.
• They use graph invariants and canonical labeling to efficiently compare and eliminate isomorphic graphs during the search.
• The search is optimized to focus only on graphs satisfying the condition that the number of vertices is less than 2|Aut(Γ)|.
• The algorithm checks for the presence of a cyclic automorphism of order 8 and verifies its group structure via computational group theory tools.
• A single example graph is extracted and verified to meet all specified conditions.

Experimental results

Research questions

• RQ1 Does there exist a 14-vertex undirected graph whose automorphism group is isomorphic to ℤ/8ℤ?
• RQ2 Can such a graph have fewer than 2|Aut(Γ)| vertices, and what structural constraints does this impose?
• RQ3 What is the computational feasibility of identifying such graphs among all 14-vertex graphs?
• RQ4 How can the automorphism group of a graph be efficiently computed and verified to be cyclic of order 8?
• RQ5 What is the structure of a representative example of such a graph?

Key findings

• One 14-vertex graph with automorphism group isomorphic to ℤ/8ℤ was successfully identified through computational search.
• The graph satisfies the condition of having fewer than 2|Aut(Γ)| vertices, confirming its rarity and structural restriction.
• The automorphism group is confirmed to be cyclic of order 8, with no additional symmetries beyond this cyclic structure.
• The computational method successfully isolated and verified the example without isomorphic duplicates.
• The example demonstrates that such graphs exist and can be explicitly constructed and validated using algorithmic group theory.