Polyhedral graphs via their automorphism groups

A polyhedral graph is a three connected simple planar graph. An automorphism of a graph is a bijection on its vertices which preserves the edge set. In this paper, we compute the automorphism group of cubic polyhedral graphs whose faces are triangles, quadrangles, pentagons and hexagons. In continuing, we classify all cubic polyhedral graphs with Cayley graph structure.