*4.2. Application 2*

Let *H* be a set of countries with the most reported cases and death in the world. Consider *H* = {USA, Brazil, India, Russia, Mexico, UK, Italy}. Thus, the SBG of *G* = &*H*, *E*' is defined as follows


Introduce the hyperoperation "⊕" for all *x*, *y* ∈ *H*, as follows:

*x* ⊕ *y* = The country or set of countries that causes disease outbreak from country *x* to country *y*

The couple (*H*, ⊕) is a hypergroupoid, as given in Table 4.



Consider the relation *γ* given below:

$$\text{x}\gamma\_n y \Longleftrightarrow \exists (a\_1, \dots, a\_n) \in H^n, \exists \sigma \in S\_n \colon x \in \prod\_{i=1}^n a\_{i\prime} \ y \in \prod\_{i=1}^n a\_{\sigma(i)}$$

we continue the procedure for all elements of *H*, according to Table 4, that is

1*γ*22 ⇐⇒ 1 ∈ 3 ⊕ 1, 2 ∈ 1 ⊕ 3 1*γ*23 ⇐⇒ 1 ∈ 3 ⊕ 5, 3 ∈ 5 ⊕ 3 1*γ*24 ⇐⇒ 1 ∈ 4 ⊕ 1, 4 ∈ 1 ⊕ 4 1*γ*25 ⇐⇒ 1 ∈ 3 ⊕ 5, 5 ∈ 5 ⊕ 3 1*γ*26 ⇐⇒ 1 ∈ 1 ⊕ 6, 6 ∈ 6 ⊕ 1 1*γ*27 ⇐⇒ 1 ∈ 1 ⊕ 7, 7 ∈ 7 ⊕ 1 2*γ*23 ⇐⇒ 2 ∈ 3 ⊕ 4, 3 ∈ 4 ⊕ 3 2*γ*24 ⇐⇒ 2 ∈ 3 ⊕ 4, 4 ∈ 4 ⊕ 3 2*γ*25 ⇐⇒ 2 ∈ 4 ⊕ 5, 5 ∈ 5 ⊕ 4 2*γ*26 ⇐⇒ 2 ∈ 6 ⊕ 2, 6 ∈ 2 ⊕ 6 2*γ*27 ⇐⇒ 2 ∈ 1 ⊕ 7, 7 ∈ 7 ⊕ 1 3*γ*24 ⇐⇒ 3 ∈ 3 ⊕ 4, 4 ∈ 4 ⊕ 3 3*γ*25 ⇐⇒ 3 ∈ 3 ⊕ 5, 5 ∈ 5 ⊕ 3 3*γ*26 ⇐⇒ 3 ∈ 6 ⊕ 3, 6 ∈ 3 ⊕ 6 3*γ*27 ⇐⇒ 3 ∈ 7 ⊕ 3, 7 ∈ 3 ⊕ 7 4*γ*25 ⇐⇒ 4 ∈ 4 ⊕ 5, 5 ∈ 5 ⊕ 4 4*γ*26 ⇐⇒ 4 ∈ 6 ⊕ 7, 6 ∈ 7 ⊕ 6 4*γ*27 ⇐⇒ 4 ∈ 6 ⊕ 7, 7 ∈ 7 ⊕ 6 5*γ*26 ⇐⇒ 5 ∈ 5 ⊕ 6, 6 ∈ 6 ⊕ 5 5*γ*27 ⇐⇒ 5 ∈ 5 ⊕ 7, 7 ∈ 7 ⊕ 5 6*γ*27 ⇐⇒ 6 ∈ 6 ⊕ 7, 7 ∈ 7 ⊕ 6

Therefore, *E* = {*e*1, ... ,*e*21} and the corresponding SBG of *G* is demonstrated in Figure 6b. By applying Proposition 4, the degree of each vertex is |*γ*∗(*z*)| = 6, and *G* is complete, and 6-regular. It also has an Eulerian circuit because of connectivity and has an even degree of each vertex; therefore, graph *G* is Eulerian. The SBG of *G* is connected and Hamiltonian and the relation *γ* is transitive.

## **5. Conclusions**

The neoteric structure of a semihypergroup-based graph (SBG) is established using a fundamental relation to advance the mathematical concept of an algebraic hypercompositional structure, namely the hypergroup, in the form of graph theory. Additionally, to model and analyze the links in social systems, the developed SBG approach is recommended to intuitively simplify the complicated procedure. Some significant characteristics of SBG are proposed, including connected, complete, regular, Eulerian, isomorphism, and Cartesian products along with illustrative examples and graphical attitude. As per the engagement of all nations and individuals after the global COVID-*n* pandemic, the resulting SBG is applied to address the trend of transmission of the coronavirus disease in social systems, particularly countries and individuals. The next phase can be the development of fuzzy SBG and intuitionistic fuzzy SBG with further applicable platforms.

**Author Contributions:** Conceptualization, N.F. and R.A.; methodology, N.F. and R.A.; software, N.F. and H.B.; validation, A.A. and H.B.; formal analysis, N.F.; investigation, N.F. and R.A.; resources, N.F. and R.A.; data curation, N.F.; writing—original draft preparation, N.F.; writing—review and editing, N.F., R.A., A.A. and H.B.; visualization, N.F. and R.A.; supervision, R.A. and A.A.; project administration, N.F.; funding acquisition, H.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We gratefully acknowledge BioRender software, Canada, for assisting in the figure drawing.

**Conflicts of Interest:** The authors declare no conflict of interests.
