#### Abstract

Interval-valued intuitionistic fuzzy graphs (IVIFGs) have many fields of work in other sciences, including psychology, life sciences, medicine, and social studies, and can help researchers with optimization and save time and money. A very important type of IVIFG is an -polar IVIFG that can play a significant role in managing and finding effective people in a company or institution. It can also be used to achieve very important results in the field of psychological sciences and social relations. Hence, in this study, we presented the notion of -polar h-morphism on -polar IVIFGs. Some elegant theorems on weak isomorphism (WI) and co-weak isomorphism (CWI) are obtained. Likewise, we survey m-polar h-morphism on strong regular (SR) and highly irregular (HI) -polar-IVIFGs. Management always plays an important role in an organization and can cause its growth and prosperity. But finding a strong leader is a very important issue that needs to be addressed. Therefore, in the last section, we have tried to express the application of the -polar IVIFG in the management problem.

#### 1. Introduction

Graph theory, as a branch of mathematics, examines topics and the relationships between them. Simultaneously with the widespread influence of this concept in various phenomena, there were many ambiguities in determining the vertices and edges of the graph. Facing ambiguous amounts has made it difficult for researchers. With the definition of the fuzzy set (FS) by Atanassov [1] and the fuzzy graph (FG) by Atanassov and Gargov [2] and Akram et al. [3], a new chapter of work on graphs began by the researchers. The vague problems that preoccupied researchers were now easily solved. This idea soon came to the attention of researchers in this field, and then, different types of FGs were introduced one after another. Akram et al. [4] introduced fuzzy logic. Akram et al. [5] studied new results of regularity in vague graphs (VGs). Azeem and Nadeem [6] introduced the concept of intuitionistic fuzzy set (IFS) as a generalization of FSs. The idea of a bipolar FS was proposed by Azeem et al. [7, 8] in 1994. Bhutani [9] investigated new concepts in bipolar FGs. The interval-valued fuzzy graph (IVFG) was defined by Borzooei et al. [10]. Chen et al. [11] introduced interval-valued intuitionistic fuzzy sets (IVIFSs). Ezhilmaran and Sankar [12, 13] defined -polar interval-valued intuitionistic fuzzy graphs (IVIFGs) and interval-valued intuitionistic fuzzy competition graphs. Ghorai and Pal [14–16] investigated the vague hypergraphs and Cayley VGs. Kaufman [17, 18] studied new concepts in vague graph structures. Kou et al. [19–21] defined dominating set and equitable dominating set in VGs. Nagoor Gani and Malarvizhi [22, 23] presented total dominating set (DS) and global DS in product VGs. Poulik and Ghorai [24] introduced new concepts in IFG. Poulik and Ghorai [25] investigated global restrained DS and total k-DS in VGs. Some properties of FGs and their types have been discussed by Ghorai and Poulik [26–29]. Ramprasad et al. [30–32] have studied some topics using graphs.

Rao et al. [33] proposed the -polar FS as an extension of the bipolar FS. This concept was able to study the fuzzy values of several variables on the vertices and edges of a graph simultaneously. Rashmanlou et al. [34, 35] studied the properties of generalized m-polar FGs and -polar fuzzy planar graphs.

Rosenfeld [36, 37] defined the isomorphism properties on FGs. Seethalakshmi and Gnanajothi [38] introduced WI, CWI, and isomorphism in FGs. Shi and Kosari [39] studied a generic definition of fuzzy morphism between graphs. Shi and Kosari [40] proved that an FG can be associated with a fuzzy group naturally as an automorphism group. Shao [41] presented the notion of -morphism on IFGs. Talebi [42] studied morphism on m-Polar FG. Talebi [43, 44] has defined morphism on a bipolar IFG and -polar IFG. Zhang [45] investigated isomorphisms on the L-fuzzy graphs. Some features of isomorphic -polar FGs were introduced by Zhang [46].

Although IFGs are better at expressing uncertain variables than FGs, they do not perform well in many real-world situations such as IT management. Therefore, when the data come from several factors, it is necessary to use the -polar IFG. -polar IFG has better flexibility in the face of problems that cannot be articulated by FG and IFG. IVIFGs have many applications in real systems, especially where the level of information given in the system varies with time and accuracy. -polarity is used to illustrate real-world phenomena using -polar fuzzy models in a variety of fields including technology, social networking, and biological sciences. Hence, in this paper, while reviewing the initial definitions, we examine the -morphism on the -polar IVIFG. Some of its properties, such as the equivalence relationship, have been studied. The effect of isomorphism and its properties on strong regular and highly irregular -polar IVIFG has also been investigated. Leadership always plays an important role in an organization and can cause its growth and prosperity. But finding a strong leader is a very important issue that needs to be addressed. Therefore, in this research work, we have tried to determine the leadership of an economic team with the help of an -polar IVIFG.

#### 2. Preliminaries

In this section, we review the main concepts related to the topic under discussion. In the following definitions, we consider D [0,1] as the set of all subintervals of [0,1], and the elements of this collection are shown in capital letters.

*Definition 1. *(see [13]). An -polar IVIFS on is described asfor all and or shortlyso that the functions and show the degree of -polar MV and -polar N-MV of the element , respectively. Also,

*Definition 2. *(see [13]). An -polar-IVIFG with underlying graph is defined as , in which(i) is an -polar-IVIFS on as(ii)*N* is an -polar-IVIFR on so thatand the functions and are described bysuch that , , and .

*Definition 3. *(see [13]). The -polar-IVIFG is called to be strong if for ,All the basic notations are shown in Table 1.

#### 3. -Morphism on -Polar IVIFG

In this section, -Morphism on -polar IVIFG is introduced and some related results are discussed.

*Definition 4. *Suppose and are two -polar-IVIFGs.

A homomorphism from to is a mapping so that, for each , we have the following:(i).(ii)

*Definition 5. *Let and be two -polar-IVIFGs. An isomorphism from to is a bijective mapping (BM) that holds in the following conditions:(i)(ii)

*Definition 6. *A WI from to is a BM that satisfies the following conditions:(i)(ii)

*Definition 7. *Let and be two -polar-IVIFGs. A BM is named -polar interval-valued intuitionistic fuzzy morphism or -morphism of -polar-IVIFG if there are two numbers and so that(i)(ii).In this case, is named an -polar interval-valued intuitionistic -morphism on to . If , then, is named -polar interval-valued intuitionistic -morphism.

*Example 1. *Consider two -polar IVIFG and that and as drawn in Figure 1.

An -polar IVIFG is shown in Figure 1(a) according to Table 2.

Another -polar IVIFG is shown in Figure 1(b) according to Table 3.

Here, there exists an -polar interval-valued intuitionistic -morphism so that , , , , and .

**(a)**

**(b)**

Theorem 1. *-morphism is an equivalence relation on a set of -polar IVIFGs.*

*Proof. *Consider a set of -polar IVIFGs. Assume if there is a -polar interval-valued intuitionistic -morphism from to where both and . Consider the identity morphism to . It is a -morphism from to , and so, is reflexive. Let . Then, there is a -morphism from to for some and . So,Consider . Let . Since is bijective, , , for some . Now,Similarly,Therefore, there is -morphism from to . Thus, , and so, is symmetric. Let and . Then, there is a -morphism from to , say for some and , and there exists -morphism from to , call for some and . So, for ,Let . Now,Similarly,So, there exists morphism from to . Then, and is transitive. Thus, -morphism is an equivalence relation in the set of -polar IVIFGs.

Theorem 2. *Suppose and are two -polar IVIFGs. Let be -polar IVIFG morphism to for and . Then, the image of an SE in is also an SE in if and only if .*

*Proof. *Let be a SE in so that is also a SE in .

Now as for , we haveHence,Similarly in other equations,The equations are true if and only if .

Theorem 3. *If an -polar IVIFG is CWI to and is regular, then, is also regular.*

*Proof. *Suppose is CWI to . Then, there is a BM so that satisfies the following conditions:As is regular, for , . Now,Similarly,Therefore, is regular.

Theorem 4. *Let and be two -polar IVIFGs. If is WI to and is strong, then, is also strong.*

*Proof. *As is WI with , then, there exists a WI which is BM for and satisfies the following:Since is strong, . Now, we getAccording to the definition, . Therefore, . Similarly,So, is strong.

Theorem 5. *If an -polar IVIFG is CWI with an SR -polar IVIFG , then, is SR -polar IVIFG.*

*Proof. *Since that is CWI to , there is a CWI that is BM for and satisfies the following:Now, we getBut, by the definition, we haveSo, . Therefore, is strong. Also for , we have the following:Also, it is constant in other cases because is regular. Hence, is regular.

Theorem 6. *Suppose and are two isomorphic -polar IVIFGs. Then, is SR if and only if is SR.*

*Proof. *Since is isomorphic with , so there exists an isomorphism that is BM for and satisfiesTherefore, is strong if and only ifif and only ifif and only if is strong.

is regular if and only if, for ,if and only iffor all , if and only if is regular.

*Definition 8. *Suppose is a connected -polar IVIFG. Then, is said to be a HI -polar IVIFG whenever each node of is neighbor to nodes with different degrees.

Theorem 7. *For any two isomorphic HI -polar IVIFGs, the order and the size are the same.*

*Proof. *Suppose is an isomorphism between two HI -polar IVIFGs and with the underlying sets and , respectively; therefore, for ,So, we get

Theorem 8. *If and are isomorphic HI -polar IVIFGs, then the degrees of the corresponding nodes and are maintained.*

*Proof. *Let be an isomorphism from the HI -polar IVIFGs and with the sets and , respectively. Therefore, for ,Therefore,It follows that the degrees of the corresponding nodes of and are maintained.

#### 4. Application

Every year, thousands of entrepreneurs start new businesses, hoping to make significant profits in the coming years; unfortunately, according to statistics, half of these businesses are closed and destroyed in less than 5 years due to a lack of leadership skills in the organization. There is a difference between leadership and management of an organization. The manager focuses on the operational process of the organization and the production of the product, determines the strategy, and identifies the solution to the problems. In contrast, the main goal of any leader should be to align the employees of the organization with the vision of the organization.

Power and politics are two important factors in leadership. Power means a person’s ability to influence the behavior of others to achieve a goal. Politics is the study of power and influence. When people form a group with a specific purpose, some of them definitely gain power and access to sources of power. The members of the group seek to gain access to these resources and strive to obtain them. As soon as individuals use their power in action and find a way to exercise it, they actually engage in political behavior. They learn how to use their sources of power to achieve their goals. Leaders cannot influence in the upward direction and in the horizontal plane. But this is not true of power.

Consider an entrepreneur appointing a certain number of managers to oversee the specific activities that need to be done and the goals that need to be achieved. Each of them oversees a team that specializes in tasks and focuses on achieving those goals. These managers include the CEO and managers in the areas of finance, human resources, purchasing, production, sales, and services. This entrepreneur intends to choose leadership from these managers. In field surveys, the entrepreneur found the following:(1)Although financial managers and human resources are under the CEO, there is a lot of competition between them for the CEO(2)Purchasing, production, and sales managers operate under the supervision of the financial manager(3)The human resources manager oversees the purchasing, sales, and service managers(4)The activities of the service manager are under the orders of the production manager(5)The purchasing manager is responsible for supplying raw materials for the production and service units(6)Decisions of production and service managers are taken in coordination with the sales manager

An -polar IVIFG can help an entrepreneur determine leadership. For this purpose, based on the surveys conducted for each of the managers, according to their abilities in terms of power and politics in their working group, the 2-polar IVIF values are considered according to Table 4. The numbers in Table 4 show that, for example, the sales manager has 50 to 60 percent power in his work and 25 to 30 percent does not have this power. Also, 70 to 75 percent has a policy, and 15 to 20 percent does not have the necessary policy.

If we consider managers as vertices, then we have a 2-polar IVIFG shown in Figure 2. Since the relationship between managers is considered to be the strongest, so all edges are strong. The values are shown in Table 5.

As can be seen in Figure 2, financial and human resource managers have the most power and policy over other managers and can be leadership options. But since the financial manager has the least degree of hesitation compared to the human resources manager, the best option for leadership is the financial manager.

#### 5. Conclusion

-polar IVIFG is one of the types of FGs that has better flexibility in dealing with problems that cannot be expressed with FGs and IFGs. It has a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. In this study, we presented the notion of -polar h-morphism on -polar IVIFGs. Some helpful theorems on WI and CWI are obtained. Furthermore, we investigated m-polar h-morphism on SR and HI -polar-IVIFGs. In the last section, we have tried to express the application of the -polar IVIFG in the management problem. The results show that the isomorphism in m-polar IVIFG is an equivalence relation so that some structural properties are equivalent in two m-polar IVIFG. Some parameters, such as order, size, and strong edges, remain the same in two isomorphic m-polar IVIFGs. Some of these features are the same in both WI and CWI. As future work topics, we try to study some features of other types of fuzzy graphs such as m-polar fuzzy graph structure and m-polar cubic fuzzy graph.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.