#### 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.

#### 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 Section 3, 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) of a nonempty set is a function,

*Definition 2 [2]. *An intuitionistic fuzzy set (IFS) of a universe of discourse is a triplet of the form , where the functions and are the membership function (degree of truthfulness) and nonmembership functions (degree of falsity), respectively. These functions must fulfill the condition

*Definition 3 [36]. *The object of the form
is a complex intuitionistic fuzzy set (CIFS) defined on universe of discourse .

,

*Definition 4 [27]. *An intuitionistic fuzzy graph is of the form on the crisp graph with vertex set and edge set , where
(1) and such that and denote the membership value (MV) and nonmembership value (NMV) of the element , respectively, such that for all (2) and where and are defined by and such that

*Definition 5 [35]. *A complex intuitionistic fuzzy graph (CIFG) with an underlaying vertex set and edge set is defined to be a pair , where is a CIFS on and is a CIFS on such that
for all .

*Definition 6 [35]. *Let be the given CIFG. The degree of a vertex in is defined by

#### 3. Direct Product of Two CIFGs

*Definition 7. *The direct product of two CIFGs, and such that is defined to be CIFG where

The MV and NMV for the vertex in are given by

The NM and NMV for the edge in are given by

Now we define the strong CIFG.

*Definition 8. *A CIFG is called strong CIFG if

Theorem 9. *Let and be two strong CIFGs; then, is also a strong CIFG.*

*Proof. *As and are strong CIFGs, so by (9) and (10), we have
for all and

Now from (7) and (8), we have
In addition for nonmembership,
This completes the proof.

Theorem 10. *Let and be two CIFGs, such that is strong; then, at least one of or must be strong CIFG.*

*Proof. *Suppose and are not strong CIFGs. Thus, there exists at least one such that
Let ; then,
Therefore,
Again, let ; then,
Therefore,
This shows that is not strong, which is contradiction.

This completes the proof.

#### 4. Semistrong Product of Two CIFGs

*Definition 11. *The semistrong product of two CIFGs and such that is defined to be the CIFG as , where

The MV and NMV of the vertex in are given as

The MV and NMV for the edge and in are given by as follows:

Theorem 12. *If and are strong CIFGs, then is also strong.*

*Proof. *If , then using (19) and (20), we have
Similar, we can show that .

Again, if , then using (22) and (24), we have
Similarly, we can show that
This completes the proof.

Theorem 13. *If are two CIFG, such that is strong, then at least one of or must be strong.*

#### 5. Strong Product of Two CIFGs

*Definition 14. *The strong product of two CIFGs is and such that is defined to be where and . The MV and NMV for the vertex in are given by

The MV and NMV for edges in are given by

Theorem 15. *If and are strong CIFGs, then is also strong.*

Theorem 16. *If and are two CIFGs, such that is strong, then at least one of or must be strong.*

*Definition 17. *A CIFG is said to be complete if
for all

Theorem 18. *If and are two CIFGs, then is complete.*

*Proof. *As a strong product of CIFGs is CIFG, and every pair of vertices is adjacent. If , then using (29) and (30), we have
And by (31), it follows that
If , then
Similarly,
Again if , then
Similarly,
This completes the proof.

#### 6. Modular Product of CIFGs

*Definition 19. *Let and be two CIFGs with underlying vertex sets and and edge sets and , respectively. Then, modular product of and is with underlying vertex set and underlying edge set or with
Here, and .

The MV and NMV for edges in are given by