2025

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Browsing 2025 by Author "Sana Zaman"

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    Computing local fractional metric dimension of path-wheel graphs
    (UMT Lahore, 2025-06-17) Sana Zaman
    Metric dimension is a useful tool for studying many distance-based issues in the field of electrical networking, robotics, computer networking, integral programming, telecommunication and robotics. The most recently created variant of the metric dimension, known as the fractional metric dimension, is widely applied to non-integral linear programming issues. In this thesis we will compute the local fractional metric dimension (LFMD) of path-wheel graphs (P_W(m,h)), in which h is the number of isomorphic copies of the wheel graph and m+1 is the number of vertices in one wheel W_m. All the obtained results are discussed by the examples of particular graphs belonging to the understudied families of graphs. Graph theory is a field of mathematics that focuses on the study of graphs—structures made up of vertices (or nodes) connected by edges (or links). Graphs can be directed (where edges have a direction) or undirected (where edges have no direction). Graph theory explores the properties, patterns and behaviors of these graphs and is widely used to solve problems involving networks, relationships and connections in areas such as computer science, transportation, biology and social sciences. The metric dimension of a graph is the smallest number of selected vertices such that the distances from any other vertex to these selected vertices uniquely identify it. These selected vertices form what’s called a resolving set. For every pair of different vertices in the graph, there must be at least one vertex in the resolving set that has a different distance to each of them. The size of the smallest such set is the metric dimension of the graph. The local fractional metric dimension of a graph is a variation of the metric dimension that focuses only on adjacent pairs of vertices and allows the use of fractional weights instead of whole numbers when choosing resolving vertices. In this concept, each vertex is assigned a weight between 0 and 1. The sum of weights for vertices that can distinguish every pair of neighboring (adjacent) vertices must be at least 1 for each such pair. The local fractional metric dimension is the smallest possible total weight of all the vertices under these conditions.