Study on multidimensional fuzzy graphs through modified partial ordering

• Published in: Mathematics in applied sciences and engineering Vol. 6; no. 1; pp. 45 - 69
• Main Authors: Jomal Josen, Sunil Jacob John, Sovan Samanta, Tofigh Allahviranloo, Baiju, T.
• Format: Journal Article
• Language: English
• Published: Western Libraries 01.03.2025
• Subjects: Multidimensional fuzzy graphs, edge powered multidimensional fuzzy graphs, operations on multidimensional fuzzy graphs, vertex degrees, path strength
• ISSN: 2563-1926; 2563-1926
• DOI: 10.5206/mase/20391

This paper introduces the concepts of multidimensional fuzzy graphs and edge-powered multidimensional fuzzy graphs, which employ a hybrid structure that combines multidimensional fuzzy sets and graphs. This study redefines the axioms of multidimensional $t-$ norms and $t-$ conorms by providing a more general partial order that can link more components of the range set $\mathcal{J}_{\infty}\big([0,1]\big)$. More studies on various operations such as direct product, composition, tensor product, join, etc. are conducted with relevant illustrations. A novel complement operator approach is also investigated to link the multidimensional fuzzy graph and the edge-powered multidimensional fuzzy graph. Finally, defining the infimum and supremum of an arbitrary family in $\mathcal{J _{\infty}\big([0,1]\big)$ introduces many notions such as vertex degree, $min-$ vertex degree, $max-$ vertex degree, path strength, etc.