Generating topologies using edges and vertices in graphs and some applications
The issue of topologizing discrete structures is highlighted by several researches. In that regards, graph theory is one of the major aspects of discrete structures, and the topological graph theory is a crucial branch of it. The investigation of topology on graphs is motivated by the embedding o...
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| Format: | Thesis |
| Language: | English |
| Published: |
2018
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/83578/1/FS%202018%20104%20-%20ir.pdf |
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| Summary: | The issue of topologizing discrete structures is highlighted by several
researches. In that regards, graph theory is one of the major aspects of discrete
structures, and the topological graph theory is a crucial branch of it. The
investigation of topology on graphs is motivated by the embedding of digital
images in a discrete space, interpreted as a graph. Applications in various
aspects had been found for topology on graphs, such as in digital geometry,
contractions, and strong maps.
In this study, a combination between graph theory and topology has been
made. The research adopted a new approach in the investigation of topology
on graphs. This is through studying topology on the set of edges of different
undirected graphs. It encompasses both simple and non-simple graphs such
as multigraph and pseudograph. A subbasis family is introduced to generate
a topology on the set of edges of undirected graphs, called the edges topology.
Further, properties of this topology are also investigated. In particular,
functions between graphs, connectivity, and dense subsets are discussed in
this topology. A fundamental step towards studying some properties of
undirected graphs by their corresponding topological spaces is displayed.
Additionally, in this research, the new approach is applied to directed graphs
by introducing two subbases families to generate two non-similar topologies
on the set of edges of any directed graph, called compatible and incompatible
edges topologies. Furthermore, the characteristics of these topologies were
examined in detail. The relation between directed graphs and their
corresponding topologies is presented as well.
In the same vein, the present study generalised the graphic topology defined
on the set of vertices of any locally finite simple graph in which every vertex
has a finite degree. This is done by presenting a subbasis family to generate a
new topology on the set of vertices of simple graphs with vertices of
finite/infinite degree, which is called the incidence topology. Accordingly, this
study investigated the properties of the incidence topology and made a useful
comparison between the two topologies. Moreover, by considering the
graphic topology and the incidence topology, this research explored
bitopological space on the set of vertices of locally finite simple graphs which
was not studied before. Therefore, properties of this bitopological space were
discussed in detail. The relation between locally finite graphs and their
corresponding bitopological spaces is introduced as well.
Lastly, the edges topology on undirected graphs is used to solve graph
problems. This is through identifying all paths between any two distinct
vertices, determining all spanning trees (or spanning paths), and finding all
Hamilton cycles in simple graphs. In addition, a MATLAB code is written to
represent previous applications and allows them to be appropriate for large
graphs. |
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