Energy of Vague Fuzzy Graph Structure and Its Application in Decision Making

: Vague graphs (VGs), belonging to the fuzzy graphs (FGs) family, have good capabilities when faced with problems that cannot be expressed by FGs. The notion of a VG is a new mathematical attitude to model the ambiguity and uncertainty in decision-making issues. A vague fuzzy graph structure (VFGS) is the generalization of the VG. It is a powerful and useful tool to ﬁnd the inﬂuential person in various relations. VFGSs can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problems where fuzzy graphs may fail to reveal satisfactory results. Moreover, VGSs are very useful tools for the study of different domains of computer science such as networking, social systems, and other issues such as bioscience and medical science. The subject of energy in graph theory is one of the most attractive topics that is very important in biological and chemical sciences. Hence, in this work, we extend the notion of energy of a VG to the energy of a VFGS and also use the concept of energy in modeling problems related to VFGS. Actually, our purpose is to develop a notion of VFGS and investigate energy and Laplacian energy (LE) on this graph. We deﬁne the adjacency matrix (AM) concept, energy, and LE of a VFGS. Finally, we present three applications of the energy in decision-making problems.


Introduction
In this modern epoch of technology, modeling uncertainties in engineering, computer sciences, social sciences, medical sciences, and economics is growing extensively. Classical mathematical methods are not always useful for dealing with such problems. FG models are advantageous mathematical tools for solving problems in various aspects. Fuzzy graphical models are obviously better than graphical models because of the natural existence of vagueness and ambiguity. The subject of a fuzzy set (FS) was introduced by Zadeh [1] in 1995. After the introduction of fuzzy sets, FS theory has included a large research field. Since then, the theory of FSs has become a vigorous area of research in different disciplines including life sciences, management, statistic, graph theory, and automata theory. The subject of FGs was proposed by Rosenfeld [2]. Kaufmann [3] presented the definitions of FGs from the Zadeh fuzzy relations in 1973. Akram et al. [4][5][6] introduced several concepts in FGs. Some of these product operations on FGs were presented by Mordeson and Peng [7]. Gau and Buehrer [8] proposed the concept of vague set (VS) in 1993 by replacing the value of an element in a set with a subinterval of [0, 1]. One type of FG is VG. VGs have a variety of applications in other sciences, including biology, psychology, and medicine. Moreover, a VG can concentrate on determining the uncertainties coupled with the inconsistent and indeterminate information of any real-world problems where FGs may not lead to adequate results. Ramakrishna [9] introduced the concept of VGs and studied some of their properties. After that, Akram et al. [10] introduced vague Definition 3 ([9,21]). Suppose G * = (W, E) is a graph. A pair G = (Q, R) is named a VG on graph G * = (W, E), where Q = (t Q , t Q ) is a VS on W and R = (t R , f R ) is a vague relation on W such that, for all v, z ∈ W . Note that R is called vague relation on Q. A VG G is named strong if for all v, z ∈ W. Definition 4 ([12]). Suppose G = (Q, R) is a VFG on G * , the degree of vertex v is defined as The order of G is defined as
Definition 7. Two vertices that are connected by an edge are named adjacent. The AM A = [v pq ] for a graph G * = (W, E) is a matrix with n rows and m columns, n = |V|, and its entries are defined by Definition 8. The spectrum of a matrix is defined as a set of its eigenvalues, and we denote it with SP(G). The eigenvalues γ p , p = 1, 2, ..., l of the AM of G are the eigenvalues of G. The spectrum γ 1 , γ 2 , ..., γ l of the AM of G is the SP(G); the eigenvalues of the graph satisfy the following relations: Definition 9. The energy of a graph G is denoted by E (G) and is defined as the sum of the absolute values of the eigenvalues of A, that is, where γ p is an eigenvalues of A.
Theorem 1. Suppose that G is a graph with l vertices and k edges and A is the AM of G then All the essential notations are shown in Table 1.

Energy of a Vague Fuzzy Graph Structure
In this section, we express a new notion of the extension of the energy of an FGS called VFGS. We define the notion of energy of a VFGS which can be used in real science.
represent the strength of relationship between z p and z q , respectively. Definition 11. The energy of a VFGS G = (Q, R 1 , R 2 , ..., R n ) is defined as the following: Suppose Q, R 1 , R 2 , and R 3 is a vague fuzzy subset of W, E 1 , E 2 , and E 3 , respectively, then, G = (Q, R 1 , R 2 , R 3 ) is a VFGS on G * as shown in Figure 2, such that The AMs and energy of each degree of G are obtained as follows: Therefore, the energy of a VFGS G = (Q, and A( f R i (v p v q )), respectively, then, is a symmetric matrix with zero trace, its eigenvalues are real with a sum equal to zero.
pq ] of G is an n × n diagonal matrix, which is defined as: are the degrees matrix and AM of a VFGS, respectively.

Definition 14.
The LE of a VFGS G = (Q, R 1 , R 2 , ..., R n ) is defined as the following: Example 3. Consider a GS G * = (W, E 1 , E 2 ) , where W = {x, w, z, y, m}, E 1 = {xw, wy, zy}, and E 2 = {wz, ym}. Suppose Q, R 1 , and R 2 are a vague fuzzy subset of W, E 1 , and E 2 , respectively, then, G = (Q, R 1 , R 2 ) is a VFGS on G * as shown in Figure 3, such that The AMs and energy of each degree of G are obtained as follows: The degree matrix and LE are as follows: After computing, we have LE (A(t R 1 )) = 1.39 and LE (A( f R 1 )) = 4.2.
(II) By tracing the properties of the matrix, we have . . .
Similarly, the other relations are fixed.
Theorem 7. Suppose G = (Q, R 1 , R 2 , ..., R n ) is a VFGS on n vertices and L R i (G) is the LM of G, then (I) Proof.
Proof. Using the Caushy-Schwarz inequality, we obtain Similarly, we can prove cases (II).

Designing an Organizational Communication System
In the real world, communication is very important in every sector, and one of the things we want to talk about is organizational communication. Organizational communication has attracted the attention of many behavioral and organizational science thinkers to the extent that many organizational difficulties have been analyzed and suitable solutions have been found for them. Some thinkers of organizational communication, such as management consultants who have been studying organizational inadequacies in recent years, believe that many of the issues and problems governing organizations are a result of incorrect communication context and lack of attention to the subtleties of organizational communication. If the managers were aware of these issues, they would probably perform their work more effectively and efficiently. With the continuation of interactions between employees, communication networks are formed naturally. Because duties, relations, and memberships are changing, the connections are not fixed and permanent. According to these concepts, we present an example of multiple organizational relationships and examine the importance and impact of multiple relationships in increasing the efficiency and success of an organization.
In this example, we consider education organization as a graph whose vertices include organization management (z 1 ), financial vice president (z 2 ), education unit (z 3 ), educational vice president (z 4 ), technology unit (z 5 ), and research unit (z 6 ). In this educational organization, we want to examine the three desired relationships between the introduced units' efficient manpower (R 1 ), improving the scientific and educational level (R 2 ), and the relationship between salaries and benefits in raising the quality and efficiency of the organization (R 3 ).
In this application, we can clearly see that if the amount of energy in the relationships between the units is greater, the units have a greater impact on each other. Here, it is clear that the energy in R 3 is more than others. Therefore, the educational vice president unit and technology unit, education unit and research unit, education unit and educational vice president unit , and education unit and research unit have a greater effect on each other.

Role of Virtual Social Networks on Cultural Communication
Virtual space has entered many areas of life in different human societies in such a way that it is used for various purposes, including business, games and entertainment, and similar work activities, and the beneficiaries of individuals and institutions use these virtual spaces to facilitate work or provide special services. Currently, social networks are the inhabitants of the turbulent ocean of the Internet. Networks play an essential role in the world's media equations with virtual socialism. The virtual space is formed depending on social constructions, and technological growth, media convergence, and related issues are different outputs in different social conditions. Virtual social networks, such as Twitter, Instagram, Facebook, WhatsApp, Telegram, etc., which provide the opportunity to meet people from different cultures with different languages and ethnicities, are very important in intercultural communication, and since in Iran the application of virtual social networks is widespread, these virtual social networks are considered an important source for the intercultural communication of Iranians. Due to the fact that today's era is the era of communication and virtual space, it is not possible to communicate in this space without accepting cultures and accepting cultures without taking into account customs and beliefs and, ultimately, creating a common culture. Therefore, the main issue of this application is the role of virtual social networks in cultural communication in Iran.
. Thus, the best platform is WhatsApp.

Role of Advertising Tools in Raising the Quality Level of Advertising Companies
An advertising company is a company that creates, plans, and manages all aspects of advertising for its customers. Advertising companies can specialize in a specific field and branch of advertising, such as interactive advertising, or comprehensively provide services and use all advertising tools such as websites, social media, online advertising, etc. Brochures, catalogs, instant messaging with direct mail, print media, television ads, sales invitations, etc., are among the advertising tools that the advertising company uses to operate in this field. In this part, four advertising companies signed contracts among themselves to raise the quality level of their work. In these contracts, the companies defined relationships between themselves. In their meeting, these four companies expressed the factors that can affect their work promotion, among which are the right price regarding the quality, the professional production group, company services, and customer orientation. We assume that there are four advertising companies with the names A, B, C, and D. We define the relationships between them as follows, Consider Q = {A, B, C, D} as a set of advertising companies and R i = { creating television teasers (R 1 ), designing and printing billboards (R 2 ), advertising photography (R 3 ) } as sets of relations between advertising companies. Now, in Figure 9, we assume G = (Q, R 1 , In Figure 9, it is clear that there are three different relationships between the advertising companies; we first obtain the energy of each relationship. The AMs and energy of each degree of G are obtained as follows: Therefore, the energy of a VFGS G = (Q, The degree matrix and LE are as follows: After computing, we have LE (A(t R 1 )) = 0.4 and LE (A( f R 1 )) = 1. In this application, we can clearly see that if the amount of energy in the relationships between the advertising companies is greater, they have a greater impact on each other. Here, it is clear that the energy in R 2 is more than others. Therefore, in order to raise the quality level of their work, two companies A and B, and also two companies C and D, can cooperate in the field of designing and printing advertising billboards.

Conclusions
Graph theory has many applications in solving different problems of several domains, including networking, planning, and scheduling. VGSs are very valuable tools for the study of various domains of computational intelligence and computer science. Optimization, neural networks, and operations research can be mentioned among the applications of VGSs in different sciences. Since many parameters in real-world networks are specifically related to the concept of energy, this concept has become one of the most extremely used concepts in graph theory. However, the energy in FG is so important because of the confrontation with uncertain and ambiguous topics. This concept becomes more interesting when we know that we are dealing with an FG called VFGS. This led us to examine the energy in VFGSs. So, in this work, we presented the notion of the energy of a VFGS and investigated some of its properties. We obtained the energy of the VFGS by using the eigenvalues of the AM and calculating its spectrum. Moreover, we expanded the concept of the LE on a VFGS. Finally, three applications of the VFGS in decision making are presented. In our future work, we will investigate the concepts of domination set, vertex covering, and independent set in VGSs and give applications of different types of domination in VGSs and other sciences.

Conflicts of Interest:
The authors declare no conflict of interest.