1 research outputs found
An analysis of the structure of trees and graphs
PhD ThesisThis thesis is concerned with the structure of trees and linear graphs. In particular an attempt is made to relate structure of
these objects to the known methods for counting them.
Although the work described here is essentially not
computer oriented, the generation and decomposition of graphs
and trees by computer is in the background, and so a short
section on the computer representation of the various objects is
included.
Trees are analysed bearing in mind the counting methods due
first to Cayley, and a later method using Polya's classical
theorem of enumerative combinatorial analysis. Various methods
of representation and generation of trees are presented and
compared.
This thesis then goes on to the substantially more
difficult problem of analysing graphs using similar techniques,
and attempts are made to relate the structure of graphs to the
known techniques for enumerating graphs. This involves a more
detailed study of Polya's theorem and an investigation into the
underlying concepts such as permutation groups as they are
applicable to the case under scrutiny. Representations are
developed to aid these investigations.
In the following section of the work, methods are
investigated for the decomposition of a linear graph, and a
number of different decompositions or factorisations are looked
at. One such factorisation considered in some detail is the
problem of extracting a spanning tree from a grapho and the ways
in which the remaining graph or co-tree graph may be
manipulated. The complete decomposition of a graph into trees
may be achieved using these methods, and the concept of the
structure tree of the decomposition is introduced and its
properties explored.
The techniques described have all been implemented, and a
discussion of the problems of the implementation together with
some estimates of timing reguirements is also included.The Science Research Counci1