An Application of Fuzzy Root Square Mean Labeling to Multi Path Networks

Abstract

Multi-path networks play a crucial role in communication, transportation, and reliability systems, where multiple internally disjoint paths exist between two terminals. In real-world applications, uncertainty in node availability and link reliability motivates the use of fuzzy graph models. This paper introduces Fuzzy Root Square Mean Labeling (FRSML) as an extension of classical root square mean labeling under fuzzy conditions. Theta-related graphs are employed as canonical models for multi-path networks. We establish labeling schemes, prove existence results for certain classes of theta graphs, and illustrate the applicability of the proposed method through mathematical examples and diagrams. The study demonstrates that fuzzy root square mean labeling effectively represents uncertainty while preserving the structural characteristics of multi-path networks.