Connectivity number in fuzzy graphs with application in mobile communication networks

This paper explores the concept of connectivity within fuzzy graphs, extending classical graph theory to model uncertainty in real-world networks. We introduce the Connectivity Number (CN) as a novel and computationally efficient metric to assess network robustness. CN quantifies the minimum strength of connectedness for any vertex in a fuzzy graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{G} = (\nu, \mu)$$\end{document}, identifying the weakest links that influence overall stability. Theoretical properties of CN are established in various fuzzy graph structures, including strong fuzzy graphs, fuzzy cycles, and complete fuzzy graphs. We further investigate the behavior of CN under six fundamental fuzzy graph operations: Cartesian product, normal product, tensor product, composition, lexicographic max product, and lexicographic min product, revealing critical insights into connectivity in composite structures. Furthermore, we demonstrate the practical relevance of CN by applying it to a mobile communication network, where it effectively identifies weak connections and informs optimization strategies. Compared to existing indices such as the Wiener index (WI) and the connectivity index (CI), CN provides a simpler and more direct measure of connectivity weaknesses, making it a valuable tool for network optimization, resilience analysis, and decision-making. Future research will explore CN in dynamic fuzzy graphs, examine its stability under network perturbations, and develop efficient algorithms for large-scale applications.