Connectivity Index of Fuzzy Rough Graphs with Applications to Uncertain Traffic Flow Networks

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.