Analysis of Fuzzy Derivative Graphs and Their Product Graphs

This paper aims to explore and develop innovative multi copies fuzzy derived graphs and their corresponding product graphs. Derived graphs are characterized by vertices weighted based on the ratio of a generalized exponential function, while edges are identified through the differentiation and integration of this function. This study provides a comprehensive analysis of the regularity and irregularity properties of multi copies fuzzy product graphs, supported by illustrative examples. Key aspects investigated include the impact of various generalized exponential functions on vertex weighting, the influence of different differentiation and integration techniques on edge identification, and the topological properties of multi copies fuzzy derived graphs. Additionally, a comparative analysis is conducted to highlight the advantages and applications of multi copies fuzzy derived graphs in contrast to traditional graph structures. Through detailed case studies, the practical applications of multi copies fuzzy derived graphs in fields such as network analysis, data clustering and decision-making processes are demonstrated. This research contributes to the existing body of knowledge in fuzzy graph theory and lays the groundwork for future studies and applications in this domain.