Steiner Geodesic Distance in Fuzzy Graphs with applications in Telecommunication Networks

Abstract

Distance in fuzzy graphs is a fundamental concept crucial in analyzing connectivity patterns and network dynamics. This article aims to extend and generalize the notion of distance beyond pair of vertices by introducing the novel concept of Steiner geodesic distance (SGD) in fuzzy graphs, which serves as a broader extension of the geodesic distance. If a fuzzy graph has n vertices, this approach provides a framework for determining the distance among k vertices, where [Formula: see text], thereby laying the foundation for analyzing interactions among multiple vertices in the network. The Steiner geodesic tree is defined and the bounds of SGD are obtained. An algorithm for finding Steiner geodesic (SG) in [Formula: see text] running time is presented. The notions of SG k - eccentricity, SG k - center, SG k - selfcentered fuzzy graphs and SG k - eccentric set are established and a characterization of SG k - selfcentered fuzzy graphs is provided. The SGD concepts are analyzed on fuzzy graphs including complete fuzzy graph, complete bipartite fuzzy graph, fuzzy cycle and fuzzy trees. The idea of SGD k - matrix is also discussed and proposed an algorithm to distinguish between SG k - selfcentered fuzzy graphs. The article also presents applications of SGD concepts in optimizing fiber optic cable routing and identifying the optimal location for a central command hub in encrypted radio communication networks.