#### Date of Award

6-1996

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Gary Chartrand

#### Second Advisor

Dr. Arthur T. White

#### Third Advisor

Dr. Allen Schwenk

#### Fourth Advisor

Dr. Yousef Alavi

#### Abstract

A basic problem in drug design consists of finding a compound that satisfies a spectrum of biological and chemical properties. Although drug design problems are central to pharmaceutical research, statisticians have yet to become involved in this area as these problems are viewed statistically as optimization problems. Before statistical optimization procedures can be defined on these spaces whose points are structures and not vectors, very basic mathematical notions of distance must be defined. Graphs have been used as mathematical models to represent the bonding arrangements of molecules for quite some time. In fact, if some of the methodology used by statisticians for optimization problems could be extended to the metric space consisting of the set of all graphs, with a fixed number of vertices and a fixed number of edges, together with a metric on this set, then some of the problems in drug design might become more accessible.

Various metrics have been studied and some of the results are presented in Chapter I. In this dissertation, we define a metric in terms of some prescribed graph H. For certain choices of the graph H, this metric is a special case of the previously studied metrics in Chapter I. In Chapter II, conditions are described for certain graphs H that allow us to determine those pairs of graphs for which this metric is defined. Further properties of these metric spaces are studied in Chapter III by means of a graphical interpretation.

Another important problem in the area of mathematical chemistry is the determination of a maximum common substructure shared by two molecular compounds. A special type of commonality that two or more graphs share is called a greatest common subgraph. These concepts have been studied extensively, and some of the results are presented in Chapter I. A certain restriction, inherent to drug design, is imposed on these common subgraphs in Chapter IV, namely, that they preserve distance. These concepts are also applied to trees in Chapter IV. Another type of common subgraph, relative to this distance constraint, is studied in Chapter V.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Gavlas, Heather D., "A Graph Theoretic Study of the Similarity of Discrete Structures" (1996). *Dissertations*. 1682.

https://scholarworks.wmich.edu/dissertations/1682

## Comments

Fifth Advisor: Dr. Mark A. Johnson