Bean, Timothy Jackson.

### Abstract:

Aspects of the fundamental concept of distance are investigated in this
dissertation. Two major topics are discussed; the first considers metrics
which give a measure of the extent to which two given graphs are removed
from being isomorphic, while the second deals with Steiner distance in
graphs which is a generalization of the standard definition of distance in
graphs.
Chapter 1 is an introduction to the chapters that follow. In Chapter
2, the edge slide and edge rotation distance metrics are defined. The edge
slide distance gives a measure of distance between connected graphs of the same order and size, while the edge rotation distance gives a measure of distance between graphs of the same order and size. The edge slide and
edge rotation distance graphs for a set S of graphs are defined and investigated.
Chapter 3 deals with metrics which yield distances between graphs
or certain classes of graphs which utilise the concept of greatest common
subgraphs. Then follows a discussion on the effects of certain graph operations on some of the metrics discussed in Chapters 2 and 3. This chapter also considers bounds and relations between the metrics defined in Chapters 2 and 3 as well as a partial ordering of these metrics.
Chapter 4 deals with Steiner distance in a graph. The Steiner distance
in trees is studied separately from the Steiner distance in graphs in general.
The concepts of eccentricity, radius, diameter, centre and periphery are generalised under Steiner distance. This final chapter closes with an algorithm which solves the Steiner problem and a Heuristic which approximates the solution to the Steiner problem.