Abstract
Neutrosophic cubic graph (NCG) belonging to FG family has good capabilities when facing problems that cannot be expressed by FGs. When an element membership is not clear, neutrality is a good option that can be well supported by a NCG. Hence, in this paper, some types of edge irregular neutrosophic cubic graphs (EI-NCGs) such as neighborly edge totally irregular (NETI), strongly edge irregular (SEI), and strongly edge totally irregular (SETI) are introduced. A comparative study between NEI-NCGs and NETI-NCGs is done. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been presented.
1. Introduction
The fuzzy set theory was introduced by Zadeh [1]. It focuses on the membership degree of an object in a particular set. Kaufmann [2] represented FGs based on Zadeh’s fuzzy relation [3, 4]. Rosenfeld [5] described the structure of FGs obtaining analogs of several graph theoretical concepts. Bhattacharya [6] gave some remarks on FGs. Several concepts on FGs were introduced by Mordeson et al. [7]. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Atanassov [8] defined an extension of fuzzy set by introducing non-membership function and defined intuitionistic fuzzy set (IFS). But after a while, Atanassov and Gargov [9] developed IFS and presented interval-valued intuitionistic fuzzy set (IVIFS). Hongmei and Lianhua [10] defined interval-valued fuzzy graph and studied its properties. Zhang et al. [11] introduced bipolar fuzzy sets and relations. Smarandache [12–14] gave the idea of neutrosophic sets. Kandasamy [15] defined neutrosophic graphs. Akram et al. [16–19] studied new results in NGs. Jun et al. [20] introduced cubic set. For more details about cubic sets and their applications in different research areas, we refer the readers to [21–23]. Rashid et al. [24] investigated cubic graphs. Jun et al. [25, 26] gave the idea of neutrosophic cubic set and defined different operations on it. Gulistan et al. [27, 28] presented complex bipolar fuzzy sets, NCGs, and some binary operations on it. Karunambigai et al. [29] discussed edge regular-IFG. Gani and Radha [30] studied the concept of regular fuzzy graphs and defined degree of a vertex in FGs. Gani et al. [31] investigated the concept of IFGs, NI-FGs, and HI-FGs in 2008. Nandhini [32] described the concept of SI-FG and studied its properties. Maheswari and Sekar defined the concepts of edge irregular-FGs and edge totally irregular-FGs [33]. Also, they analyzed some properties of NEI-FGs, NETI-FGs, SEI-FGs, and SETI-FGs [34, 35]. Rao et al. [36–38] studied dominating set, equitable dominating set, valid degree, isolated vertex, and some properties of VGs with novel application. Kou et al. [39] investigated g-eccentric node and vague detour g-boundary nodes in VGs. Shi et al. [40, 41] introduced total dominating set, perfect dominating set, and global dominating set in product vague graphs. Rashmanlou et al. [42] presented some properties of cubic graphs. Amanathulla et al. [43] studied on distance two surjective labeling of paths and interval graphs. Bhattacharya and Pal [44] gave the fuzzy covering problem of fuzzy graphs and its application. Borzooei et al. [45, 46] defined inverse fuzzy graphs and new results of domination in vague graphs. Kalaiarasi et al. [47] presented regular and irregular m-polar fuzzy graphs. Ramprasad et al. [48] investigated some properties of highly irregular, edge regular, and totally edge regular m-polar fuzzy graphs. Poulik and Ghorai [49] defined certain indices of graphs under bipolar fuzzy environment. Ullah et al. [50] introduced new results on bipolar-valued hesitant fuzzy sets. Jan et al. [51] presented some root level modifications in interval valued fuzzy graphs. Broumi et al. [52] introduced a novel system and method for telephone network planning based on neutrosophic graph. Muhiuddin et al. [53, 54] presented reinforcement number of a graph and new results in cubic graphs. Talebi et al. [55–57] presented some properties of irregularity and edge irregularity on intuitionistic fuzzy graphs and single valued neutrosophic graphs.
NCGs have many applications in psychology and medical sciences and can play a significant role in solving the vague and complex problems that exist around our lives. With the help of this fuzzy graph, the most effective person in an organization can be determined according to the amount of its performance in a specific period. Therefore, in this paper, some types of EI-NCGs such as neighborly edge totally irregular (NETI)-NCGs, strongly edge irregular (SEI)-NCGs, and strongly edge totally irregular (SETI)-NCGs are introduced. Also, we have given some interesting results about EI-NCGs, and several examples are investigated. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been presented.
2. Preliminaries
Definition 1. A graph is a mathematical model consisting of a set of nodes and a set of edges , where each is an unordered pair of distinct nodes.
Definition 2 (see [5]). A FG is a non-empty set together with a pair of functions and so that , .
All the basic notations are shown in Table 1.
3. New Concepts of Edge Irregular-NCGs
Definition 3. Let be a graph. By NCG of , we mean a pair where is the NCS representation of and is the NCS representation of so that(i).(ii).(iii).
Definition 4. Let be a NCG on . Then, the degree of a node is defined as where , . , . and .
Definition 5. Let be a NCG on . The TD of a node is defined by where , . , . and .
Definition 6. Let be a NCG on . Then:(i) is irregular, if there is a node that is neighbor to nodes with VDs.(ii) is TI, if there is a node which is neighbor to nodes with various TDs.
Definition 7. Let be a CNCG. Then, is called a(i)NI-NCG if each pair of neighbor nodes has VDs.(ii)NTI-NCG if each pair of neighbor nodes has various TDs.(iii)SI-NCG if each pair of nodes has VDs.(iv)STI-NCG if each pair of nodes has various TDs.(v)HI-NCG if each node in is neighbor to the nodes having VDs.(vi)HTI-NCG if each node in is neighbor to the nodes having various TDs.
Definition 8. Let be a NCG. The degree of an edge is defined as where , . , . and .
Definition 9. Let be a NCG. The TD of an edge is defined as where . . . . . .
Definition 10. Let be a CNCG on . Then, is called a(1)NEI-NCG if each pair of AEs has VDs.(2)NETI-NCG if each pair of AEs has various TDs.
Example 1. Consider a graph which is both NEI-NCG and NETI-NCG.
Consider where and are defined asFrom Figure 1,Clearly, is a NEI-NCG.So, is a NETI-NCG.
Therefore, is both NEI-NCG and NETI-NCG.
Example 2. NEI-NCG need not to be NETI-NCG.
Let be a NCG and be a star that includes four nodes where
and are defined asFrom Figure 2,.
Here, . Hence, is a NEI-NCG. But is not a NETI-NCG, since all edges have same TDs.
Example 3. NETI-NCG does not need to be NEI-NCG. The following shows this subject.
Let be a NCG so that is a path that consists of four nodes where
and are defined asFrom Figure 3,Here, . Hence, is not a NEI-NCG. But is a NETI-NCG, since and .
Theorem 1. Let be a CNCG on and be a CF. Then, is a NEI-NCG, iff is a NETI-NCG.
Proof. Assume that is a CF, and let , in , where is constant.
Let and be pair of AEs in . Then,Therefore, adjacent edges have various degrees if and only if they have various total
degrees. So, is a NEI-NCG iff is a NETI-NCG.
Remark 1. Let be a CNCG on . If is both NEI-NCG and NETI-NCG, then does not need to be a CF.
Example 4. Let be a NCG and be a path that consists of four nodes where and are defined asFrom Figure 4,Here, and . Hence, is a NEI-NCG. Also, and . Hence, is a NETI-NCG. But is not CF.
Theorem 2. Let be a CNCG on and be a CF. If is a SI-NCG, then is a NEI-NCG.
Proof. Let be a CNCG. Assume that is a CF, and let , in , where is constant.
Let and be any two AEs in and be a SI-NCG. Then, each pair of nodes in
has VDs, and henceTherefore, each pair of AEs has VDs. Hence, is a NEI-NCG.
Theorem 3. Let be a CNCG on and be a CF. If is a SI-NCG, then is a NETI-NCG.
Remark 2. Converse of Theorems 3 is not generally true.
Example 5. Let be a NCG so that is a path on four nodes where and are defined asFrom Figure 5,Here, is not a SI-NCG. . . . .It is noted that and . Also, and . Hence, is both NEI-NCG and NETI-NCG. But is not a SI-NCG.
Theorem 4. Let be a CNCG and be a CF. Then, is a HI-NCG if and only if is a NEI-NCG.
Proof. Let be a CNCG. Assume that is a CF, and let , in , in which is constant.
Let and be any two AEs in . Then,Therefore, every pair of AEs has VDs, iff every node neighbor to the nodes has VDs.
Hence, is a HI-NCG, iff is a NEI-NCG.
Theorem 5. Let be a CNCG and be a CF. Then, is HI-NCG iff is NETI-NCG.
Proof. It is clear.
Definition 11. Let be a CNCG. Then, is called to be a(i)SEI-NCG if each pair of edges has VDs (or no two edges have same degree).(ii)SETI-NCG if each pair of edges has various TDs (or no two edges have same TD).
Example 6. Consider a graph that is both SEI-NCG and SETI-NCG.
Let be a CNCG that is a cycle of length five where
and are defined asFrom Figure 6,So, is a SEI-NCG.Thus, is a SETI-NCG.
Therefore, is both SEI-NCG and SETI-NCG.
Example 7. SEI-NCG need not be SETI-NCG.
Let be a NCG on is a cycle of length three, , and . We define M and N as
follows:From Figure 7,Note that is SEI-NCG, since each pair of edges has VDs. Also,
is not SETI-NCG, since all the edges have same TD. Hence,
SEI-NCG need not be SETI-NCG.
Example 8. SETI-NCG need not be SEI-NCG.
Consider be a NCG so that , a cycle of length four where and defined asFrom Figure 8,Obviously, . Hence, is not SEI-NCG.
But is SETI-NCG, since .
Hence, SETI-NCG need not be SEI-NCG.
Theorem 6. Let be a CNCG on and be a CF. Then, is a SEI-NCG, iff is a SETI-NCG.
Proof. Assume is a CF. Let , for all in , in which is constant.
Let and be any pair of edges in . Then,So, each edge has different degree if and only if it has different total degrees. Hence,
is SEI-NCG iff is a SETI-NCG.
Remark 3. Let be a CNCG. If is both SEI-NCG and SETI-NCG, then need not be a CF.
Example 9. Let be a NCG so that is graph for Example 6 (Figure 9). As seen in that example,
each pair of edges in has VDs. Hence, is a SEI-NCG.
Also, note that each pair of edges in has various TDs. Hence, is a SETI-NCG. Therefore, is both SEI-NCG and SETI-NCG. But is not a CF.
Theorem 7. Let be a NCG on . If is a SEI-NCG, then is a NEI-NCG.
Proof. Let be a NCG. Assume that is a SEI-NCG. Then, each pair of edges in has VDs. So, each pair of AEs has VDs. Hence, is a NEI-NCG.
Theorem 8. Let be a NCG. If is a SETI-NCG, then, is a NETI-NCG.
Proof. Let be a NCG. Suppose that is a SETI-NCG; then, each pair of edges in has various TDs. So, each pair of AEs has various TDs. Hence, is a NETI-NCG.
Remark 4. The inverse of Theorems 7 and 8 is not generally true.
Example 10. Consider be a NCG so that is graph for Example 4 (Figure 4). As seen in that example, and . Hence, is a NEI-NCG. But is not a SEI-NCG, since . Also, note that and . Hence, is a NETI-NCG. But is not a SETI-NCG, since .
Theorem 9. Let be a CNCG and be a CF. If is a SEI-NCG, then is an irregular NCG.
Proof. Let be a CNCG. Assume that is a CF, and let , for all in , in which is constant.
Suppose is a SEI-NCG. Then, each edge has VD. Let
and be AEs in having VDs, and henceSo, there exists a node which is neighbor to nodes and having VDs. Hence, is an irregular NCG.
Theorem 10. Let be a CNCG and be a CF. If is a SETI-NCG, then is an irregular NCG.
Proof. Proof is similar to Theorem 9.
Remark 5. The inverse of Theorems 9 and 10 is not generally true.
Example 11. Let be a NCG so that is graph for Example 5 (Figure 5). As seen in that example, is an irregular NCG. Also, it is noted that . Hence, is not a SEI-NCG. Also, . Hence, is not a SETI-NCG.
Theorem 11. Let be a CNCG and be a CF. If is a SEI-NCG, then, is a HI-NCG.
Proof. Let be a CNCG. Assume that is a CF. Let , in , in which is constant.
Let be any node neighbor with , , and . Then, , , and are AEs in . Let be a SEI-NCG. Then, each pair of edges in
has VDs. So, each pair of AEs in has VDs. Hence,Therefore, the node is neighbor to the nodes with VDs. Hence,
is a HI-NCG.
Theorem 12. Let be a CNCG and be a CF. If is a SETI-NCG, then is a HI-NCG.
Proof. Proof is similar to Theorem 11.
Remark 6. The inverse of Theorems 11 and 12 is not generally true.
Example 12. Let be a NCG so that is graph for Example 5 (Figure 5). As seen in that example, is a HI-NCG. Note that . So, is not a SEI-NCG. Also, . Thus, is not a SETI-NCG.
4. Application of Neutrosophic Cubic-Influence Digraph to Find the Most Effective Person in a School
School is one of the most important places for education and training of students. The approved goals of the study courses are established and managed in accordance with the rules and instructions of the Ministry of Education. At school, a student interacts with his/her classmates and tries to learn the necessary scientific points. The physical, psychological, and educational environment of the school is one of the issues that can have an important and significant reflection on the structure of mental and intellectual growth and development, as well as creativity and mental health of students. The transfer of the basic values of the society is the main focus of the educational system, in such a way that the school commits the students to internalize the values of the society. In schools, values are taught in a variety of subjects, and the effectiveness of teaching values in each subject depends on the teacher’s understanding of the objectives of the subject. By recognizing the possibilities, the teacher equips the educational environment and by recognizing the interests and abilities of the students guides them in the right direction of learning because success in schools requires teachers to accept the opinions of others. Therefore, considering the importance of schools in shaping the personality and behavior of students, we have tried to determine the most effective person in a school according to its performance. To do this, we consider the nodes of this influence graph as the staff of a school and the edges as the influence of one employee on another employee. The names of the staff and their specialization in the school are shown in Table 2. For this school, the staff is as follows:(i)Momeni has been working with Salimi for 16 years and values his views on issues.(ii)Eskandari has been the head of the school, and not only Salimi but also Jafari is very satisfied with Eskandari’s performance.(iii)Taking care of the educational and moral affairs of students is one of the most important issues. Rouhi is an expert for this.(iv)Rouhi and Jafari have a long history of conflict.(v)Jafari is a very effective person in communicating with students’ parents and teachers in school.
Considering the above points, the influence graph can be very important. But such a graph cannot show the power of employees within a school and the degree of influence of employees on each other. Since power and influence do not have defined boundaries, they can be represented as a neutrosophic cubic set. On the other hand, there can be no fair interpretation of the power and influence of individuals because evaluations are always accompanied by skepticism. So, here we use the neutrosophic cubic degrees, which is very useful for influence and conflicts between employees. The neutrosophic cubic set of employees is shown in Table 3.
We have shown the influence of persons in the NC-digraph with an edge. This graph is shown in Figure 9.
School staff are the vertices of the NC-digraph of Figure 9. The weight of the vertices, respectively, indicates the power of speech, the degree of interaction with students, and the extent of their management in school affairs. For example, Mr. Rouhi has 20% to 30% of eloquence, but does not have 30% of the power to interact with students. Also, his power in processing school affairs is between 20% and 40%. Edges represent the extent of friendship, cultural, and political relationships, respectively. For example, Mr. Momeni has between 20% and 30% friendship with Mr. Jafari, but cultural differences between them are equal to 30%. Similarly, the rate of political relations between them is equal to 20% to 30%.
In Figure 9, it is clear that Mr. Eskandari controls the school deputy, Mr. Momeni, the representative of the parents and teachers association, Mr. Jafari, and educational instructor, Mr. Rouhi. Clearly, Mr. Eskandari has the most influence in the organization because he has an impact on three school staff and also has the highest level of management among other employees.
5. Conclusions
Neutrosophic cubic graphs have various uses in modern science and technology, especially in the fields of neural networks, computer science, operation research, and decision making. Also, they have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. Therefore, in this research, some types of EI-NCGs such as NETI-NCGs, SEI-NCGs, and SETI-NCGs are introduced. A comparative study between NEI-NCGs and NETI-NCGs is presented. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been introduced. In our future work, we will introduce connectivity index, Wiener index, and Randic index in neutrosophic cubic graphs and investigate some of their properties. Also, we will study some types of edge irregular neutrosophic cubic graphs such as neighborly edge totally irregular, strongly edge irregular, and strongly edge totally irregular neutrosophic cubic graphs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.