γ-Product of Product Fuzzy Graphs

Fuzzy logic is used to address the undetermined nature of most graph-related problems.Product fuzzy graphs are widely applied in various fields, including decision-making theory, computer science, psychology, neurological systems, and artificial intelligence.In this study, a new operation for product-fuzzy graphs, namely -product, is proposed.We demonstrated that the product of two-product fuzzy graphs results in product fuzzy graphs.Sufficient conditions for the -product of two-product fuzzy graphs to be strong were provided, and they were shown to be complete when at least one factor graph is strong.Moreover, we proved that the complement operation is distributive over this product, in contrast to most well-known products.Furthermore, the concept of unbiasedness within the class of product fuzzy graphs was explored, and the necessary and sufficient conditions for -product to be unbiased were established.We demonstrated that the density of the -product of the two unbiased product fuzzy graphs equals the density of each graph.Finally, we proved that the unbiased notion is retained under isomorphism.