Fuzzy graphic and representable matroids duality connectivity and applications

Abstract

Fuzzy matroids mark a significant advancement in mathematical theory, pro- viding an effective framework for modeling complex systems where partial truths and uncertainty prevail. Classical matroid theory often struggles to represent contexts involving vagueness, highlighting the need for more flexible structures. This research focuses on fuzzy graphic and representable matroids, offering a thorough analysis of their properties and applicability in various domains. A comprehensive conceptual basis is developed for fuzzy graphic matroids, emphasizing their distinct characteristics compared to classical graphic matroids. Key findings reveal how bases, circuits, and connectivity vary with different threshold applications, demonstrating the adaptable nature of fuzzy structures. Furthermore, the study of fuzzy representable matroids uncovers important in- sights into their representability, this challenges traditional crisp matroid theory and underscores the unique complexities inherent in fuzzy settings. The thesis also explores the construction of fuzzy duals within fuzzy graphic matroids, offering valuable insights into their connectivity and establishing a link between fuzzy duals and geometric duals. These findings advance the theoretical understanding of dualization processes in fuzzy contexts. The results emphasize the need for advanced approximation techniques in fuzzy matroid theory, particularly for applications in network theory, decision- making, and optimization under uncertainty.