(untitled) (Contemporary Mathematics, 2025)
Fuzzy Graphs (FGs) are very useful for modeling systems with uncertain relationships between elements, allowing degrees of membership between edges and vertices. They are used in fields like decision-making, bioscience, information technology, and artificial intelligence. The Picture Fuzzy Graphs (PFGs) are a refined representation of uncertainty in decision-making environments, allowing for the inclusion of positive membership and non-membership degrees, making them useful in areas like medical diagnosis, pattern recognition, and risk assessment. While a Bipolar Picture Fuzzy Graph (BPFG) is a class of FG developed to describe six types of information: positive membership, positive neutrality, positive hesitancy, negative membership, negative neutrality, and negative hesitancy. The BPFGs are a vital approach to examining systems with coexisting support, resistance, neutrality, and reluctance. They are commonly used in domains like medical diagnosis, social networks, and multi-criteria decision-making. A BPFG is an extension of a PFG due to three additional negative functions. A BPFG is better for analyzing ambiguous and inconsistent data related to real-valued problems. In this paper, we define the Maximal Product (MP) and Residue Product (RP) of BPFG with the help of examples and related theorems. We discuss isomorphism and homomorphism of BPFG with the help of theorems. We define the ideas of weak isomorphism and co-weak isomorphism in BPFG. We prove that isomorphism between BPFG is an equivalence relation. At the end, we provide an application of BPFG.