Spherical fullerene graphs that do not satisfy Andova and Škrekovski's conjecture

Abstract

A fullerene graph is a planar, cubic, 3-connected graph with only pentagonal and hexagonal faces. In 2012, Andova and Škrekovski conjectured that the diameter of every fullerene graph with n vertices is at least √5n/3 - 1. They computed this lower bound by studying a particular class of fullerene graphs named spherical with icosahedral symmetry. We denote these graphs by Gi,j, by setting two parameters i,jε N*, such that i≤j. In their study, Andova and Škrekovski offered numerous properties of hexagonal lattices and calculated the diameter of two remarkable spherical fullerene graphs: G0,j and Gj,j. Although the conjecture is valid for these two distinct classes, it remains open deciding whether the premise is proper for all spherical fullerene graphs Gi,j.In this work, we present the first class of fullerene graphs with icosahedral symmetry that do not satisfy Andova and Skrekovski's conjecture, which refutes that this conjecture is valid for all spherical graphs. We also focus on showing properties of spherical fullerene graphs and the hexagonal lattice itself. We prove that all graphs Gi,j have a reduction of the form Gi-k,j-k, where k≤i, such that their triangular faces are entirely contained in the triangular faces of Gi,j. In addition, by setting k=i, this property states a particular link among Gi,j, Gi-1,j-1,• • •,G0,j-i, creating a chain of reductions of Gi,j, which implies that diam (Gi,j)≥diam (G0,j-i).