Graphs with $2^n+6$ vertices and cyclic automorphism group of order $2^n$

TL;DR

This paper constructs undirected graphs with $2^n+6$ vertices and cyclic automorphism groups of order $2^n$, confirming an upper bound for minimal vertex count for such graphs.

Contribution

It provides explicit constructions of graphs with specified cyclic automorphism groups and verifies an existing upper bound on their minimal vertex number.

Findings

Graphs with $2^n+6$ vertices and automorphism group of order $2^n$ exist for all $n  2

Confirms the upper bound for minimal vertices of graphs with cyclic $2$-group automorphism groups

Supports previous claims about minimal vertex counts for such automorphism groups

Abstract

The problem of finding upper bounds for minimal vertex number of graphs with a given automorphism group is addressed in this article for the case of cyclic $2$-groups. We show that for any natural $n\ge 2$ there is an undirected graph having $2^n+6$ vertices and automorphism group cyclic of order $2^n$. This confirms an upper bound claimed by other authors for minimal number of vertices of undirected graphs having automorphism group $\mathbb{Z}/2^n\mathbb{Z}$.