$16$-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.