Leapfrog fullerenes and Wiener index

Fullerene graphs are cubic, 3-connected planar graphs with only pentagonal and hexagonal faces. A fullerene is called a leapfrog fullerene, Le(F), if it can be constructed by a leapfrog transformation from other fullerene graph F. Here we determine the relation between the Wiener index of Le(F) and the Wiener index of the original graph F. We obtain lower and upper bounds of the Wiener index of Lei(F) in terms of the Wiener index of the original graph. As a consequence, starting with any fullerene F, and iterating the leapfrog transformation we obtain fullerenes, Lei(F), with Wiener index of order O(n2.64) and Ω(n2.36), where n is the number of vertices of Lei(F). These results disprove Hua et al. (2014) conjecture that the Wiener index of fullerene graphs on n vertices is of order Θ(n3).