Connectivity of fuzzy incidence and influence graphs under Cartesian product

This paper develops incidence and influence fuzzy graph frameworks with emphasis on connectivity in Cartesian product. For the incidence model, we derive a formula for incidence pair connectivity and characterize [Formula: see text]-strong, [Formula: see text]-strong, and [Formula: see text]-incidence pairs, enabling their systematic identification without graphical representation. We further investigate incidence cutvertices and establish that an incidence cutvertex in the Cartesian product is related to a cutvertex in at least one of its factor graphs. In addition, we show that the Cartesian product of two fuzzy incidence blocks is again a fuzzy incidence block and study fuzzy incidence end nodes in the product. For the influence model, we introduce purely influence fuzzy graphs and their Cartesian product, define local and distributive influence pairs, and investigate their connectivity through horizontal, vertical, and hybrid influence paths. We further establish theoretical bounds for the connectivity of influence pairs in the Cartesian product. These results provide a unified theoretical framework for studying incidence and influence based connectivity in Cartesian product of fuzzy graphs.