Study on multidimensional fuzzy graphs through modified partial ordering

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 J∞([0,1]). 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 J ∞([0,1]) introduces many notions such as vertex degree, $min-$ vertex degree, $max-$ vertex degree, path strength, etc.