Every three-connected simple planar graph is a polyhedral graph and a cubic polyhedral graph with pentagonal and hexagonal faces is called as a classical fullerene. The aim of this paper is to survey some results about the symmetry group of cubic polyhedral graphs. We show that the order of symmetry group of such graphs divides 240.

Let $G$ be a connected graph. For an edge $e=uv\in E(G)$, suppose $n(u)$ and $n(v)$ are respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$. The Mostar index is a topological index which is defined as $Mo(G)=\sum_{e\in E(G)}f(e)$, where $f(e) = |n(u)-n(v)|$. In this paper, we will compute the Mostar index of a family of fullerene graphs in terms of the automorphism group.