Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization
Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weig...
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MDPI AG
2024-10-01
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| Series: | Mathematics |
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| author | Ali N. A. Koam Muhammad Faisal Nadeem Ali Ahmad Hassan A. Eshaq |
| author_facet | Ali N. A. Koam Muhammad Faisal Nadeem Ali Ahmad Hassan A. Eshaq |
| author_sort | Ali N. A. Koam |
| collection | DOAJ |
| description | Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems. |
| format | Article |
| id | doaj-art-a3a88a2199314261ba6a64ae36d691eb |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-a3a88a2199314261ba6a64ae36d691eb2024-11-08T14:37:46ZengMDPI AGMathematics2227-73902024-10-011221339710.3390/math12213397Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and OptimizationAli N. A. Koam0Muhammad Faisal Nadeem1Ali Ahmad2Hassan A. Eshaq3Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi ArabiaDepartment of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, PakistanDepartment of Computer Science, College of Engineering and Computer Science, Jazan University, Jazan 45142, Saudi ArabiaDepartment of Educational Sciences, Faculty of Arts and Humanities, Jazan University, Jazan 45142, Saudi ArabiaGraph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.https://www.mdpi.com/2227-7390/12/21/3397weighted asymmetry indexnetwork analysisdistance-based indicesoptimizationcomplex systemsasymmetry in networks |
| spellingShingle | Ali N. A. Koam Muhammad Faisal Nadeem Ali Ahmad Hassan A. Eshaq Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization Mathematics weighted asymmetry index network analysis distance-based indices optimization complex systems asymmetry in networks |
| title | Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization |
| title_full | Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization |
| title_fullStr | Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization |
| title_full_unstemmed | Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization |
| title_short | Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization |
| title_sort | weighted asymmetry index a new graph theoretic measure for network analysis and optimization |
| topic | weighted asymmetry index network analysis distance-based indices optimization complex systems asymmetry in networks |
| url | https://www.mdpi.com/2227-7390/12/21/3397 |
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