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

Peteris Daugulis

Published 2015-01-27Version 1

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 sets a new upper bound for minimal number of vertices of undirected graphs having automorphism group $\mathbb{Z}/2^n\mathbb{Z}$.