*Article* **Semihypergroup-Based Graph for Modeling International Spread of COVID-***n* **in Social Systems**

**Narjes Firouzkouhi 1, Reza Ameri 2, Abbas Amini 3,4 and Hashem Bordbar 5,\***


**Abstract:** Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroupbased graph (SBG) of *G* = &*H*, *E*' is constructed through the fundamental relation *γ<sup>n</sup>* on *H*, where semihypergroup *H* is appointed as the set of vertices and *E* is addressed as the set of edges on SBG. Indeed, two arbitrary vertices *x* and *y* are adjacent if *xγny*. The connectivity of graph *G* is characterized by *xγ*∗*y*, whereby the connected components SBG of *G* would be exactly the elements of the fundamental group *H*/*γ*∗. Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup *H* and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-*n* in social systems (i.e., countries and individuals).

**Keywords:** graph theory; hypergroup; fundamental relation; social systems; geometric space

**MSC:** 05C25; 20N20
