Study on multidimensional fuzzy graphs through modified partial ordering - Jomal Josen; Sunil Jacob John; Sovan Samanta; Tofigh Allahviranloo; Baiju, T. (Mathematics in applied sciences and engineering, 2025)
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 $\mathcal{J}_{∞}\big([0,1]\big)$. 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 $\mathcal{J _{∞}\big([0,1]\big)$ introduces many notions such as vertex degree, $min-$ vertex degree, $max-$ vertex degree, path strength, etc.