Automorphism Group and Fixing Number of (3,6)– and (4, 6)–Fullerene Graphs

The fixing number of a graph Γ is the minimum cardinality of a set S of V (Γ) such that every non-identity automorphism of G moves at least one member of S. In this case, it is easy to see that the automorphism group of the graph obtained from Γ by fixing every node in S is trivial. The aim of this paper is to investigate the automorphism group and fixing number of six families of (3, 6)-fullerene graphs. Moreover, an example of an infinite class G[n] of cubic planar n-vertex graphs is presented in which faces are triangles and hexagons. It is proved that the automorphism group of G[n] has order 2n+2 and fixing number n+1. This shows that by omitting the condition of 3-connectivity in definition of a fullerene graph, the symmetry group can be enough large.