Date of Award

May 2018

Degree Type


Degree Name

Doctor of Philosophy (PhD)




Jack E. Graver


Carbon, Fullerene, Graph Theory

Subject Categories

Physical Sciences and Mathematics


Fullerenes can be considered to be either molecules of pure carbon or the trivalent plane graphs with all hexagonal and (exactly 12) pentagonal faces that models these molecules. Since carbon atoms have valence 4 and our models have valence 3, the edges of a perfect matching are doubled to bring the valence up to 4 at each vertex. The edges in this perfect matching are called a Kekule structure and the hexagonal faces bounded by three Kekule edges are called benzene rings. A maximal independent (disjoint) set of benzene rings for a given Kekule structure is called a Clar set, and the maximum possible size of a Clar set over all Kekule structures is the Clar number of the fullerene. For any "patch" of hexagonal faces in the fullerene away from all pentagonal faces, there is a "perfect" Kekule structure: a Kekule structure for which the faces of an independent set of benzene rings are packed together as tightly as possible. Starting with such a patch and extending it as far as possible results in a "perfect" Kekule structure except for isolated regions, called clusters, containing the pentagonal faces. It has been shown that clusters must contain even numbers of pentagonal faces. It has also been shown that the Kekule structure of the patch can be extended into each of these clusters to give a full Kekule structure. However, these Kekule extensions will not admit as tightly packed benzene rings as in the patch external to the clusters. A basic problem in computing the Clar number of a fullerene is to make these extensions in a way that maximizes the number of benzene rings in each cluster. The simplest case, that of 2-clusters, has been completely solved. This thesis is devoted to developing a complete understanding of the Clar structures of 4-clusters.


Open Access