key: cord-1021046-mu2max8d authors: Sultana, Fazeelat; Gulistan, Muhammad; Ali, Mumtaz; Yaqoob, Naveed; Khan, Muhammad; Rashid, Tabasam; Ahmed, Tauseef title: A study of plithogenic graphs: applications in spreading coronavirus disease (COVID-19) globally date: 2022-04-04 journal: J Ambient Intell Humaniz Comput DOI: 10.1007/s12652-022-03772-6 sha: b3831b69c330ed40ad8d884c3e22fd878839a961 doc_id: 1021046 cord_uid: mu2max8d During the last two decades, the world has experienced three major outbreaks of Coronaviruses, namely severe acute respiratory syndrome (SARS- CoV), middle east respiratory syndrome (MERS-CoV), and the current ongoing pandemic of severe acute respiratory syndrome 2 (SARS-CoV-2). The SARS-CoV-2 caused the disease known as Coronavirus Disease 2019 (COVID-19). Since its discovery for the first time in Wuhan, China, in December 2019, the disease has spread very fast, and cases have been reported in more than 200 countries/territories. In this study, the idea of Smarandache’s pathogenic set is used to discuss the novel COVID-19 spread. We first introduced plithogenic graphs and their subclass, like plithogenic fuzzy graphs. We also established certain binary operations like union, join, Cartesian product, and composition of pathogenic fuzzy graphs, which are helpful when we discuss combining two different graphs. In the end, we investigate the spreading trend of COVID-19 by applying the pathogenic fuzzy graphs. We observe that COVID-19 is much dangerous than (MERS-CoV) and (SARS-CoV). Moreover, as the SARS-CoV and MERS-CoV outbreaks were controlled, there are greater chances to overcome the current pandemic of COVID-19 too. Our model suggests that all the countries should stop all types of traveling/movement across the borders and internally too to control the spread of COVID-19. The proposed model also predicts that in case precautionary measures have not been taken then there is a chance of severe outbreak in future. successfully for many years in problems involving uncertainty, vagueness, ambiguity, and imprecision of state. Further, Zadeh (1975) extended fuzzy sets (FS) to intervalvalued fuzzy sets in which values of membership are intervals instead of real numbers between 0 and 1. Intervalvalued fuzzy sets (IVFS) provide precise results of uncertainty than fuzzy sets (FS). Another researcher Atanassov (Atanassove, 1986) , introduced membership and nonmembership and gave the idea of the intuitionistic fuzzy set (IFS) . It generalized the idea of Zadeh's fuzzy sets. Jun et al. (2010) gave the idea of cubic sets by combining interval-valued fuzzy sets and fuzzy sets. Cubic sets have many applications in many directions (Jun et al. 2011 (Jun et al. , 2012 . Smarandache extended the idea of Atanassov and gave the concept of neutrosophic set (NS) (Smarandache 1999 (Smarandache , 2005 . Further Wang et al. (2005) introduced interval valued neutrosophic sets (INS) . In 2017, Jun et al. (2017a, b) ) presented the idea of neutrosophic cubic sets (NCS) to handle imprecise information. More recently, Smarandache (2017) and Smarandache and Broumi (2018) introduced for the first-time idea of Plithogeny in Philosophy and as its derivatives give the concept of Plithogenic set/logic/probability/statistics in mathematics and engineering. Plithogeny is the origination, formation, development, evolution of new entities and is a connection or combination of theories and ideas in any field. Plithogeny is the dynamics of many opposites, their neutrals and nonopposites, and their organic fusion. The Plithogenic set's theory generalizes previous theories of fuzzy sets. Smarandache introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosoph-ic sets), which is a set whose elements are characterized by many attributes' values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x, v) of the element x, to the set P, with respect to some given criteria. Plithogenic set theory is being extensively used in various decision-making problems as well as in many other applied fields and for more details we refer the reader (Smarandache 2018a, b, c, d; Gayen et al. 2020) . These different versions of sets have been used in the theory of graphs as well; so in the following we provide brief history of graphs. The theory of graphs: The idea of graph theory is used in many fields. Graph theory is the mathematical structure used to design pair-wise relations between objects. It is constructive in solving problems of different fields as they give a clear picture of the problem at hand. The concept of graph theory begins with the problem of Konigsberg bridge problem in 1736 . In 1973 , Kauffman (1973 introduced the idea of the fuzzy graph. Rosenfeld (1975) developed the concept of fuzzy graph obtaining analogs of several graphtheoretical concepts. Atanassov (1995) extended his concept of fuzzy sets to intuitionistic fuzzy graphs. For more details see Shannon and Atanassov (1994) . Bhattacharya (1987) , give some remarks on fuzzy graphs. Akram and Dudek (2011) gave the concept of interval-valued fuzzy graph in 2011. For more details of fuzzy graphs, readers are referred to Akram (2012) and Akram et al. (2013) . The idea of neutrosophic graphs was given by Kandasamy et al. (2015) in the book title as neutrosophic graphs. Rashid et al. (2018) give the concept of neutrosophic cubic graphs with real-life applications in industry. For more details see Gulistan et al. (2018 Gulistan et al. ( , 2019 and Huang et al. (2019) . Contributions and the motivations: The current study is essential and an excellent addition to the current scientific information and data on COVID-19. After the discovery of SARS-CoV-2, a large number of studies have been conducted to study different aspects of the virus and the disease caused like genetics, mode of transmission, epidemiology, immunotherapeutics, diagnosis, treatment, vaccine, etc. However, very little work has been done to study and plot the COVID-19, spreading of the disease, and burden using the Plithogenic graphs and other models. Thus, this study was performed to answer these quarries. This study will help the researchers, scientists, and policymakers. The same models can be used to predict the coinfections and diseases associated with COVID-19. In this study, Smarandache's plithogenic set is used to introduce the idea of plithogenic graphs and discuss the novel COVID-19 spread. We also established certain binary operations like union, join, Cartesian product, and composition of plithogenic fuzzy graphs, which are helpful when we discuss two different graphs combined. In the end, we used these concepts to find the effects of COVID-19 in other countries. Since it is a mathematical model assessing the spread of COVID-19, it has more to deal with the mathematical results than experimental work. Tthe primary purpose of this article is to develop the new mathematical model of Plithegonic graphs and test them using the real-life application of the spread of COVID-19. Organization of the paper: In Sect. 1, named ''Introduction,'' we provided a brief history and basic definitions of the related material used in this paper. Section 2 is named ''The proposed Method'' we presented the mathematical model of plithogenic fuzzy graphs with some basic operations. In Sect. 3, ''Results and analysis,'' we discussed two examples related to covid-19 and examined some exciting outcomes of the proposed mathematical model. Comparison analysis is provided in Sect. 4. Finally, we concluded our study in Sect. 5 with some future work. (Smarandache 2017) , Plithogeny is the genesis or origination, creation, formation, development, and evolution of new entities from dynamics and organic fusions of contradictory and/or neutrals and/or non-contradictory multiple old entities. At the same time, plithogenic means what is about plithogeny. A plithogenic set P is a set whose elements are characterized by one or more attributes, and each attribute may have many values. Each attribute's value v has a corresponding degree of appurtenance d(x, v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators, a contradiction (dissimilarity) degree is defined between each attribute value and the dominant (most important) attribute value. However, there are cases when such dominant attribute value may not be taken into consideration or may not exist; therefore, it is considered zero by default, or there may be many dominant attribute values. In such cases, either the contradiction degree function is suppressed, or another relationship function between attribute values should be established. The plithogenic aggregation operators (intersection, union, complement, inclusion, equality) are based on contradiction degrees between attributes' values, and the first two are linear combinations of the fuzzy operators' t-norm and t-conorm. Plithogenic set is a generalization of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic set, since these four types of sets are characterized by a single attribute value (appurtenance): which has one value (membership)-for the crisp set and fuzzy set, two values (membership, and nonmembership)-for intuitionistic fuzzy set, or three values (membership, nonmembership, and indeterminacy)-for the neutrosophic set. So we first provide the defintions of fuzzy set, intuitionistic fuzzy set and neutrosophic set. Definition 1 (Zadeh 1996 ) Let S be a universe of discourse then the set is called the fuzzy set where k T : S ! ½0; 1 are truth (membership value), such that 0 k T ðxÞ 1: Definition 2 (Atanassov 1986) Let S be a universe of discourse then the set is called the intuitionistic fuzzy set where k T ; k F : S ! ½0; 1 are truth and falsity membership degrees respectively and 0 k T ðxÞ þ k F ðxÞ 1: Definition 3 (Smarandache 1999) Let S be a universe of discourse then the set is the neutrosophic set where k T ; k I ; k F : S ! ½0; 1 are truth, indeterminancy and falsity membership degrees respectively and 0 k T ðxÞ þ k I ðxÞ þ k F ðxÞ 3: (Smarandache 2017 ) Plithogenic set is a generalization of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic set, since these four types of sets are characterized by a single attribute (appurtenance): which has one value (membership)-for the crisp set and for fuzzy set, two values (membership, and nonmembership)-for intuitionistic fuzzy set, or three values (membership, nonmembership, and indeterminacy)-for neutrosophic set. Definition 4 (Smarandache 2017) Let S be a universal set and P S. A plithogenic set denoted as P S is defined as where v is an appurtenance or attribute, P v is corresponding range of attribute's value, p df : P  P v ! ½0; 1 s is the degree of appurtenance function (DAF) and p CF : P v  P v ! ½0; 1 t is the corresponding degree of contradiction function (DCF) which will satisfy following axioms: for all a; b ð Þ 2 P v  P v ; p Cf a; a ð Þ ¼ 0 and p Cf a; b ð Þ ¼ p Cf b; a ð Þ: Here s; t 2 f1; 2; 3g: For s ¼ t ¼ 1; P S is called plithogenic fuzzy set and is denoted by P FS ; also for s ¼ 2; t ¼ 1; P S is called plithogenic intuitionistic fuzzy set and is denoted by P IFS and for s ¼ 3; t ¼ 1 P S is called plithogenic neutrosophic set and is denoted by P NS . Definition 5 (Rosenfeld 1975 ) A fuzzy graph with set of vertices V is defined to be a pair G ¼ ðh; dÞ;where h is a fuzzy function in V and m is a fuzzy function in E V ÂV, such that dðxyÞ min hðxÞ; hðyÞ ð Þ f gfor all xy 2 E: We call h the fuzzy vertex function of V, m the fuzzy edge function of E, respectively. Note that m is a symmetric fuzzy relation on h. Thus, G ¼ ðh; mÞ is a fuzzy graph of G à ¼ ðV; EÞ if dðxyÞ minðhðxÞ; ðyÞÞ for all xy 2 E. Definition 6 (Atanassov 1995) An intuitionistic fuzzy graph is of the form for all x; y 2 V and xy 2 E: Based on the above literate, it is quite natural to extend the notions of neutrosophic graphs in the environment of plithogenic set as under, 2 The proposed method (plithogenic fuzzy graphs) In this section, we define a more general class of fuzzy graphs known as plithogenic graphs. We also discuss plithogenic fuzzy graphs and their basic operations like union, join, cartesian product, and composition. Definition 8 Let G à ¼ ðV; EÞ be a crisp graph. A plithogenic graph denoted as P G is defined as P G ¼ ðP M ; P N Þ where P M ¼ ðM; l; M l ; a df ; a Cf Þ is plithogenic set for vertices; where M & V, l is an attribute, M l is the corresponding range of attribute values such that a df : M  M l ! 0; 1 ½ s is the degree of appurtenance function (DAF) for vertices defined as a df ðx, aÞ 2 0; 1 ½ s :and a Cf : M l  M l ! 0; 1 ½ t is degree of contradiction function (DCF) for vertices. Also P N ¼ ðN; m; N m ; b df ; b Cf Þ is plithogenic set for edges, where N & E , m is some attribute, N m is the corresponding range of attribute values such that ðM l ; N v Þ is a graph with vertices M l and edges N v . Also b df : N  N m ! 0; 1 ½ s is the degree of appurtenance function for edges and b Cf : N m  N m ! 0; 1 ½ t is degree of contradiction function for edges. Then P G is plithogenic graph iff for all ðx; aÞ&ðy; bÞ 2 M  M l ; b df ððx; aÞðy; bÞÞ minfa df ðx; aÞ; a df ðy; bÞg; ð11Þ b Cf ðða; bÞðc; dÞÞ minfa Cf ðða; bÞ; a Cf ðc; dÞg ð12Þ for all ðða; bÞðc; dÞÞ 2 N m  N m ; where b Cf ðða; bÞða; bÞÞ ¼ 0 as a Cf ðða; aÞ ¼ 0 ¼ a Cf ðb; bÞÞ: Here s; t 2 f1; 2; 3g: Here we discuss a subclass of plithogenic graphs known as plithogenic fuzzy graphs. Definition 9 If we take s ¼ t ¼ 1 in the Definition 8, then we define the plithogenic fuzzy graph P FG as follows; A plithogenic fuzzy graph of a crisp graph G à ¼ ðV; EÞ denoted by P FG is defined as P FG ¼ ðP FM ; P FN Þ where P FM ¼ ðM; l; M l ; a Fdf ; a Fcf Þ is plithogenic fuzzy set P FM for vertices; where M & V, l is an attribute, M l is the corresponding range of attribute values such that a Fdf : M  M l ! 0; 1 ½ is the fuzzy degree of appurtenance function (FADF) for vertices defined as a Fdf ðx; bÞ 2 0; 1 ½ and a Fcf : M l  M l ! ½0; 1 is fuzzy degree of contradiction function (FDCF) for vertices. Also sponding range for some attribute l ¼ typical symptoms and N ¼ fxy; yz; xzg be their relationship with each other and N v be reasons for spreading , then ðM l ; N v Þ is a graph with vertices M l ¼ fa; b; c; dg and edges N v ¼ fab; bc; cd; ac; adg: Also let a Fdf : M  M l ! 0; 1 ½ and a FCf : M l  M l ! ½0; 1 is the FDAF and FDCF for vertices defined as in Table 1 (i) and (ii). Here second column of FDAF in Table 1 shows that 30% people get effected from fever, 40% effected from cough, 20% from dyspnoea and 10% from fatigue, which then converted into coronavirus (COVID-19) in a country x. Similarly we have observations for country y and for counyrty z. Also the FDAF and FDCF for edges is defined as in Table 2 (i) and (ii). Using the Definition 9, it is obvious that P FG is a plithogenic fuzzy graph as shown in Fig. 1a . Remark 1 If we have only one attribute i.e, fever or cough, then plithogenic fuzzy graph provided in Fig. 1a reduces to fuzzy graph as shown in Fig. 1b so by doing so we have lost a lot of information. 2. If we increase the range of attributes, we may get more reliable information, which is main theme of plithogenic sets. Definition 10 Let G à 1 ¼ ðV 1 ; E 1 Þ and G à 2 ¼ ðV 2 ; E 2 Þ be any two crisp graphs. Also suppose that P FG 1 and P FG 2 be any two plithogenic fuzzy graphs such that ½ and is defined as is union of plithogenic fuzzy sets for edges, where for all ðða; bÞðc; dÞÞ 2 N m 1 [ N m 2 ð Þ : Example 2 Consider any two plithogenic graphs such thatðP 1 ; Q 1 Þ is a graph with vertices P 1 ¼ fx; y; zg and edges Q 1 ¼ fxy; yz; xzg; ðl 1 ; m 1 Þ be an attribute, ðP l 1 ; Q m 1 Þ be a graph with vertices P l 1 ¼ fa; b; dg and edges Q m 1 ¼ fab; bdg: Also let a 1 Fdf : P 1  P l 1 ! ½0; 1 and a 1 FCf : P l 1  P l 1 ! ½0; 1 be FDAFand FDCF for vertices defined as Also b 1Fdf : Q 1  Q m 1 ! ½0; 1 and b 1 FCf : Q m 1  Q m 1 ! ½0; 1 be FDAF and FDCF for edges defined as Then using the Definition 9, it is obvious that P FG 1 is a plithogenic fuzzy graph as shown in Fig. 2 , Similarly we have a plithogenic fuzzy graph P FG 2 ; with P FV 2 ¼ P 2 ; l 2 ; P l 2 ; a 2 Fdf ; a 2 FCf Á such that ðP 2 ; Q 2 Þ is a graph with vertices P 2 ¼ fx; z; rg and edges Q 2 ¼ fxz; zr; xrg; ðl 2 ; m 2 Þ be an attribute; ðP l 2 ; Q m 2 Þ is a graph with vertices P l 2 ¼ fa; c; dg and edges Q m 2 ¼ fac; cdg: Also let a 2 Fdf : P 2  P l 2 ! ½0; 1 and a 2 FCf : P l 2  P l 2 ! ½0; 1 be FDAFand FDCF for vertices defined as Also b 2Fdf : Q 2  Q m 2 ! ½0; 1 and b 2 FCf : Q m 2  Q m 2 ! ½0; 1 be FDAFand FDCF for edges defined as Then using the Definition 9, it is obvious that P FG 2 is a plithogenic fuzzy graph as shown in Fig. 3 . Then their union is defined as Here we have P 1 [ P 2 ¼ fx; y; z; rg, Q 1 [ Q 2 ¼ fxy; yz; xz; zr; xrg such that ðP 1 [ P 2 ; Q 1 [ Q 2 Þ is a graph, ðl 1 [ l 2 ; m 1 [ m 2 Þ is an attribute, P l 1 [l 2 ¼ fa; b; c; dg is range of attribute for vertices and Q m 1 [m 2 ¼ fab; bd; ac; cdg is range of attribute for edges so that ðP l 1 [l 2 ; Q m 1 [m 2 Þ is a graph. Here ða 1 Fdf [ a 2 Fdf Þ : ðP 1 [ P 2 Þ Â P l 1 [l 2 ! ½0; 1 and ða 1 FCf [ a 2 FCf Þ : P l 1 [l 2  P l 1 [l 2 ! ½0; 1 are FDAFand FDCF for vertices defined as in Table 7 (i) and (ii). FDCF for edges and is defined as in Table 8 (i) and (ii). Using the Definition 10, it is obvious that P FG 1 [FG 2 is a plithogenic fuzzy graph as represented as in Fig. 4a . Remark 2 If we have only one attribute in plithogenic fuzzy graphs P FG 1 and P FG 2 then union of plithogenic fuzzy graphs provided in Fig. 4a reduces to union of fuzzy graphs as shown in Fig. 4b . Definition 11 Consider any two plithogenic graphs P FG 1 ¼ ðP FV 1 ; Q FE 1 Þ and P FG 2 ¼ ðP FV 2 ; Q FE 2 Þ as given in Definition 10 of crisp graphs G à 1 ¼ ðV 1 ; E 1 Þ and G à 2 ¼ ðV 2 ; E 2 Þ: We define their join as P FG 1 þFG A study of plithogenic graphs... attributes for vertices and Q m 1 [m 2 is range of attributes for edges so that ðP l 1 [l 2 ; Q m 1 [m 2 Þ is a graph. Here FDAF for vertices of P FG 1 þFG 2 i.e. for P 1 [ P 2 is ða 1 Fdf þ a 2 Fdf Þ : and FDCF for vertices is ða 1 FCf þ a 2 FCf Þ : P l 1 [l 2  P l 1 [l 2 ! ½0; 1 such that ða 1 FCf þ a 2 FCf Þða; aÞ ¼ 0 for all ða; aÞ 2 P l 1 [l 2  P l 1 [l 2 and ða 1 FCf þ a 2 FCf Þða; bÞ ¼ ða 1 FCf þ a 2 FCf Þðb; aÞ for all ða; bÞ 2 P l 1 [l 2  P l 1 [l 2 : Also FDAF for edges of P FG 1 þFG 2 i.e. for Q 1 [ Q 2 [ Q 0 ; where Q0 stands for the set of all the edges joining the nodes of P 1 and P 2 given by ; 1 such that ðb 1 FCf þ b 2 FCf Þðða; bÞða; bÞÞ ¼ 0 for all ðða; bÞða; bÞÞ 2 Q l 1 [l 2  Q l 1 [l 2 : Also we have Example 3 Consider any two plithogenic fuzzy graphs such thatðP 1 ; Q 1 Þ is a graph with vertices P 1 ¼ fx; y; zg and edges Q 1 ¼ fxy; yz; xzg; ðl 1 ; m 1 Þ be an attribute, ðP l 1 ; Q m 1 Þ be a graph with vertices P l 1 ¼ fa; bg and edges Q m 1 ¼ fabg: Also let a 1 Fdf : P 1  P l 1 ! ½0; 1 and a 1 FCf : P l 1  P l 1 ! ½0; 1 be FDAFand FDCF for vertices defined as also b 1Fdf : Q 1  Q m 1 ! ½0; 1 and b 1 FCf : Q m 1  Q m 1 ! ½0; 1 be FDAFand FDCF for edges defined as Then using the Definition 9, it is obvious that P FG 1 is a plithogenic fuzzy graph as shown in Fig. 5 . Also for plithogenic fuzzy graph P FG 2 ; we have P FV 2 ¼ ðP 2 ; l 2 ; P l 2 ; a 2 Fdf ; a 2 FCf Þ&Q FE 2 ¼ ðQ 2 ; m 2 ; Q m 2 ; b 2Fdf ; b 2FCf Þ such that ðP 2 ; Q 2 Þ is a graph with vertices P 2 ¼ fx; z; rg and edges Q 2 ¼ fxz; zr; xrg; ðl 2 ; m 2 Þ be an attribute; ðP l 2 ; Q m 2 Þ is a graph with vertices P l 2 ¼ fa; cg and edges Q m 2 ¼ facg: Also let a 2 Fdf : P 2  P l 2 ! ½0; 1 and a 2 FCf : P l 2  P l 2 ! ½0; 1 be FDAFand FDCF for vertices defined as also b 2Fdf : Q 2  Q m 2 ! ½0; 1 and b 2 FCf : Q m 2  Q m 2 ! ½0; 1 be FDAF and FDCF for edges defined as Then using the Definition 9, it is obvious that P FG 2 is a plithogenic fuzzy graph as shown in Fig. 6 . Then their join is defined as P FG 1 þFG 2 ¼ ðP FV 1 þFV 2 ; Q FE 1 þFE 2 Þ where P FV 1 þFV 2 ¼ ðP 1 [ P 2 ; l 1 [ l 2 ; P l 1 [l 2 ; ða 1 Fdf þ a 2 Fdf Þ; ða 1 FCf þ a 2 FCf ÞÞ and Q FE 1 ; 1 are FDAFand FDCF for vertices defined as in Table 13 (i) and (ii). and FDCF for edges and is defined as in Table 14 (i) and (ii). Using the Definition 11, it is obvious that P FG 1 þFG 2 is a plithogenic fuzzy graph as represented in Fig. 7a . Remark 3 If we have only one attribute i.e if a ¼ b in plithogenic fuzzy graph P FG 1 and a ¼ c in plithogenic fuzzy graph P FG 2 then join of plithogenic fuzzy graphs provided in Fig. 7a reduces to join of fuzzy graphs as shown in diagram Fig. 7b, Definition 12 Consider any two plithogenic graphs P FG 1 ¼ ðP FV 1 ; Q FE 1 Þ and P FG 2 ¼ ðP FV 2 ; Q FE 2 Þ as given in Definition 10 of crisp graphs G à 1 ¼ ðV 1 ; E 1 Þ and G à 2 ¼ ðV 2 ; E 2 Þ: We define their cartesian product as P FG is the corresponding range of attribute values and a 1 Fdf  a 2 Fdf : Þ 2 0; 1 ½ :and is defined as ÞÞ is the cartesian product of plithogenic fuzzy sets for edges, where Þis the corresponding range of attribute values such that is FDCF for edges defined as for all ððða; bÞðc; dÞÞ; ððe; f Þ; ðg; hÞÞ 2 N m 1  N m 2 ð Þ : Example 4 Let G à 1 ¼ ðV 1 ; E 1 Þ and G à 2 ¼ ðV 2 ; E 2 Þ be any two crisp graphs. Also suppose that P FG 1 and P FG 2 be any two plithogenic fuzzy graphs such that P FG 1 ¼ ðP FM 1 ; P FN 1 Þ; whereP FM 1 ¼ ðM 1 ; l 1 ; M l 1 ; a 1 Fdf ; a 1 FCf Þ & P FN 1 ¼ ðN 1 ; m 1 ; N m 1 ; b 1Fdf ; b 1 FCf Þ and suppose that M 1 ¼ fx; yg & V 1 and M l 1 ¼ fa; bg is the corresponding range for some attribute. Also N 1 ¼ fxyg & E 1 and m 1 be some attribute for edges N m 1 ¼ fabg be range of attribute. Then a 1 Fdf : M 1  M l 1 ! ½0; 1 and b 1Fdf : N 1  N m 1 ! ½0; 1 is the FDAF for vertices V 1 and edges E 1 defined as and FDCF for vertices V 1 and edges E 1 is defined as Using the Definition 9, it is obvious that P FG 1 is a plithogenic fuzzy graph as represented in Fig. 8 . Also suppose that M 2 ¼ fx; zg & V 2 and M l 2 ¼ fa; cg is the corresponding range for some attribute. Also N 2 ¼ fxzg & E 1 and m 2 be some attribute for edges N m 2 ¼ facg be range of attribute. Then a 2 Fdf : M 2  M l 2 ! ½0; 1 and b 2Fdf : N 2  N m 2 ! ½0; 1 is the FDAF for vertices V 2 and edges E 2 defined as and FDCF for vertices V 2 and edges E 2 is defined as Using the Definition 9, it is obvious that P FG 2 is a plithogenic fuzzy graph as represented in Fig. 9 . Hence Using the Definition 12, the FDAF of cartesian product for vertices a 1  a 2 ð Þ Fdf : and so on. Hence P FG 1 ÂFG 2 is a plithogenic fuzzy graph as shown in Fig. 10 . Consider the Example 4 then the composition of plithogenic fuzzy graphs P FG 1 and P FG 2 is represented in Fig. 11 . Coronaviruses (CoV) are a large family of viruses that cause illnesses ranging from the common cold to more severe diseases such as Middle East Respiratory Syndrome (MERS-CoV) and Severe Acute Respiratory Syndrome (SARS-CoV). A novel coronavirus (nCoV) is a new strain that has not been previously identified in humans. It has been discussed and tried to verify how these viruses affected human beings during their outbreaks. We are usig the database taken from the following source https://cor onavirus.jhu.edu/map.html. Example 5 We are discussing how these viruses affected the world at different times for different durations. Let G à ¼ ðV; EÞ be a crisp graph where V is the set of different types of fatalic viruses and P & V be types of coronavirus. Let us denote these types (1) Let v ¼ attribute which denote the effects of these coronaviruses in different years for different durations on the whole world and range of this attribute is a 1 ¼ number of countries effected, a 2 ¼ number of people effected, a 3 ¼ number of casualities, a 4 ¼ duration. Then we have P ¼ fV S ; V M ; V n g and range of attribute for vertices P v ¼ fa 1 ; a 2 ; a 3 ; a 4 g. Also Q ¼ fV S V M ; V M V n ; V S V n g E V  V and m be some attribute for edges, such that ( P v ; Q m Þ forms a graph. Then we have Let the FDAF for vertices be a Fdf : P  P v ! 0; 1 ½ defined by a Fdf ð V i Vm ; aj a m Þ; where i ¼ S, M, n ; m ¼ largest number with maximum effect and j ¼ 1; 2; 3: where as for a 4 we have membership function as duration for which these coronavirses proved to be more fatalic. So we have FDAF and FDCF for vertices as in Table 19 (i) and (ii). Also FDAF and FDCF for edges which is defined as main reason for spreading of these viruses i.e.for V S through bats, V M through bats and animal flesh and V n through all the previous reasons including human to human interaction, also low immunity, see Table 20 (i) and (ii). Hence P FG ¼ ðP FV ; Q FE Þ is plithogenic graph as shown in Fig. 12a . Example 6 Let us consider the case of the novel Coronavirus , and we want to measure its overall effect on the whole world. There are a lot of factors to discuss, but here we discuss some of them. Let G à ¼ ðV; EÞ be a crisp graph where V is the set of different countries effected by new Coronavirus COVID-19 and let Pð& VÞ consisting of highly effected countries having casualities. Let us denote countries Following results have been derived from the mathematical study: 1. We Provided the mathematical existence of Plithogenic' graphs through examples that generalize fuzzy, intuitionistic, and neutrosophic graphs. 2. We established different binary opertions of Plithogenic graphs for practical use in real-life applications. 3. COVID-19 was found much dangerous than (MERS-CoV) and (SARS-CoV). 4. Figure 11 shows that COVID-19 seems to be Chimera of two viruses. 5. As the SARS-CoV and MERS-CoV outbreaks were controlled, there are greater chances to overcome the current pandemic of COVID-19. 6. The global spread of COVID-19 is associated with traveling, as the edges of Fig. 12a suggest. 7. If the value of an edge is 0, it means that there is no traveling between two particular countries, so the spread of COVID-19 is 0. 8. This model suggests that all the countries should stop all types of traveling/movement across the borders and inside the country. 9. We predict if precautionary measures have not been taken, then there is a chance of severe outbreaks in the future. 10. The performance metrics of our proposed mathematical model indicated that the COVID-19 is zoonotic, and the human transmission is very fast in the case of frequent travel. In this paper, we extend the plithogenic sets to plithogenic graphs. As we see that in plithogenic graphs. we have P G ¼ ðP M ; P N Þ ,where P M ¼ ðM; l; M l ; a df ; a Cf Þ is plithogenic set P S for vertices; where M & V, l is an attribute, M l is the corresponding range of attribute values such that a df : M  M l ! 0; 1 ½ s is the DAF for vertices defined as a df ðv 1 , l 1 Þ 2 0; 1 ½ s :and a Cf : M l  M l ! 0; 1 ½ t is DCF for vertices. Also P N ¼ ðN; m; N m ; b df ; b Cf Þ is plithogenic set for edges, where N & E , m is the some attribute, N m is the corresponding range of attribute values for edges such that b df : N  N l ! 0; 1 ½ s is the DAF for edges and b Cf : N m  N m ! 0; 1 ½ t is DCF for edges. Then P G is plithogenic graph iff for all ðv 1 ; l 1 Þ& ðv 2 ; l 2 Þ 2 M  M l ; b df ððv 1 ; l 1 Þðv 2 ; l 2 ÞÞ a df ððv 1 ; l 1 Þ^a df ðv 2 ; l 2 ÞÞ. Here s; t 2 f1; 2; 3g: If we take the set of edges as null set then our model reduces to plithogenic set P S ¼ ðP; v; P v ; p df ; p CF Þ where v is an appurtenance or attribute, P v is corresponding range of attribute's value, p df : P  P v ! ½0; 1 s is the degree of appurtenance function and p CF : P v  P v ! ½0; 1 t is the corresponding degree of contradiction function. Here s; t 2 f1; 2; 3g: 1. Consider the Example 5. (a) if we have only one attribute i.e., a 1 ¼ number of countries effected then plithogenic fuzzy graph provided in Fig. 12a reduces to fuzzy graph as shown in Fig. 12b. (b) If we have two attributes suppose a 1 ¼ number of countries effected and a 2 ¼number of people effected then plithogenic fuzzy graph provided in Fig. 12a reduces to intuitionistic fuzzy graph as shown in Fig. 12c (c) If we take three attributes say a 1 ¼ number of countries effected and a 2 ¼number of people effected a 3 ¼ number of casualties then plithogenic fuzzy graph provided in Fig. 12a reduces to neutrosophic graph as shown in Fig. 12d . 2. Consider the Example 6. (a) If we take only one attribute say a 1 ¼confirmed cases having COVID-19 then plithogenic fuzzy graph provided Fig. 13a reduces to fuzzy graph as shown in Fig. 13b. (b) If we have two attributes suppose a 1 ¼ confirmed cases with COVID-19 and a 2 ¼ serious, critical cases having COVID-19 then plithogenic fuzzy graph provided in Fig. 13a reduces to intuitionistic fuzzy graph as shown in Fig. 13c . (c) If we consider three attributes i.e. a 1 ¼confirmed cases with COVID-19, a 2 ¼ critical cases having Coronavirus COVID-19, a 3 ¼ recovered cases of COVID-19, then plithogenic fuzzy graph provided in Fig. 13a reduces to neutrosophic graph as shown in Fig. 13d . Thus our model of plithogenic graph is better than already existing graphs as it can capture more information. Initially, the virus was zoonotic in origin, and then it was spread by human interactions (person to person). Our model verified that its spread across the borders was mainly due to travelers. To control the COVID-19, it is strongly recommended to go for self-isolation to break the chain by making the edges of Plithogenic graphs equal to 0. The animals or their food which is thought to be the source of this virus, should be banned in the markets, and personal hygiene should be maintained to reduce the spread. This mathematical model can be applied to assess the spread of COVID-19 in any region of the world. In the future, we aim to make more different types of graphs in the circumstances of plithogenic theory by taking more attributes and other approaches. This model has been initially applied on database taken from the following source https://coronavirus.jhu.edu/map.html. We are aiming to use this model on some other database in the future. 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Conflict of interest Authors declare that there is no conflict of interest.Ethical approval The present article does not contain any studies with human participants or animals performed by any of the authors.