# $16$-vertex graphs with automorphism groups $A_{4}$ and $A_{5}$ from icosahedron

## TL;DR

This paper constructs two 16-vertex graphs with automorphism groups A4 and A5, improving known bounds by leveraging the icosahedron's structure.

## Contribution

It presents explicit 16-vertex graphs with automorphism groups A4 and A5, surpassing previous bounds and using projectivisation of the icosahedron's vertex-face graph.

## Key findings

- Graphs have automorphism groups A4 and A5
- Improves Babai's bound for A4
- Enhances graphical regular representation bound for A5

## Abstract

The article deals with the problem of finding vertex-minimal graphs with a given automorphism group. We exhibit two undirected $16$-vertex graphs having automorphism groups $A_{4}$ and $A_{5}$. It improves the Babai's bound for $A_{4}$ and the graphical regular representation bound for $A_{5}$. The graphs are constructed using projectivisation of the vertex-face graph of icosahedron.