Smallest graphs with given automorphism group

TL;DR

This paper investigates the minimal size of graphs with a given automorphism group, establishing upper bounds and computing exact values for specific groups, advancing understanding of graph symmetries.

Contribution

It proves an upper bound on the smallest graph size for a given automorphism group, identifies exceptions, and computes exact values for previously unknown cases.

Findings

Proved that lpha(G) |G| with specific exceptions.

Identified four infinite families and 17 small groups as exceptions.

Computed lpha(G) for groups where the value was previously unknown.

Abstract

For a finite group $G$, denote by $\alpha(G)$ the minimum number of vertices of any graph $\Gamma$ having $\text{Aut}(\Gamma)\cong G$. In this paper, we prove that $\alpha(G)\leq |G|$, with specified exceptions. The exceptions include four infinite families of groups, and 17 other small groups. Additionally, we compute $\alpha(G)$ for the groups $G$ such that $\alpha(G)> |G|$ where the value $\alpha(G)$ was previously unknown.