Date of Award

5-11-2025

Date Published

June 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Jack Graver

Keywords

Clar number;encircling chain;fullerene

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

Fullerenes are purely carbon molecules which can be represented by a 3-regular planar graph consisting of only hexagonal and (exactly 12) pentagonal faces. Since carbon atoms have valence 4 but our graphs are trivalent we need to double the edges in a perfect matching to bring the valence up to 4. The edges in a perfect matching form a Kekulé structure and the hexagonal faces bound by three Kekulé edges are called benzene rings. A maximal independent set of benzene rings for a given Kekulé structure is called a Clar set, and the maximum possible size of a Clar set over all Kekulé structures is the Clar number of the fullerene. A perfect Kekulé structure can be extended through isolated regions, called clusters, which contain the pentagons in a way the creates a chain decomposition. These chain decompositions can be used to help find the Clar number for a given fullerene through the idea of the Clar deficit. In this paper we will look to extend the notion of chains to a new type of chain called an encircling chain. Then, we will look to answer if encircling chains around larger clusters improves the Clar deficit. We will answer this question for the case of 4-clusters. Then, for the case of 6-clusters we will look at examples that get us closer to this answer.

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Open Access

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Mathematics Commons

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