Abstract
Neutrosophic set (NS) is an extensively used framework whenever the imprecision and uncertainty of an event is described based on three possible aspects. The association, neutral, and nonassociation degrees are the three unique aspects of an NS. More importantly, these degrees are independent which is a great plus point. On the contrary, neutrosophic graphs (NGs) and single-valued NGs (SVNGs) are applicable to deal with events that contain bulks of information. However, the concept of degrees in NGs is a handful tool for solving the problems of decision-making (DM), pattern recognition, social network, and communication network. This manuscript develops various forms of edge irregular SVNG (EISVNG), highly edge irregular SVNG (HEISVNG), strongly (EISVNG), strongly (ETISVNG), and edge irregularity on a cycle and a path in SVNGs. All these novel notions are supported by definitions, theorems, mathematical proofs, and illustrative examples. Moreover, two types of DM problems are modelled using the proposed framework. Furthermore, the computational processes are used to confirm the validity of the proposed graphs. Furthermore, the results approve that the decision-making problems can be addressed by the edge irregular neutrosophic graphical structures. In addition, the comparison between proposed and the existing methodologies is carried out.
1. Introduction
The theory of fuzzy sets (FSs) is one of the communalized notions of classical set theory. There are merely two prospects of a statement in classical set theory; the statement/event is either true or not. However, there are many statements that cannot be dealt with only these two prospects. FSs can be accurately employed to manage such statements that have variable values. Zadeh [1] developed the concept of FSs to manage the issues with uncertainties. FS theory has an important role in complicated process that could not simply categorized by classical set theory. Some years later, Atanassov [2] suggested the concept of intuitionistic FS (IFS) as a communalization of FS. Additionally, he also gives a novel element which demonstrates the falsity membership grade in the description of FS. The notion of IFS is more significant in addition to exhaustive because of truth membership grade and falsity membership grade, in which the indeterminacy membership degree of IFS is its hesitation membership grade. To some extent, both the truth and falsity membership degrees are independent from each other with the condition that the summation of both these degrees does not exceed one. By joining the nonstandard analysis, Smarandache [3] developed the notion of neutrosophic sets (NS).
In mathematics, NS is an instrument which is the generalization of classical set theory that is used to handle practical issues consisting of imprecise, indeterminate and varying information. Like the theories of FSs and IFSs, the theory of NSs is beneficial in several fields, such as topology, medicines, decision-making (DM) problems, and in many others practical issues. To manage NS more easily with daily life problems, Wang et al. [4] established the concepts of single-valued NSs (SVNS). An SVNS has three elements: truth, indeterminacy, and falsity membership degrees. These degrees are independent in an SVNS, and their values are enclosed in the standard unit interval [0, 1]. The SVNS is indeed an oversimplification of an IFS. The SVNS has been a very significant research topic recently, and several researchers have considered SVNS in their works [5–9]. Other related works, such as, Majumdar and Samanta [10], examined the entropy and similarity of SVNS. Correlation coefficients of SVNS were suggested by Ye [11, 12] and utilized it to SVN-DM problems.
Apart from that, the idea of graphs can be related to NS. Graph theory has turned out to be an influential framework to model and solve the joint problems that occur in many fields, such as mathematics, engineering, and computer sciences. An SVN graph (SVNG) has many characteristics which are the origin of various techniques that are employed in modern mathematics as it is the generalization of graphs. A lot of studies on FS, fuzzy graphs (FGs), and intuitionistic FGs (IFGs) [13–21] have been explored and every single one have considered the set of vertices and the set of edges as FSs and/or IFSs. However, the FG and IFG are unsuccessful when the relations between nodes (or vertices) in problems are not determined or not recognised. For this reason, Smarandache [3] introduced four major classes of the neutrosophic graphs (NGs). Two of these are built on literal indeterminacy, i.e., NGs are I-edge NG and I-vertex NG. The vast range of applications in decision-making problems made the NGs the hot topic for the researchers of the field. Since then, many attempts have been made to extend the notion of NGs. The work of Broumi et al. [22] stands alone, which is the introduction of a novel concept of SVNG. Besides that, Mohanta et al. [23] described the types of products of NGs and neutrosophic algebraic structures. Ramia et al. [24] defined the ideas of complimentary domination in SVNGs. The notion of operations of SVNG and interval-valued SVNG are discussed in the literature, see [25]. Abu Saleem [26] worked on the neutrosophic folding and a neutrosophic retraction on a SVNG. Lu and Ye [27] discussed SVN hybrid arithmetic and geometric aggregation operators. Lately, Shahzadi et al. [28] presented an application that carried out a medical diagnosis by using the concepts of SVNS.
However, the literature has great capacity when comes to the SVNGs and the types of their edges. Henceforth, this study intends to define the concepts of edge irregular SVNG and totally edge irregular SVNG. In addition, the path and cycle of an edge irregular SVNG will also be established. Instead of considering a general NG in which edges and vertices must be considered, our proposed works depart more formally from the three degree aspects of NG to interval three degree aspects of SVNG. Moreover, the proposed notions of SVNGs are applied to a couple of decision-making problems. The first problem is to select the best company among a collection of companies. To this end, the weighted averaging and weighted geometric aggregation operators were used as tool for the solution. While the second problem, which was targeted to select the best combination of subjects for a student of high school, was modelled and solved by the idea of edges in SVNGs. In order to provide strength to our study, we carried out a detailed comparison between the proposed framework and other contenders in the field. The experiments verified the validity of our method. The benefits of the proposed framework are as follows: (i) it is capable of modelling a complex situation, (ii) it can handle the events by describing three degrees, i.e., association, neutral, and nonassociation degrees, (iii) the decision maker can independently assign values to the degrees, and (iv) there is no constraints and limitations of these structures. Considering these benefits, we chose the SVNGs for our study.
In Section 2, some basic definitions are given which provide some base to construct further ideas. In Section 3, edge regular and highly edge regular SVNGs are defined. Section 4 defines the strong edge irregular SVNGs and strong edge totally irregular SVNGs. In Section 5, the edge irregularity is discussed on a path and on a cycle in SVNGs. Then, the applications of the proposed concepts are presented in Section 6. Section 6 also contains the comparison of our method with the other methods. And finally, the concluding remarks are given in Section 7.
2. Preliminaries
Some basic definitions related to our graphical work such as IFG, SVNG, and degree of SVNG are presented in this section. Some examples are also presented to illustrate the notions.
Definition 1 (see [14]). A pair is called IFG, where , is an IFS on , and such that and with the condition , for all
Example 1. Let be an IFG, where is the collection of vertices and is the collection of edges. Figure 1 shows an IFG.
Definition 2 (see [4]). A pair is known as SVNG, where , is an SVNS on , and such that , and with the condition , for all
Example 2. An SVNG is shown in Figure 2.
Definition 3 (see [4]). A pair is called strong SVNG, where , is an SVNS on , and such that , , and with the condition , for all
Definition 4 (see [4]). The degree of a vertex in a SVNG is denoted and defined by , where , , and . Here, denotes the membership degree, denotes the indeterminacy degree, and denotes the nonmembership degree.
Example 3. Let be a SVNG, where is the collection of vertices and is the collection of edges
Figure 3 is an NG which is explained below.
This graph contains four vertices , and the values between their vertices is called edges. Furthermore, by Definition 4, we find the degrees of its vertices of Figure 3 which is given below.
Degree of vertices of Figure 3 is
Definition 5 (see [27]). The SVN-weighted aggregation (SVNWA) operator is denoted and defined by , , where represents the weight vector.
Definition 6 (see [27]). The SVN-weighted geometric (SVNWG) operator is denoted and defined bywhere represent the weight vector.
Definition 7 (see [28]). The single-valued neutrosophic Hamming distance between two SVNSs is defined by
Definition 8 (see [27]). The score function in a SVNS is denoted and defined by , where represents the membership, indeterminacy, and nonmembership grades, respectively.
3. Edge Irregular and Highly Edge Irregular SVNG
We propose the definitions of edge irregular and highly edge irregular SVNG in this section.
Definition 9. A connected graph is called the edge irregular SVNG (EISVNG) if at least single edge is neighboring to the edges with different degrees.
Definition 10. A connected graph is called an edge totally irregular SVNG (ETISVNG) if at least single edge is neighboring the edges with different total degrees.
Definition 11. A connected graph is called highly edge irregular SVNG (HEISVNG) if each edge is neighboring to the edges with different degrees.
Example 4. Let be a SVNG, where is the collection of edges and is the collection of vertices.
Figure 4 contains four vertices , and the values between their vertices are called edges. Furthermore, by Definition 4, we find the degrees of its vertices of Figure 4.
Here, , , and . Degrees of edges areWe observe that every edge is neighboring to the edges with different degrees. Consequently, is HEISVNG and also EISVNG.
Definition 12. A connected graph is called HETISVNG if each edge is neighboring to the edges with different total degrees.
We also propose the following theorems as statements that have been proven to be true.
Theorem 1. If is a connected HEISVNG, then is an EISVNG.
Proof. Let us assume that is a connected HEISVNG; then, each edge in neighbors the edges with different degrees; consequently, there exist at least single edge that is neighboring the edge with distinct degrees. Hence, is an EISVNG.
Theorem 2. If is a connected HETISVNG, then is an ETISVNG.
Proof. It follows Theorem 1, thus omitted.
Remark 1. A HEISVNG may not be a HETISVNG.
Example 5. This example supports Remark 1.
Let be a SVNG.
Figure 5 contains four vertices , and the values between their vertices is called edges. Furthermore, by Definition 4, we find the degrees of its vertices are given as below.
Here, , , , and . Degrees of edges are , , and and , and Clearly, we note that is HEISVNG, but is not HETISVNG. Therefore, all edges are with the same total degrees.
Remark 2. HETISVNG might not be an HEISVNG.
Example 6. This example supports Remark 2.
Let be an SVNG.
Figure 6 contains four vertices , and the values between their vertices are called edges. Furthermore, by Definition 4, we find the degrees of its vertices of Figure 6.
Here, , , , and . Degrees of edges are , , and and , and
Theorem 3. If a connected SVNG is HISVNG and is constant function, then is HETISVNG.
Proof. Suppose that is constant function. Assume that , , and ; for all , , and are constants. Now, consider is EHISVNG. Then, every edge is neighboring to the edges; it has distinct degrees. Suppose be an edge that is neighboring to the edges , and these edges that are incident at the vertex and are the edge incident with the vertex Then, , , and , where , and are neighboring to the vertex Next, Again, and . Hence, is HETISVNG.
Theorem 4. If a connected SVNG is EISVNG and is constant function, then is ETISVNG.
Proof. It follows the proof Theorem 3, and thus, it is omitted.
Theorem 5. If a connected SVNG is ETISVNG and is constant function, then is EISVNG.
Proof. It follows the proof of Theorem 3, and thus, it is omitted.
Remark 3. If a connected SVNG is both HEISVNG and HETISVNG. Then, may not be considered a constant function.
Example 7. The following example supports Remark 3.
Let be a SVNG.
Figure 7 contains four vertices , and the values between their vertices are called edges. Furthermore, by Definition 4, we find the degrees of its vertices of Figure 7.
Here, , , , , and Degrees of edges are , , and and , , and .
Theorem 6. If a connected SVNG is EISVNG and is constant function, then is an ISVNG.
Proof. Suppose that is constant function. Assume that , , and , for all , , and are constants. Now, consider is EHISVNG. Then, every edge neighbors the edges with distinct degrees. Suppose be an edge such that it is neighboring the edges , and these edges are incident at the vertex and is the edge incident to the vertex . Then, , , and , where , and are neighboring to the vertex . Now, . Again, and .
Consequently, there is a vertex neighboring the vertices and with different degrees. Thus, is an ISVNG. In Section 4, we present several definitions and examples to explain the degree of edge irregularity.
4. Strongly Edge Irregular and Strongly Edge Totally Irregular SVNG
Definition 13. A connected graph is known as strongly EISVNG if each pair of edges has different degrees.
Definition 14. A connected graph is called strongly ETISVNG if each pair of edges has different total degrees.
Example 8. The following example supports Remark 3. Let be a SVNG.
Figure 8 contains five vertices , and the values between their vertices are called edges. Furthermore, by Definition 4, we find the degrees of its vertices of Figure 8.
Here, , , , , and . Degrees of edges are , , and and , , and .
Theorem 7. If is a strongly connected EISVNG, then is an HEISVNG.
Proof. Let us assume that is a connected strongly EISVNG; then, all pairs of edges in have distinct degrees; therefore, every edge neighbors the edge with a distinct degree. Hence, is an HEISVNG.
Theorem 8. If a connected SVNG is strongly EISVNG and is constant function, then is strongly ETISVNG.
Proof. Suppose that is constant function. Assume that , , and ; for all , , and are constants. Now, assume that is strongly EISVNG. Then, every pair of edge is neighboring to the edges with distinct degrees. Suppose and be some pair of edge in . Now, for any pair of and in Similarly, for any pair of and in and for any pair of and in . Therefore, . Hence, is strongly ETISVNG.
Theorem 9. If a connected SVNG is strongly ETISVNG and is a constant function, then is strongly EISVNG.
Proof. Suppose that is constant function. Assume that , , and ; for all , , and are constants. Let be strongly ETISVNG. Then, all pairs of edges are with distinct total degrees. Suppose and be any pair of edge in . Then, for any pair of edge and in Similarly, for any pair of edge and in and for any pair of edge and in Therefore, . Hence, is strongly EISVNG.
Theorem 10. If a connected SVNG is strongly EISVNG and is a constant function, then is strongly ISVNG.
Proof. Suppose that is a constant function. Assume that , , and ; for all , , and are constants. Now, assume that is strongly EISVNG. Then, every pair of edges is with distinct degrees. Suppose and be neighboring to the edges with distinct degrees Then, . Also,