A Comprehensive Study on Complex Linear Diophantine Fuzzy Graphs and Application in Decision‐Making

The linear Diophantine fuzzy set (LDFS) is a generalization of the complex LDFS (CLDFS), developed as a mathematical framework for optimization, soft computing, and decision‐making. This paper integrates graph theory with CLDFS by introducing complex linear Diophantine fuzzy graphs (CLDFGs), which generalize several existing concepts, including Pythagorean fuzzy graphs (PFGs), intuitionistic fuzzy graphs (IFGs), and linear Diophantine fuzzy graphs (LDFGs). In this work, we extend key properties of classical graph theory to CLDFGs, focusing on operations such as composition, join, union, and the Cartesian product. Each of these operations is demonstrated with examples, and significant results are obtained for the composition, Cartesian product, and join of two CLDFGs. Moreover, we explore homomorphisms in CLDFGs, and equivalence relations via isomorphisms between CLDFGs are defined. In addition, important results regarding the complement of a CLDFG are presented. Finally, we demonstrate an application of CLDFGs in predicting earthquake risks and intensities. Overall, this research strengthens the theoretical foundation of fuzzy graph theory and offers innovative approaches to complex decision‐making, thereby advancing the field of CLDFSs.