A comprehensive study on m-polar picture fuzzy graphs and its application

In real-life problems, information collected from multiple agents often exhibits “multipolar information,” which cannot be accurately modeled by existing graph models. This paper introduces the [Formula: see text]-polar picture fuzzy graph (mPPFG) as a more efficient model for such problems. The mPPFG offers a precise, flexible and comparable system compared to other graph models. The paper defines the mPPFG model and introduces new terms and properties such as order, degree, weighted arithmetic average (WAA) order, WAA degree, complement, homomorphism, isomorphism, eccentricity, radius, diameter and more. It also explores the concepts of [Formula: see text]-polar antipodal picture fuzzy graphs, eccentric mPPFG, self-centered mPPFG and self-median mPPFG, highlighting their definitions and properties. The [Formula: see text]-polar antipodal picture fuzzy graph represents a graph where each vertex is associated with an [Formula: see text]-polar antipodal picture fuzzy set. The eccentric mPPFG focuses on highly eccentric vertices, while the self-centered mPPFG considers central vertices. The self-median mPPFG represents median vertices in the graph. The study of these graph types and their isomorphism properties contributes to a deeper understanding of the structural variations within mPPFGs. Furthermore, the paper proposes an application of the mPPFG model in evaluating the performance of ministers in India during their tenure. An algorithm is also developed for computing the performance of ministers.