A study on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si864.svg" display="inline" id="d1e2751"> <mml:mi>m</mml:mi> </mml:math> -polar fuzzy influence graph and its application in highway ramping system

The concept of m-polar fuzzy graphs (mPFGs) generalizes fuzzy graph (FG) theory by assigning m membership components to vertices and edges, offering a richer framework than bipolar or intuitionistic FGs. This flexibility enables efficient modeling of complex systems with multiple attributes, driving recent research on new theorems, properties, and examples. Within this framework, the m-polar fuzzy (mPF) incidence graph and m-polar fuzzy influence graph (mPFIG) are introduced, along with notions such as path strength, influence connectivity, α-strong, β-strong, and δ-pair in the mPF environment. These models reformulate vertex–edge relationships in fuzzy incidence and influence graphs, making them suitable for representing natural flow networks, including conditions with illegal flows, with applications in social networks and traffic systems. Connectivity analysis is extended through the study of multimin, locamin, m-polar fuzzy influence (FI) cycles, and special influence vertices, highlighting interactions among influence parameters. To demonstrate practical relevance, a real-world application is presented: modeling highway ramping systems. Here, influence connectivity in an mPF environment effectively captures vehicle flow, including congestion scenarios. Overall, mPFIGs provide a comprehensive framework for analyzing multi-attribute networks.