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 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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