Date of Award

6-1991

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Gary Chartrand

Second Advisor

Dr. Mark A. Johnson

Third Advisor

Dr. Kapoor

Fourth Advisor

Dr. Stoddart

Abstract

Graphs can be used to represent the atomic structure of chemical compounds where the vertices of the graph represent the individual atoms and the edges of the graph represent the valence bonds between a pair of atoms. M. A. Johnson (1991) introduced a graph-theoretic way to represent structural changes in chemical compounds. Thus, certain labelings of graphs called transitional labelings can be thought as representing chemical equations. Associated with these labelings, we introduce a new invariant of a graph G called the transitional value of G. The transitional value of a graph G gives an indication of how dramatic a structural change occurs in a chemical reaction represented by a transitional labeling of G.

Although the problem of determining the transitional value of an arbitrary graph is open, the transitional values of some families of graphs are determined in Chapters II and III. Chapter II is mainly devoted to the study of the existence of optimal transitional labelings of complete graphs.

Transitional labelings of graphs can be classified as polarizations or quasipolarizations. Complete graphs are the only graphs that do not accept polarizations. Thus, the value of a complete graph is achieved only by means of quasipolarizations. In Chapter III we show the existence of noncomplete graphs whose transitional values are achieved only by quasipolarizations. Using an algorithm, we prove that the situation for trees is the opposite, that is, for any tree T it is always possible to find an optimal transitional labeling of T that is a polarization. Therefore, graphs are divided into two classes, one containing the trees and the other containing the complete graphs. Another consequence of the aforementioned algorithm is a lower bound for the transitional value of a tree.

Given a transitional labeling t of a graph G, three graphs are naturally defined. They are the negative graph of t, the positive graph of t, and the linking graph of t. We begin Chapter TV with a discussion about the existence of transitional labelings with prescribed negative graph, positive graph, and linking graph.

Johnson (1991) introduced a formalism to represent chemical reactions pathways. The main goal of Chapter IV is to present this formalism from another point of view. A particular type of transitional labeling plays a fundamental role in the modeling introduced by Johnson. They are called transforms. We characterize transforms by proving that the concepts of transforms and quasipolarizations are equivalent.

Motivated by ideas developed in Chapter TV, we introduce the concepts of cores and induced cores of a graph in Chapter V and present properties and some related ideas.

Comments

Fifth Advisor: Dr. White

Access Setting

Dissertation-Open Access

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