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

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.