Diameter of Fullerene Graphs with Full Icosahedral Symmetry

Fullerene graphs are 3-connected cubic planar graphs with only pentagonal and hexagonal faces. A class of fullerene graphs with icosahedral symmetry is called icosahedral fullerenes, and they are determined by a vector (i, j), 0 ≤ i ≤ j and j> 0. For i = j or i = 0, the fullerenes are of full icosahedral symmetry. We show that the diameter of an (i, j)-icosahedral fullerene graph of full icosahedral symmetry is 4i +6 j − 1. We conclude the paper by stating a conjecture on the diameter of any fullerene graph.