Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models

Muhammad Akram; Saba Siddique; Majed G. Alharbi; Muhammad Akram; Saba Siddique; Majed G. Alharbi

doi:10.3934/mbe.2022021

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 1: 420-455. doi: 10.3934/mbe.2022021

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Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models

1.
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan
2.
Department of Mathematics, College of Science and Arts, Al Mithnab, Qassim University, Saudi Arabia

Academic Editor:Antonello Rizzi

Received: 11 September 2021 Accepted: 05 November 2021 Published: 16 November 2021

In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.
- $ m $-polar fuzzy sets,
- strong degree,
- vertex connectivity,
- edge connectivity,
- clustering
Citation: Muhammad Akram, Saba Siddique, Majed G. Alharbi. Clustering algorithm with strength of connectedness for $ m $-polar fuzzy network models[J]. Mathematical Biosciences and Engineering, 2022, 19(1): 420-455. doi: 10.3934/mbe.2022021

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Abstract

In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.

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