Department of Mathematics

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Author: search.filters.author.Dr. Hani ShakerSubject: search.filters.subject.SP21

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    Metric Dimension of Wheel Based Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muzammal Ashraf; CIIT/SP21-BSM-037/LHR; Dr. Hani Shaker; LHR TP 9894
    The concept of metric dimension is essential for solving a variety of structural and opti- mization issues. The smallest size of a set of vertices that allows the original vertex to be uniquely determined by the shortest distances from any vertex to these selected vertices is known as the graph’s metric dimension. This idea finds use in the construction of effective routing algorithms, network navigation, and locating issues. In this study, we investigate the metric dimension of different classes of graphs, such as trees, pathways, cycles, and more intricate structures like grids and hypercubes. We explore methods for computing the metric dimension and present novel approaches to obtain accurate values or constraints for particular graph families. The study sheds light on the behavior of the metric dimen- sion under various graph settings, emphasizing how it varies with respect to graph factors including degree, vertex connectivity, and diameter. Furthermore, we examine the compu- tational difficulty of figuring out the metric dimension and suggest effective strategies for real-world uses. In order to provide more efficient solutions for graph-based problems in practical settings, we seek to optimize the determination of metric dimensions by utilizing recent developments in computational techniques.
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    Metric Dimension of Related Wheel Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muhammad Zubair Aslam; CIIT/SP21-BSM-033/LHR; Dr. Hani Shaker; LHR TP 9893
    The metric dimension of a graph is a central concept in graph theory, which deals with de- termining the minimum number of vertices required to uniquely identify every other vertex in the graph through their distances to the selected set of vertices, known as a resolving set. This thesis explores the metric dimension of related wheel graphs, a family of graphs ob- tained by attaching a cycle graph to a central vertex, thus forming a wheel like structure. We examine the metric dimension in both classical and related wheel graphs, where modifica- tions such as additional connections or altered structures are introduced. Through a detailed analysis, we aim to provide bounds on the metric dimension for different classes of related wheel graphs, propose efficient algorithms for finding the resolving set, and compare the metric dimension of these graphs with other well-studied graph families. The results of this study contribute to a deeper understanding of the interplay between graph structure and metric dimension, with potential applications in network design, sensor placement and graph labeling.
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    Metric Dimension of Related Wheel Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Zubair Aslam; SP21-BSM-033; Dr. Hani Shaker; LHR TP 9893
    The metric dimension of a graph is a central concept in graph theory, which deals with de termining the minimum number of vertices required to uniquely identify every other vertex in the graph through their distances to the selected set of vertices, known as a resolving set. This thesis explores the metric dimension of related wheel graphs, a family of graphs ob tained by attaching a cycle graph to a central vertex, thus forming a wheel like structure. We examine the metric dimension in both classical and related wheel graphs, where modifica tions such as additional connections or altered structures are introduced. Through a detailed analysis, we aim to provide bounds on the metric dimension for different classes of related wheel graphs, propose efficient algorithms for finding the resolving set, and compare the metric dimension of these graphs with other well-studied graph families. The results of this study contribute to a deeper understanding of the interplay between graph structure and metric dimension, with potential applications in network design, sensor placement and graph labeling.