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Browsing MS or Mphil by Author "Adeem Ahmad Bashir"

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    Classes of cycle related graphs with constant edge metric dimension
    (UMT Lahore, 2023-01-19) Adeem Ahmad Bashir
    Graph theory serves as a powerful tool for understanding the structural intricacies of various systems, and in this thesis, we delve into the complex dimensions of this mathematical domain. Specifically, our exploration centers around the concept of edge metric dimension within connected graphs, aiming to unravel its complexities and implications. We begin by defining a connected graph R = (V (R), E(R)), where each vertex m, n ∈ V (R) is associated with the minimum path length d(m, n). This fundamental metric provides the basis for understanding the spatial relationships between different nodes in the graph. Extending our inquiry to edges, let e = mn ∈ E(R) be an edge, and μ ∈ V (R) be a vertex. We introduce the minimum distances d(μ, e) and d(e, μ) as crucial measures, representing the shortest paths from μ to the vertices m and n respectively, enhancing our comprehension of the graph’s connectivity. The thesis introduces an innovative idea that involves vertices μ distinguishing between edges e and e∗ based on the inequality d(e, μ) ̸ = d(e∗, μ). This introduces an additional layer of discrimination within the graph, contributing to the understanding of its topology. To systematically characterize this distinction, a set WE = {ω1, ω2, ..., ωp} within V (R) is utilized. The resulting characterization r(e/WE ) of an edge e with respect to WE becomes a p-tuple (d(e, ω1), d(e, ω2), ..., d(e, ωp)). This tuple provides a detailed representation of how the edge is identified from different vertices, surrounding the complex relationships within the graph. A key contribution of this work is the introduction of an edge metric generator WE , which serves as a set uniquely characterizing each edge in R. The minimum cardinality of this generator, referred to as the edge metric basis edim(R), becomes a key metric in our analysis. We explore into the detailed exploration of edge metric dimensions, focusing on various uni-cyclic and bi-cyclic graphs. For instance, structures like C′ M,ę,6 and C′ M,6 are examined, revealing intriguing dimensions of 2 and 3 respectively.