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 main aim of this paper is to combine the connections between graph theory and rough set theory. We created graph ideal ASs by proposing the interior and closure operators, utilizing the concept of graph ideals. With the help of various graphical examples, we applied some topological concepts to the induced graph ideal ASs, including the subspace, continuous functions, lower separation axioms, and connectedness. However, simple directed graphs with or without loops are the ones that are discussed throughout the study. The obtained results are valid for any type of graph: multi-graphs or simple graphs, connected or disconnected graphs, with loops or without loops, and undirected or directed graphs.

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.