Journals Information
Mathematics and Statistics Vol. 13(3), pp. 136 - 144
DOI: 10.13189/ms.2025.130302
Reprint (PDF) (313Kb)
Edge Metric and Edge Metric Topology on Graph
Roy John 1,*, Athira Chandran 2, G. Suresh Singh 2
1 Department of Mathematics, St.Stephen's College, Pathanapuram-689695, Kerala, India
2 Department of Mathematics, University of Kerala, Thiruvananthapuram-695581, Kerala, India
ABSTRACT
The method for generating a topology on an edge set of a graph using a distance function on the graph's trail involves the conceptualization of a graph as a topological space. This approach employs a distance function defined over the graph's trail, which is a set of edges that form paths between vertices, to induce a topology on the edge set. The first step of the method is to define a distance metric that measures the "closeness" or "similarity" between different edges based on their positions within the graph's trails. This distance function is typically defined in terms of the structural properties of the graph, such as the number of common vertices, the shortest path distance, or the traversal distance between edges. This provides a powerful tool for analysing the structure and behaviour of graphs from a topological perspective. By considering how edges relate to each other through common trails, this method allows for a deeper understanding of the graph's geometric and connectivity properties. Once the distance function is defined, it is used to generate a family of open sets on the edge set by determining which sets of edges are close to one another. This leads to the generation of a topology that satisfies the standard properties of topological spaces, including the existence of open sets, the union and intersection of these sets, and the presence of limit points. In this article, we present a topology on an edge set of a graph by using a distance function on the trail of a graph. We also describe the graphs that produce discrete topology and indiscrete topology. Topologies produced by union, join and corona graphs are also discussed.
KEYWORDS
Edge Metric, Edge Metric Topology, Open Edge Neighbourhood, Closed Edge Neighbourhood, Union, Join, Corona
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Roy John , Athira Chandran , G. Suresh Singh , "Edge Metric and Edge Metric Topology on Graph," Mathematics and Statistics, Vol. 13, No. 3, pp. 136 - 144, 2025. DOI: 10.13189/ms.2025.130302.
(b). APA Format:
Roy John , Athira Chandran , G. Suresh Singh (2025). Edge Metric and Edge Metric Topology on Graph. Mathematics and Statistics, 13(3), 136 - 144. DOI: 10.13189/ms.2025.130302.