Department of Mathematics

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    Distance Based Invariants on Embedded Wheel Graphs
    (Library Information Services COMSATS University Lahore Campus, 2024-03-17) Muhammad Shoaib Younas; FA22-RMT-039; Prof. Dr. Muhammad Husssain; LHR TP 9382
    This study’s main goal is to investigate and understand the mathematical character istics of topological indices. Graph theory heavily depends on these indices, which are mathematical characteristics that measure the topological attributes of chemical compounds. These indicators have been applied across diverse domains such as drug discovery, property forecasting, molecular categorization, and the design of polymers, among other related fields. Our research specifically concentrates on the computation of distance-based topological indicators, with a particular focus on those derived from the eccentricity of both chemical and non-chemical graphs. Our aim is to explore their correlation with a broad spectrum of physical and chemical properties. We aim to calculate several distance-related topological indicators, like hosoya polynomial H(G, x), winner index W(G), modified winner index Wλ (G), hyper winner index WWλ (G), modified hyper winner index WWWλ (G), schultz index Sc(G) and its modified form S ∗ c(G), schultz polynomial Sc(G, x), modified schultz polynomial S ∗ c(G, x), harary polynomial h(G, x), generalized harary index ht(G), multiplicative winner index π(G), eccentric connectivity index ξ (G) and its polynomial ξ (G, x)
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    Computational aspects of SSC of Wheel Graphs
    (Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Rimsha Amber; CIIT/FA20-BSM-058/LHR; Dr. Imran Ahmed; LHR TP 9923
    Simplicial complexes represent crucial mathematical structures with broad applications. Stanley-Reisner ideals encode combinatorial information about these complexes, bridg- ing combinatorial geometry and commutative algebra. Simplicial complexes, comprised of simplices, capture complex topological and combinatorial relationships, serving as founda- tional tools in numerous mathematical disciplines. Shellable complexes, a significant sub- class, allow systematic decomposition into ”shells,” revealing insights into their topological and homological properties. Studying the spanning simplicial complexes of wheel graphs elucidates the connection between graph theory and algebraic topology. Investigation into the shellability of line simplicial complexes sheds light on their combinatorial intricacies, contributing to a deeper understanding of their geometric and algebraic nature. Through these explorations, we uncover the rich interplay between combinatorial geometry, alge- braic structures, and topological properties inherent in simplicial complexes, facilitating broader comprehension and application across mathematical domains.
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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.