Samrin AkbarCIIT/SP24-RMT-016/LHRDr. M. Faisal NadeemLHR TP 100772026-05-132025https://repository.cuilahore.edu.pk/123456789/3900Real world networks are frequently described by information that is not only imprecise but also of varying reliability. Classical graphs represent relations crisp, while fuzzy graphs incorporate graded memberships; however, standard fuzzy graph models typically do not distinguish between the imprecision of a statement and the credibility of that statement. Motivated by Zadeh’s Z numbers, this thesis investigates a reliability aware network model called the fuzzy Z-graph. In this framework, vertices and/or edges are labeled by Z-numbers of the form Z = (A,B), where A is a fuzzy restriction describing the primary uncertain value (e.g., strength, cost, risk, or affinity) and B ∈ [0, 1] represents the reliability (confidence) associated with that restriction. We develop a rigorous mathematical foundation for fuzzy Z-graphs by providing pre- cise definitions, consistent notation, and canonical constructions for vertex labeled, edge labeled, and mixed Z-labeling schemes. Suitable aggregation operators and orderings on Z-numbers are introduced to enable comparison and optimization in Z-labeled networks. Fundamental structural properties are then established, including closure behavior under basic graph operations, well-defined Z-degree concepts under admissible aggregation, and path-based calculus with reliability compounding. Algorithmic aspects are also addressed through Z-weight evaluation and scalarization strategies that connect fuzzy Z-graphs to classical combinatorial optimization methods. Finally, the thesis outlines potential applica- tion settings such as risk reliability modeling, decision support, and uncertain infrastructure networks demonstrating how fuzzy Z-graphs offer a principled way to encode both “what is believed” and “how reliably it is believed” within a single graph-theoretic object.enDepartment of MathematicsSP24MathematicsFuzzy Z-graphsFuzzy graph theoryUncertainty modelingDr. M. Faisal NadeemThesis