Statistical Central Tendency Measures in Fuzzy Graph Theory: A Unified Analytical, Computational, and Interdisciplinary Framework

This paper presents an analytical study of four central tendency measures Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM), and Median within the framework of fuzzy graph theory, incorporating triangular, trapezoidal, and Gaussian membership functions. Formal definitions and key theorems, including the chain inequality FHM ≤ FGM ≤ FAM, are established with rigorous proofs. A Monte Carlo simulation framework (500 iterations per configuration) is developed to evaluate the statistical behaviour of these measures under varying fuzziness parameters. Results demonstrate that the Harmonic Mean exhibits superior noise-resistance in sparse graphs, the Arithmetic Mean provides the most efficient approximation in dense networks, and the Median is the most robust against membership outliers. The framework is validated through interdisciplinary applications in network reliability, bioinformatics, traffic optimization, epidemiology, and financial portfolio analysis. These findings offer new theoretical insights and practical tools for researchers across mathematics, computer science, and engineering.