**1. Introduction**

Graph theory with its systematic structure is applied to different complicated problems such as physical, biological, and social systems. By employing graph theory, social network structures can be modeled and analyzed to provide simplified knowledge of such systems, where nodes (vertices) are users and lines (edges) are the links among users. Graph theory was first proposed by Euler to solve Konigsberg's seven-bridge problem [1]. After that, he established a novel graph structure called an Eulerian graph [2]. The concepts of a complete graph [3] and a bipartite graph was defined along with tree structure and coloring problems [4]. With the integration of graph theory and fuzzy set theory, the notion of fuzzy graph theory was proposed by Kaufmann. Then, this theory was developed by Rosenfeld, where fuzzy relations on fuzzy sets were introduced to improve graph-theoretic concepts (e.g., bridges and trees) [5]. To eliminate new problems in science, especially combinatorics, hypergraph theory was initiated and formulated by Berge [6] as the generalization of graph theory, where the edges are arbitrary subsets of the vertices to effectively analyze and simplify complex relations in various spectra for real-world problems [7].

**Citation:** Firouzkouhi, N.; Ameri, R.; Amini, A.; Bordbar, H. Semihypergroup-Based Graph for Modeling International Spread of COVID-*n* in Social Systems. *Mathematics* **2022**, *10*, 4405. https:// doi.org/10.3390/math10234405

Academic Editor: Patrick Solé

Received: 19 September 2022 Accepted: 17 November 2022 Published: 22 November 2022

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<sup>1</sup> Department of Mathematics, Golestan University, Gorgan 15759-49138, Iran

Algebraic hypercompositional structure theory, with its dynamic multi-valued systems, is enumerated as the extension of a classical algebraic structure. Marty introduced a hyperoperation (hypercomposition) on a nonvoid set *H*, which is a map from *H* × *H* to the power set *P*(*H*) of *H*, such that with associative property and reproductivity, *H* would be hypergroup [8]. Then, the hypercompositional structure theory was improved in terms of theory and applications by Corsini et al. [9]. Freni determined a novel characterization of the derived hypergroup via strongly regular equivalence relation *γ* on a hypergroup *H*, and a binary operation on the quotient set *H*/*γ*<sup>∗</sup> so that *H*/*γ*<sup>∗</sup> is a group with relation *γ*<sup>∗</sup> as a fundamental relation (*γ*<sup>∗</sup> is the transitive closure of *γ* and *H*/*γ*<sup>∗</sup> is the fundamental group) [10,11]. Indeed, a fundamental relation is a powerful gadget for the derivation of universal algebra (group, ring, module, etc.) on algebraic hypercompositional structures as well as fuzzy algebraic hypercompositional structures. The present authors studied and formulated the fundamental relations on the fuzzy hypergroup, fuzzy hyperring, and fuzzy hypermodule, where their fundamental relations have the smallest equivalence relation resulting in their quotients being a group, ring, and module, respectively, [12–14]. In other studies, they appointed the fundamental functor between the category of fuzzy hyperrings (hypermodules) and the category of rings (modules) [15,16].

The relevance between graphs/hypergraphs and hypergroups has been investigated by many scholars such as Corsini [17] and Leoreanu [18]. Farshi et al. studied hypergroups associated with hypergraphs and established a *ρ*-hypergroup with a given hypergraph by describing a relation *ρ* which resulted in the fundamental relation of an *ρ*-hypergroup [19]. Kalampakas et al. surveyed path hypergroupoids, especially commutativity and graph connectivity, along with the directed graph isomorphism classes of *C*-hypergroupoids [20]. Nikkhah et al. developed hypergroups constructed from hypergraphs using a hyperoperation upon the set of vertice degrees of a hypergraph, where the established hypergroupoid is *Hv*-group [21]. Recently, the present authors proposed a Caley graph related to a semihypergroup (hypergroup) with some important features including the category of Cayley graphs and a functor with an application in social networks [22].

With dynamic and potential applications of graph theory in various fields of science, i.e., computer science, linguistics, physics, chemistry, social sciences, biology, mathematics, bioinformatics, etc., many studies have been conducted [23]. For example, Savinkov et al. analyzed and modeled human lymphatic systems via graph theory [24]. The systematic converter derivation/modeling and advanced control in an emerging/challenging power electronics converter was simulated by graph theory as a powerful mathematical structure [25]. Park et al. indicated important insights from complex travel mobility networks with graph-based spatiotemporal analytics [26]. In another work, an effective transductive learning technique was proposed by employing variational nonlocal graph theory for hyperspectral image classification [27]. Recently, the authors presented a soft hypergraph as the generalization of graph theory with the pragmatic application for modeling global interactions in social media networks [28].

The COVID-19 (coronavirus disease 2019) pandemic is considered the most fatal global health catastrophe to date with its serious negative and destructive impact on human life, i.e., social, economical, and environmental challenges. After its detection, the virus extended globally and caused innumerable death. At present, there is no definitive treatment of clinical antiviral drugs or vaccines against the virus [29]. Almost whole nations attempted to decline the transition of the disease via examination and treating patients, quarantining suspected persons through contact tracing, limiting large gatherings, maintaining complete or partial lockdowns, etc. The impact of COVID-19 on various societies and useful ways for controlling viral disease were investigated in [30].

The principal objective of this study is to establish a novel framework of a graph called SBG using a specific relation of algebraic hypercompositional structures in the context of social systems, i.e., the spread trend of the coronavirus disease among societies and individuals. After the Introduction and the Preliminary sections, in Section 3, we appoint a neoteric graph *G* = &*H*, *E*' by applying a fundamental relation *γ*<sup>∗</sup> on a semihypergroup *H*. The elements of *H* are vertices and two vertices *x* and *y* are adjacent if *xγny*, that is, they are considered edges. The connectivity SBG of *G* is defined as *xγ*∗*y*, where the connected components of *G* are precisely the elements of the fundamental group *H*/*γ*∗. Certain fundamental properties of graph theory such as complete, regular, Eulerian, isomorphism, and Cartesian products are proposed. In addition, elucidatory examples are applied to demonstrate the relationship between semihypergroup (hypergroup) *H* and its associated graph. The mathematical notions of geometric space, block, polygonal, and connected components are discussed. In the end, in Section 4, the developed SBG is utilized to model the global outbreak of COVID-*n* in social systems (i.e., individuals as well as countries) (Figure 1).

 

 

**Figure 1.** SBG for modeling global spread of COVID-*n*.
