# On Certain Products of Complex Intuitionistic Fuzzy Graphs

Authors: Abida Anwar; Faryal Chaudhry
Journal: Journal of Function Spaces
 (2021)

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## Abstract

A complex intuitionistic fuzzy set (CIFS) can be used to model problems that have both intuitionistic uncertainty and periodicity. A diagram composed of nodes connected by lines and labeled with specific information may be used to depict a wide range of real-life and physical events. Complex intuitionistic fuzzy graphs (CIFGs) are a broader type of diagram that may be used to manipulate data. In this paper, we define the key operations direct, semistrong, strong, and modular products for complex intuitionistic fuzzy graphs and look at some interesting findings. Further, the strong complex intuitionistic fuzzy graph is defined, and several significant findings are developed. Furthermore, we study the behavior of the degree of a vertex in the modular product of two complex intuitionistic fuzzy graphs.


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## Body

## 1. Introduction

Obscurity is a common occurrence in everyday life. This is not a world of precise calculations and ideas. For human intellect, this judgment mistake is extremely tough. To tackle this problem, a variety of mathematical techniques and ideas, such as fuzzy sets and complex fuzzy sets, have been developed. A group system with uncertain information was used to create the complex fuzzy logic. Due to the elastic potential of advanced intuitionistic fuzzy sets (IFSs) to control unreliability, this event is considered wonderfully great for humanistic logic underlying wrong reality and infinite knowledge. Because it allows for more erroneous information to be given, this theory is a cornerstone of classical complex fuzzy sets because it provides for more suitable answers to a range of situations. In cases when we must deal with relatively limited alternatives, such as yes or no, these specialized sets generated beneficial models. Another essential feature of this knowledge is that it allows man to evaluate the negative and positive elements of erroneous ideas. To deal with uncertainty, Zadeh [1] introduced the fuzzy set theory. Following that, a number of academics looked into the theory of fuzzy sets and fuzzy logic in order to deal with a variety of real-world problems involving an uncertain and ambiguous environment. Atanassov [2] came up with the concept of intuitionistic fuzzy sets (IFSs), which are fuzzy sets with a new component. With the addition of the degree of truth and falsity membership, the concept of IFSs has become more relevant and vivid. The applications of these sets have gotten a lot of attention in fields like multicriteria decision-making and image processing. Furthermore, when data is phase-shifted, the ambiguity and uncertainty in the data come from everyday life. As a result, taking this information into account is theoretically insufficient, and information is lost as a result of the procedure. To handle this uncertainty, Ramot et al. [3] introduced the elongated form the fuzzy set by including a phase term part, called complex fuzzy set. The competency of complex fuzzy logic in the sense of membership has a very significant role to address concrete problems. It is not only a vital source for measuring unevenness but also very effective mode to deal with ambiguous ideas. Besides its usefulness, we still have massive problems regarding the physical properties of complex membership-based functions. It is highly demanding to design an additional theory of complex fuzzy set in the sense of set knotty members. This logic is straight development of conventional fuzzy logic that naturally develops problem basing fuzzy logic which is not suitable for the artificial function of membership. This certain set has core role in various applications especially in modern commanding systems which foreshadows periodic phenomenon wherein a series of fuzzy variables are interlinked in a complicated way and it cannot be properly run by fuzzy operations. Owing to the feature of handling the information regarding both periodicity and uncertainty, the complex fuzzy sets gained the special attentions in the latest trends of fuzzy sets, bipolar fuzzy sets, and IFSs. By using these models, both periodicity and uncertainty may be presented in a single set. Atanassov [2] added new components to the concept of a fuzzy set that specifies the degree of nonmembership. Fuzzy sets provide membership degrees, while IFSs provide both membership and nonmembership degrees, which are more or less independent of one another. The sole requirement is that the total of the two degrees is less than one. IFSs have been used in economics, chemistry, medicine, engineering, and computer science. Therefore, the studies regarding complex fuzzy sets got a broad spectrum both in theoretical aspects as well as application aspects. Many researchers investigated the extensive applications of complex fuzzy sets in signal processing applications, time series, solar activity, and forecasting problems (for instance, see [4–8]). Complex numbers and complex fuzzy sets were utilized by Buckley [9]. Alkouri and Salleh [10] developed the concept of complex Atanassov’s intuitionistic fuzzy relation and complex Atanassov’s intuitionistic fuzzy sets. Rosenfeld coined the term “fuzzy subgroups” and established a connection between group theory and fuzzy set theory. As a result, several academics developed fuzzy algebraic structures based on fuzzy sets, intuitionistic fuzzy sets (IFSs), and CIFs (for detail, see [11–18]). Several real and tangible circumstances can be illustrated using a diagram composed of a collection of nodes with lines joining specific pairs of these nodes. The nodes could select individuals, with lines connecting pairs of friends, or primary health care facilities, with lines representing beneficiaries’ streets or roads in the region. Fuzzy graph modeling is a useful mathematical tool for solving combinatorial issues in a variety of fields, such as image capturing, computer network, electric network, operations research, social science, road network, topology, optimization, algebra, computer science, environmental science, and scheduling problem. Fuzzy graph theory has an intuitive and aesthetic appeal because of the diagrammatic representation. Due to the natural presence of vagueness and ambiguity, fuzzy graphical models are far superior to graphical models. We needed fuzzy set theory at first to deal with numerous complicated phenomena that had inadequate information. Based on Zadeh’s fuzzy connection, Kauffman [19] was the first to coin the term “fuzzy graph.” Rosenfeld [20] went on to invent fuzzy vertex, fuzzy edge, and theoretical fuzzy graph ideas like routes, connectedness, and cycle, among other things. Following Mordeson and Chang-Shyh’s [21] discussion of fuzzy graph operations, Bhutani and Battou’s [22] research of M-strong fuzzy graphs was published. Following that, Eslahchi and Onagh [23], Gani and Malarvizhi [24], Mordeson and Nair [25], and Mathew and Sunitha [26] propose a slew of concepts and definitions, primarily under the headings of vertex strength of fuzzy graphs, fuzzy trees, isomorphism on fuzzy graphs, fuzzy subgraphs, fuzzy paths, and complement of a fuzzy graph. Because the membership function was insufficient to express the complexity of object features, a nonmembership function was created. By combining the nonmembership and hesitation qualities, Atanassov [2] constructed the intuitionistic fuzzy set theory, which was an elaboration of the basic set theory. This idea has been used to a variety of domains, including computer programming, medical fields, decision-making problems, marketing evaluation, and banking issues. In 2006, Parvathi and Karunambigai [27] proposed an intuitionistic fuzzy graph as a variant of Atanassov’s IFG. Thirunavukarasu et al. [28] built on this concept by incorporating complex fuzzy graphs. Shannon and Atanassov [29] defined and discussed intuitionistic fuzzy graphs. Later on, a number of authors worked on intuitionistic fuzzy graphs and made several important contributions to the subject (for instance, see [30–33]). Sahoo and Pal discussed different types of products on intuitionistic fuzzy graphs in [34]. Using the concept of a complex intuitionistic fuzzy set, Yaqoob et al. [35] constructed complex intuitionistic fuzzy graphs (CIFGs).This paper’s structure is as follows: the second section dives into some basic definitions. In Section3, we define the direct product of two CIFGs. We define strong CIFG. We show that the direct product of two CIFGs is a CIFG as well. At the end of this section, we show that if the direct product of two CIFGs is strong, then at least one of them is strong. In Section 4, we define the semidirect product of two CIFGs. This section demonstrates that the semidirect product of two CIFGs is also a CIFG. At the end of this section, we demonstrate that if the semidirect product of two CIFGs is strong, then at least one of them is strong. The strong product of two CIFGs is defined in the fifth section. We demonstrate that the strong direct product of two CIFGs is CIFG. Furthermore, we demonstrated that if the strong product of two CIFGs is strong, then at least one of them is strong. In the last section of this paper, we define the modular product of two CIFGs and examine some intriguing results. We also investigate how the degree of vertex behaves in the modular product of two CIFGs.


## 2. Preliminaries

We go over some basic definitions that will assist us in our future discussions.Definition 1 [1].
A fuzzy set (FS)X of a nonempty set A is a function, X:A⟶0,1.Definition 2 [2].
An intuitionistic fuzzy set (IFS)X of a universe of discourse A is a triplet of the form X=a,mXa,nXa∣a∈A, where the functions mXa:A⟶0,1 and nXa:A⟶0,1 are the membership function (degree of truthfulness) and nonmembership functions (degree of falsity), respectively. These functions must fulfill the condition 0≤mXa+nXa≤1.Definition 3 [36].
The object of the form(1)X=a,mXaeiαXa,nXaeiβXa:a∈A,is a complex intuitionistic fuzzy set (CIFS) defined on universe of discourse A.
Here,
(2)i=−1,mXa,nXa∈0,1,αXa,βXa∈0,2πand0≤mXa+nXa≤1.Definition 4 [27].
An intuitionistic fuzzy graph is of the formG=B,C,X,Y on the crisp graph G∗ with vertex set B and edge set C, where
(1)
B=b1,b2,⋯,bn and X=mX,nX such that mX:B⟶0,1 and nX:B⟶0,1 denote the membership value (MV) and nonmembership value (NMV) of the element bi∈B, respectively, such that mXbi+nXbi≤1 for all bi∈B(2)
C⊆B×B and Y=mY,nY where mY:C⟶0,1 and nY:C⟶0,1 are defined by mYbi,bj≤mXbi∧mXbj and nYbi,bj≤nXbi∨nXbj such that mYbi,bj+nYbi,bj≤1,∀bi,bj∈CDefinition 5 [35].
A complex intuitionistic fuzzy graph (CIFG) with an underlaying vertex setB and edge set C is defined to be a pair G=B,C,X,Y, where X is a CIFS on B and Y is a CIFS on C⊆B×B such that
(3)mYbi,bjeiαYbi,bj≤minmXbi,mXbjeiminαXbi,αXbj,nYbi,bjeiβYbi,bj≤maxmXbi,nXbjeimaxβXbi,βXbj,for all bi,bj∈B.Definition 6 [35].
LetG=B,C,X,Y be the given CIFG. The degree of a vertex bi in G is defined by
(4)degGbi=∑bi,bj∈CmYbi,wbjeiαYbi,bj,∑bi,bj∈CnYbibjeiβYbi,bj.


## 3. Direct Product of Two CIFGs

Definition 7.
The direct product of two CIFGs,G1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 such that B1∩B2=ϕ is defined to be CIFG G1óG2=B,C,X1óX2,Y1óY2 where
(5)B=B1×B2andC=C1×C2=bi1,bj1,bi2,bj2bi1,bi2∈C1,bj1,bj2∈C2.The MV and NMV for the vertexbi1,bi2 in G1óG2 are given by
(6)mX1eiαX1ómX2eiαX2bi1,bi2=mX1bi1eiαX1bi1∧mX2bi2eiαX2bi2=minmX1bi1,mX2bi2eiminαX1bi1,αX2bi2,nX1eiβX1ónX2eiβX2bi1,bi2=nX1bi1eiβX1bi1∧nX2bi2eiβX2bi2=maxnX1bi1,nX2bi2eimaxβX1bi1,βX2bi2.The NM and NMV for the edgeu=bi1,bj1,v=bi2,bj2 in G1ΠG2 are given by
(7)mY1eiαY1ΠmY2eiαY2u,v=mY1ueiαmY1u∧mY2veiαmY2v=minmY1u,mY2veiminαY1u,αY2v,(8)nY1eiβY1ΠnY2eiβY2u,v=nY1ueiβY1u∨nY2veiβY2v=maxnY1u,nY2veimaxβY1u,βY2v.Now we define the strong CIFG.Definition 8.
A CIFGG=B,C,X,Y is called strong CIFG if
(9)mYu,veiαYu,v=mXueiαXu∧mXveiαXv=minmXu,mXveiminmXu,mXv,(10)nYu,veiβBu,v=nXueiβXu∨nXveiβXv=maxnXu,nXveimaxβXu,βXv.Theorem 9.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be two strong CIFGs; then, G1ΠG2 is also a strong CIFG.Proof.
AsG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 are strong CIFGs, so by (9) and (10), we have
(11)mY1u1,v1eiαY1u1,v1=minmX1u1,mX1v1eiminαX1u1,αX1v1,nY1u1,veiβY1u1,v1=maxnX1u1,nX1v1eimaxβX1u1,βX1v1,mY2u2,v2eiαY2u2,v2=minmX2u2,mX2v2eiminαX2u2,αX2v2,nY2u2,v2eiβY2u2,v2=maxnX2u2,nX2v2eimaxβX2u2,βX2v2,for all u1,v1∈E1 and u2,v2∈E2.
Now from (7) and (8), we have
(12)mY1eiαY1ΠmY2eiαY2bi1,bj1,bi2,bj2=mY1bi1,bi1eiαY1bi1,bj1∧mY2bj2,vj2eiαY2bi2,bj2=mX1bi1eiαX1bi1∧mX1bj1eiαX1bj1∧mX2bi2eiαX2∧mX2bj2eiαX2bj2=mX1bi1eiαX1bi1∧mX2bj2eiαX2bj2∧mX1bj2eiαX1bj2∧mX2bj2eiαX2bj2=mX1eiαX1Πmx2eiαX2bi1,bj1∧mX1eiαX1ΠmX2eiαX2bi2,bj2.
In addition for nonmembership,(13)nY1eiβY1ónY2eiβY2bi2,bj1,bi2,bj2=nY1bi1,bj1eiβY1bi1,bj1∨nY2bi2,bj2eiβY2bi2,bj2=nX1bi1eiβX1bi1∨nX1bi2eiαX1bj1∨nX2bi2eiβX2∧nX2bj2eiβX2bj2=nX1bi1eiβX1bi1∨nX2bj2eiβX2bj2∨nX1bj2eiβX1bj2∧nX2bj2eiβX2bj2=nX1eiβX1ΠnX2eiβX2bi1,bj1∨nX1eiαX1ΠnX2eiβX2bi2,bj2.
This completes the proof.Theorem 10.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be two CIFGs, such that G1ΠG2 is strong; then, at least one of G1 or G2 must be strong CIFG.Proof.
SupposeG1 and G2 are not strong CIFGs. Thus, there exists at least one bi1,bj1∈B1,bi2,bj2∈B2 such that
(14)mY1bi1,bj1eiαY1bi1,bj1<mX1bi1eiαX1bi1∧mX1bj1eiαX1bj2,nY1bi1,bj1eiβY1bi1,bj1<nX1bi1eiβX1bj1∨nX1bj1eiβX1bj1,mY2bi2,bj2eiαY2bi2,bj2<mX2bi2eiαX2bi2∧mX2bj2eiαX2bj2,nY2bi2,bj2eiβY2bi2,bj2<nX2bi2eiβX2bj2∨nX2bj2eiβX2bj2.
Letu=bi1,bj1,v=bi2,bj2∈B1×B2; then,
(15)mY1eiαY1ΠmY2eiαY2u,v=mY1ueiαY1u∧mY2veiαY2v<mX1bi1eiαX1bi1∧mX1bj1eiαX1bj1∧mX2bi2eiαX2bi2∧mX2bj2eiαX2bi2=mX1bi1eiαX1bi1∧mX2bi1eiαA2bj1∧mA1bi2eiαA1bi1∧mA2bi2eiαA2bi2=mX1eiαX1ΠmX2eiαX2bi1,bj1∧mX1eiαX1ΠmX2eiαX2bi2,bj2.
Therefore,(16)mY1eiαY1ΠmY2eiαY2u=bi1,bj1,v=bi2,bj2<mX1eiαX1ΠmX2eiαX2bi1,bj1∧mX1eiαX1ΠmX2eiαX2bi2,bj2.
Again, letu=bi1,bj1,v=bi2,bj2∈B1×B2; then,
(17)nY1eiβY1ΠnY2eiβY2u,v=nY1ueiβY1u∨nY2veiβY2v<nX1bi1eiβX1bi1∨nX1bj1eiβX1bj1∨nX2bi2eiβX2bi2∨nX2bj2eiβX2bi2=nX1bi1eiβX1bi1∨nX2bi1eiβX2bj1∨nX1bi2eiβX1bi1∨nX2bi2eiβX2bi2=nX1eiβX1ΠnX2eiβX2bi1,bj1∨nX1eiβX1ΠnX2eiβX2bi2,bj2.
Therefore,(18)nY1eiβY1ΠnY2eiβY2u=bi1,bj1,v=bi2,bj2<nX1eiβX1ΠnX2eiβX2bi1,bj1∨nX1eiβX1ΠnX2eiβX2bi2,bj2.
This shows thatG1ΠG2 is not strong, which is contradiction.
This completes the proof.


## 4. Semistrong Product of Two CIFGs

Definition 11.
The semistrong product of two CIFGsG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 such that B1∩B2=ϕ is defined to be the CIFG as G1♦G2=B,C,X1♦X2,Y1♦Y2, where B=B1×B2andC=bi1,bi2,bj1,bj2∣bi1,bj1∈B1andbi2=bj2orbi1,bj1∈B1andbi2,bj2∈B2.
The MV and NMV of the vertexu,v in G1♦G2 are given as
(19)mX1eiαX1♦mx2eiαX2u,v=mX1ueiαX1u∧mX2veiαX2v=minmX1u,mX2veiminαX1u,αX2v,(20)nX1eiβX1♦nX2eiβX2u,v=nX1ueiβX1u∨nX2veiβX2v=maxnX1u,nX2veimaxβX1u,βX2v.The MV and NMV for the edgeu,v1,u,v2 and u1,v1,u2,v2∈C in G1♦G2 are given by as follows:
(21)mY1eiαY1♦mY2eiαB2u,v1,u,v2=mX1ueiαA1u∧mY2v1,v2eiαY2v1,v2,(22)mY1eiαB1♦mY2eiαB2u1,v1,u2,v2=mY1u1,u2eiαB1u∧mY2v1,v2eiαY2v1,v2,(23)mY1eiαB1♦mY2eiαY2u,v1,u,v2=mX1ueiαX1u∧mY2v1,v2eiαY2v1,v2,(24)mY1eiαY1♦mY2eiαY2u1,v1,u2,v2=mY1u1,u2eiαY1u∧mY2v1,v2eiαY2v1,v2.Theorem 12.
IfG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 are strong CIFGs, then G1♦G2 is also strong.Proof.
Ifu,v1,u,v2∈C, then using (19) and (20), we have
(25)mY1eiαY1♦mY2eiαY2u,v1,u,v2=mX1ueiαX1u∧mX2v1,v2eiαY2v1,v2=mX1ueiαX1u∧mX2v1eiαX2v1∧mX2v2eiαX2v2=mX1ueiαX1u∧mX2v1eiαX2v1∧mX2ueiαA2u∧mX2v2eiαA2v2=mX1eiαX1♦mX2eiαX2u,v1∧mX1eiαX1♦mX2eiαA2u,v2.
Similar, we can show thatnB1eiβB1♦nB2eiβB2u,v1,u,v2=nA1eiαA1♦nA2eiαA2u,v1∨nA1eiαA1♦nA2eiαA2u,v2.
Again, ifu1,v1,u2,v2∈E, then using (22) and (24), we have
(26)mY1eiαY1♦mY2eiαY2u1,v1,u2,v2=mY1u1,v1eiβY1u1,v2∧mY2u2,v2eiβY2u2,v2=mX1u1eiαX1u1∧mX2v1eiαX2v1∧mX2u2eiαX2u2∧mX2v2eiαX2v2=mX1eiαX♦mX2eiαX2u1,v1∧mX1eiαA1♦mX2eiαX2u2,v2.
Similarly, we can show that(27)nY1eiβY1♦nY2eiβY2u1,v1,u2,v2=nX1eiβX1♦nX2eiβX2u1,v1∨nX1eiβX1♦nX2eiβX2u2,v2.
This completes the proof.Theorem 13.
IfG1=B1,C1,X1,Y1andG2=B2,C2,X2,Y2 are two CIFG, such that G1♦G2 is strong, then at least one of G1 or G2 must be strong.


## 5. Strong Product of Two CIFGs

Definition 14.
The strong product of two CIFGs isG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 such that B1∩B2=ϕ is defined to be CIFGG1⊗G2=B,C,X1⊗X2,Y1⊗Y2 where B=B1×B2 and C=u,v1,u,v2∣u∈C1,v1,v2∈C2∪u1,v,u2,v∣u1,u,v2∈C1∪u1,u2,v1,v2∣u1,u2∈C1,v1,v2∈C2. The MV and NMV for the vertex u,v in G1⊗G2 are given by
(28)mX1eiαX1⊗mX2eiαX2u,v=mX1ueiαX1u∧mX2veiαA2v,(29)nX1eiβX1⊗nX2eiβX2u,v=nX1ueiβX1u∨nX2veiβX2v.The MV and NMV for edges inG1⊗G2 are given by
(30)mY1eiαY1⊗mY2eiαY2u,v1,u,v2=mX1ueiαX1u∧mY2v1,v2eiαY2v1,v2,mY1eiαY1⊗mY2eiαY2u1,v,u2,v=mY1u1,u2eiαY1u1,u2∧mX2veiαX2v,mY1eiαY1⊗mY2eiαY2u1,v1,u2,v2=mY1u1,v2eiαY1u1,v2∧mY2u1,v2eiαY2u1,v2,(31)nY1eiβY1⊗nY2eiβY2u,v1,u,v2=nX1ueiβX1u∨nY2v1,v2eiβY2v1,v2,nY1eiβY1⊗nY2eiβY2u1,v,u2,v=nY1u1,u2eiβY1u1,u2∨nX2veiβx2v,nY1eiβY1⊗nY2eiβY2u1,v1,u2,v2=nY1u1,v2eiβY1u1,v2∨nY2u1,v2eiβY2u1,v2.Theorem 15.
IfG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 are strong CIFGs, then G1⊗G2 is also strong.Theorem 16.
IfG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 are two CIFGs, such that G1⊗G2 is strong, then at least one of G1 or G2 must be strong.Definition 17.
A CIFGG=B,C,X,Y is said to be complete if
(32)mYu,veiαYu,v=mXueiαXu∧mXveiαXvnYu,veiβYu,v=nXueiαXu∧nXveiαXv.for all u,v∈B.Theorem 18.
IfG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 are two CIFGs, then G1⊗G2 is complete.Proof.
As a strong product of CIFGs is CIFG, and every pair of vertices is adjacent. Ifu,v1,u,v2∈C, then using (29) and (30), we have
(33)mY1eiαY1⊗mY2eiαY2u,v1,u,v2=mX1ueαX1u∧mY2v1,v2eαY2v1,v2=mX1ueαX1u∧mX2v1eiαX2v1∧mX2v2eiαX2v2=mX1ueiαX1u∧mX2v1eiαX2v1∧mX1ueiαX1u∧mX2v2eiαX2v2=mX1eiαX1⊗mX2eiαX2u,v1∧mX1eiαX1⊗mX2eiαX2u,v1.And by (31), it follows that
(34)nY1eiβY1⊗nY2eiβY2u.v1,u,v2=nX1ueβX1u∨nY1v1,v2eβY2v1,v2=nX1ueβX1u∨nY1v1,v2eβY2v1,v2=nX1ueβX1u∨nX2v1eiβX2v1∨nX2v2eiβX2v2=nX1ueiβX1u∨nX2v1eiβX2v1∨nX1ueiβX1u∨nX2v2eiβX2v2=nX1eiβX1⊗nX2eiβX2u,v1∨nX1eiβX1⊗nX2eiβX2u,v1.
Ifu1,v,u2,v∈C, then
(35)mY1eiαY1⊗mY2eiαY2u1,vu2,v=mY1u1,v2eiαY1u1,v2∧mX2veiαX2v=mX1u1eiαX1u1∧mX1v1eiαX2u2∧mX2veiαX2v∧mX2veiαX2v=mX1eiαX1⊗mX2eiαX2u1,v∧mX1eiαX1⊗mX2eiαX2u2,v.
Similarly,(36)nY1eiβY1⊗nY2eiβY2u1,vu2,v=nX1eiβX1⊗nX2eiβX2u1,v,∨nX1eiβX1⊗nX2eiβX2u2,v.
Again ifu1,v1,u2,v2∈C, then
(37)mY1eiαY1⊗mY2eiαY2u1,v1u2,v2=mY1u1,v1eiαY1u1,v1∧mY2v1,v2eiαY2u2,v2=mX1u1eiαX1u1∧mX1v2eiαX2v2∧mX2v1eiαX2v1∧mX2v2eiαX2v2=mX1u1eiαX1u1∧mX2u1eiαX2v1∧mX1u2eiαX2u2∧mX2v2eiαX2v2=mX1eiαX1⊗mX2eiαX2u1,v1∧mX1eiαX1⊗mX2eiαX2u2,v2.
Similarly,(38)nY1eiβY1⊗nB2eiβY2u1,v1u2,v2=nX1eiβX1⊗nX2eiβX2u1,v1,∨nX1eiβX1⊗nX2eiβX2u2,v2.
This completes the proof.


## 6. Modular Product of CIFGs

Definition 19.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be two CIFGs with underlying vertex sets B1 and B2 and edge sets C1 and C2, respectively. Then, modular product of G1 and G2 is G1eG2=B1eB2,C1eC2,X1eX2,X1eX2 with underlying vertex set B1eB2=x1,y1x1∈B1,y1∈B2 and underlying edge set C1eC2=x1,y1x2,y2x1,x2∈C1,y1,y2∈C2 or x1,x2∉C1,y1,y2∉C2 with
(39)mX1eiαX1emX2eiαX2x1,y1=mX1x1eiαX1x1∧mX2y1eiαX2y1=minmX1x1,mX2y1eiminαX1x1,αX2y1,nX1eiβX1enX2eiβX2x1,y1=nX1x1eiβX1x1∨nX2y1eiβX2y1=maxnX1x1,nX2y1eimaxβX1x1,βX2y1.
Here,x1∈B1 and y1∈B2.The MV and NMV for edges inG1eG2 are given by
(40)mY1eiαY1emY2eiαY2x1,y1,x2,y2=mY1x1,x2eiαY1x1,x2∧mY2x1,x2eiαY2x1,x2,ifx1,x2∈C1,y1y2∈C2,mX1x1eiαX1x1x1∧mX1x2eiαX1x2∧mX2y1eiαX2y1∧mY2y2eiαY2x1,ifx1,x2∉C1,y1,y2∉C2,(41)nY1eiβY1enY2eiβY2x1,y1,x2,y2=nY1x1,x2eiβY1x1,x2∨nY2x1,x2eiβY2x1,x2,ifx1,x2∈C1,y1,y2∈C2,nX1x1eiβX1x1x1∨nX1x2eiβX1x2∨nX2y1eiβX2y1∨nY2y2eiβY2x1,ifx1,x2∉C1,y1,y2∉C2.Theorem 20.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,Y2,Y2 be the CIFGs; then, modular product G1⨀G2 is also a CIFGs.Proof.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be the CIFGs. We have to prove that G1⨀G2 is CIFG. By the definition, X1⨀X2 is CIFS on B1⨀B2 and Y1⨀Y2 is CIFS on C1⨀C2. From (40) and (41), we have
(42)mY1eiαY1emY2eiαY2x1,y1,x2,y2=mY1x1,x2eiαY1x1,x2∧mY2x1,x2eiαY2x1,x2ifx1,x2∈C1,y1y2∈C2≤mX1x1eiαX1x1x1∧mX1x2eiαX1x2∧mX2y1eiαX2y1∧mY2y2eiαY2x1.
SinceG1 and G2 are CIFGs,
(43)nY1eiβY1⨀nY2eiβY2x1,y1,x2,y2=nY1x1,x2eiβY1x1,x2∨nY2x1,x2eiβY2x1,x2ifx1,x2∈C1,y1y2∈C2≤nX1x1eiβX1x1x1∨nX1x2eiβX1x2∨nX2y1eiβX2y1∨nY2y2eiβY2x1.
SinceG1 and G2 are CIFGs,
(44)mY1eiαY1⨀mY2eiαY2x1,y1,x2,y2=mX1x1eiαX1x1x1∧mX1x2eiαX1x2∧mX2y1eiαX2y1∧mY2y2eiαY2x1ifx1,x2∉C1,y1,y2∉C2=mY1eiαY1⨀mY2eiαY2x1,y1,x2,y2.
SinceG1 and G2 are CIFGs,
(45)nY1eiβY1⨀nY2eiβY2x1,y1,x2,y2=nX1x1eiβX1x1x1∨nX1x2eiβX1x2∨nX2y1eiβX2y1∨nY2y2eiβY2x1ifx1,x2∉C1,y1,y2∉C2=nY1eiβY1⨀nY2eiβY2x1,y1,x2,y2.
Hence,(46)mY1eiαY1⨀mY2eiαY2x1,y1,x2,y2≤mX1eiαX1⨀mX2eiαX2x1,y1∧mX1eiαX1⨀mX2eiαX2x2,y2,nY1eiβY1⨀nY2eiβY2x1,y1,x2,y2≤nX1eiβX1⨀mX2eiβX2x1,y1∨nX1eiβX1⨀nX2eiβX2x2,y2.
Hence,G1eG2 is CIFG.Theorem 21.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be strong CIFGs. Then, modular product G1⨀G2 is also a strong CIFG.Theorem 22.
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be two complete CIFGs.
(1)
IfmY1x≤mY2x,αY1x≤αY2x and nY1x≥nY2x,βY1x≥βY2x, then degG1eG2x,y=degG1x(2)
IfmY1x≥mY2x,αY1x≥αY2x and nY1x≤nY2x,βY1x≤βY2x, then degG1eG2x,y=degG2xProof.
(1)
LetG1=B1,C1,X1,Y1 and G2=B2,C2,X2,Y2 be two complete CIFGs. The degree of vertex x,y∈B1⨀B2 is degG1⨀G2x,y=deg1G1⨀G1x,y,deg2G1⨀G1x,y. From (4), we have(47)d1G1⨀G2x1,y1=∑x1x2∈C1,y1y2∈C2mY1x1,x2∧mY2y1,y2eαY1x1,x2∧mY2y1,y2+∑x1x2∉C1,y1y2∉C2mX1x1∧mX1x2∧mX2y1∧mX2y1eαX1x1∧αX1x2∧αX2y1∧αX2y2=∑x1x2∈C1,y1y2∈C2mY1x1,x2∧mY2y1,y2eαY1x1,x2∧αY2y1,y2.
Since bothG1 and G2 are complete CIFGs, it follows that deg1G1⨀G2x1,y1=deg1G1⨀G2x1.
Similarly,(48)deg1G1⨀G2x1,y1=∑x1,x2∈C1,y1,y2∈C2mY1x1,x2∧mY2y1,y2eαY1x1,x2∧mY2y1,y2+∑x1,x2∉C1,y1,y2∉C2mX1x1∧mX1x2∧mX2y1∧mX2y1eαX1x1∧αX1x2∧αX2y1∧αX2y2=∑x1,x2∈C1,y1,y2∈C2mY1x1,x2∧mY2y1,y2eαY1x1,x2∧αY2y1,y2.
Since bothG1 and G2 are complete CIFGs, it follows that deg1G1⨀G2x1,y1=deg1G1⨀G2x1.
(2)
Omitted


## 7. Conclusions

Graphs are a strong and adaptable data structure for representing real-world connections between different types of data. Graph operations take existing graphs and build new ones. In this investigation, we looked at some interesting results from the key operations direct, semistrong, strong, and modular products for complex intuitionistic fuzzy graphs. A strong complex intuitionistic fuzzy graph is also defined, as well as a number of noteworthy findings. We also look at how a vertex’s degree behaves in the modular product of two complex intuitionistic fuzzy graphs.



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*Source: 6515646-2021-12-16.xml*