\(\mathcal{P}\)-topological spaces in simple graphs

Volume 40, Issue 3, pp 330--340 https://dx.doi.org/10.22436/jmcs.040.03.03

Publication Date: July 08, 2025 Submission Date: March 04, 2025 Revision Date: March 09, 2025 Accteptance Date: May 19, 2025

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Authors

H. Alzubaidi - Department of Mathematics, AL-Qunfudhah University College, Umm Al-Qura University, KSA. D. Djurcic - Faculty of Technical Sciences Cacak, University of Kragujevac, 32102 Cacak, Serbia. L. D. R. Kocinac - Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia. H. A. Othman - Mathematics Department, Albaydha University, Albaydha, Yemen. - Department of Information Technology, 21 September University of Medical and Applied Sciences, Sana’a, Yemen.

Abstract

This paper introduces and develops the concept of topological spaces within graph theory, with a particular focus on pathing vertices and \(\mathcal{P}\)-topological spaces. We define pathing vertices as those that facilitate the formation of paths within a graph, enabling the creation of \(\mathcal{P}\)-topological spaces. Our research presents key contributions, including the proof of openness properties in these topologies and the establishment of relationships between homeomorphisms in \(\mathcal{P}\)-topological spaces and graph isomorphisms, particularly in the context of connectedness. Furthermore, we explore the application of \(\mathcal{P}\)-topological spaces in the study of \(\mathcal{N}\)-star graphs, demonstrating their utility in understanding graphic topological structures. This work significantly advances the integration of topological concepts in graph theory, offering new insights and methodologies for future research.

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Keywords

• Simple graph
• \(\mathcal P\)-topology
• continuity
• connectedness
• density

MSC

•  05C20
•  05C99
•  65C10
•  54D99

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