Graded hyperideals-based graphs on weak graded multiplicative hyperrings

In this paper, we introduce the notion of weak monoid-graded multiplicative hyperrings and present their properties, special the relation to monoid-graded multiplicative hyperrings. We bring forward the concept of an intersection graph of graded hyperideals of monoid-graded multiplicative hyperrings. Indeed, for any given monoid and multiplicative hyperring, it is introduced the notion of weak monoid-graded multiplicative hyperring, so concerning any non-avoided subset of given multiplicative hyperring, we established special nontrivial graded hyperideals as the vertices and any two vertices are adjacent if have a nonzero intersection. We proved that the graded hyperideals of this class of weak monoid-graded multiplicative hyperring are cyclic and computed the set of all graded hyperideals of any finite weak monoid-graded multiplicative hyperrings. For any given natural number, based on the fundamental theorem of arithmetic, and any given prime, we characterize the intersection graph of graded hyperideals of finite monoid-graded multiplicative hyperrings.