In this paper, we introduce the concept of the “graph primal,” which serves as the dual structure to the “graph grill”. We present several results associated with graph primal operations. Moreover, we introduce two new graph-local functions on graph adjacency topological spaces (graph ATSs). We then explore the basic properties of the proposed graph-local functions and describe a method for generating two new graph ATSs from existing ones via graph primals. In addition, we examine several fundamental properties and connections of the resulting topologies, supported by some counterexamples. Furthermore, we characterize the nature of the open sets of these new topologies in terms of closure operators. Finally, we assess the compatibility of graph ATSs with the graph primal concept.

The essential objectives of this study are to propose enhancements and modifications to the bipolar soft rough sets methodology by incorporating ideals. The paper introduces two distinct types of ideal bipolar soft approximation operators, which serve as extensions to the existing bipolar soft rough approximation operator. Furthermore, two approaches are employed to establish and investigate a novel type of bipolar approximation space, referred to as the bi-ideal bipolar soft approximation space. This work also explores the relationships between these proposed techniques and previous methods, detailing their respective characteristics and advantages. By enlarging the ideal bipolar lower approximations and reducing the ideal bipolar upper approximations, these strategies significantly reduce the ambiguity and uncertainty within the decision-making process. The paper additionally outlines several key metrics related to ideal bipolar soft spaces, enriching the theoretical understanding of these structures. A practical application of the proposed spaces is presented in the context of multi-attribute group decision-making (MAGDM) problems. To support this, an algorithm is developed to facilitate the selection of the most optimal alternative from a range of options, accompanied by a practical example to demonstrate its effectiveness. The analysis highlights the reliability, adaptability, and superiority of the proposed MAGDM framework. Furthermore, a concise comparison with existing methodologies is provided, showcasing the advantages and robustness of the proposed approach in addressing complex decision-making challenges.

This paper introduces a connectivity index for fuzzy rough graphs that effectively integrates the mathematical frameworks of fuzzy set theory and rough set theory within graph-theoretic structures. We develop a comprehensive theoretical foundation by constructing fuzzy graph approximation spaces based on fuzzy minimal neighborhoods, enabling the definition of lower and upper approximation operators for both vertices and edges simultaneously. Building upon this framework, we establish rigorous definitions for connectivity measures in fuzzy rough graphs, including strongest directed paths, strength of connectedness, and various arc classifications. The proposed connectivity index quantifies the overall connectedness of fuzzy rough graphs by aggregating vertex membership degrees with path strengths across both approximation boundaries. We demonstrate that this index exhibits monotonicity properties with respect to fuzzy rough subgraphs and provides critical insights into network robustness through bridge identification. To validate the practical utility of our theoretical contributions, we present a compelling application in traffic flow network analysis, where the connectivity index successfully identifies the busiest intersections in urban transportation systems under conditions of uncertainty and incomplete information. The integration of fuzzy rough set theory with graph connectivity measures represents a significant advancement in handling real-world problems where relationships exist with varying degrees of certainty rather than binary connections.