Regularity of Pythagorean neutrosophic graphs with an illustration in MCDM

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Introduction
Atanassov established the idea of an Intuitionistic set [1] by introducing a generalization of fuzzy set [2]. Each element in the set is assigned a membership and non-membership degree with the constraint that the addition of these values lies between 0 to 1. Researchers have studied Intuitionistic fuzzy sets (IFS) and have been implemented in various fields, including decision making [3,4], cluster analysis [5], pattern recognition [6], market prediction [7], medical diagnosis [8]. Smarandache [9] initiated the neutrosophic set theory in which each element is independently assigned a truth, indeterminacy, and falsity membership degree in the non-standard interval ]0 − , 1 + [. Wang et al. [10] presented the concept of a single valued neutrosophic set (SVNS) as a special case of a neutrosophic set. These sets have been widely used in a various of fields, including image processing [11], medical diagnosis [12], decision making [13], information fusion [14], control theory [15], and graph theory [16,17] among others.
To deal with complex imprecision and uncertainty Pythagorean fuzzy sets PFS was pioneered by Yager [18][19][20] such that the addition of the squares of M and NM degrees lies in 0 and 1. Consequently, in comparison to IFSs, PFSs account for a greater amount of uncertainty. Smarandache introduced and developed the degree of dependence among components of fuzzy sets and neutrosophic sets. One special case with independent indeterminacy and dependent truth and falsity is chosen out of three membership functions of neutrosophic sets with the constraint addition of squares of M, I, and NM lies between 0 and 2, and it is known as the Pythagorean Neutrosophic set (PNS) [21].
To deal with structural information, graph representations are widely used in domains such as networks, economics, systems analysis, image interpretation, operations research, and pattern recognition. Based on the fuzzy relations Kauffman [22] presented the Fuzzy graphs. Rosenfeld [23] established the structure of fuzzy graphs to derive numerous basic theoretical concepts. Bhattacharya [24] introduced various concepts on fuzzy graphs and Radha and Kumaravel investigated edge regular fuzzy graphs in [25]. Atanassov [26] introduced intuitionistic fuzzy graphs (IFG) with IFS as edge sets and vertex sets, which were later developed by Akram [27].
The idea of edge regular IFGs was defined by Karunambigai et al. [28,29]. Borzooei et al. [30] recently developed the notion of fuzzy graph regularity to vague graph regularity. Interval-valued IFGs were first pioneered by Mishra and Pal [31]. Kandasamy et al. [32] proposed the notion of neutrosophic graphs in which the edge weights are neutrosophic numbers. Broumi et al. [33,34] also proved the existence of some of the properties of SVNGs and their extensions.
In [35], the fuzzy graph was extended to Pythagorean fuzzy graphs. By combining concepts of PNSs and fuzzy graphs, the new Pythagorean neutrosophic graph (PNG) [36] was developed. The PNGs are a graphical representation that is the same as the structure of the graphs but the sum of the membership grades of the vertices is less than 2 and the same goes for the edges of the graph.
Decision-making problems are complex to deal with and examining them using a single criterion to take optimum decisions can lead to an unrealistic decision. Simultaneous consideration of all the factors to the problem is a mere good approach. Multicriteria Decision Making (MCDM) is an advancement of operation research which is for the development of decision methodologies to make the decision problems simple involving multiple criteria, goals of conflicting nature [37,38]. Problems with a finite set of alternatives and evaluated to rank them and to select the most appropriate one are called discrete MCDM problems while problems with an infinite set of alternatives are continuous MCDM problems. Discrete MCDM problems are addressed through the multiattribute decision making (MADM) [39][40][41] methods while continuous MCDM problems are addressed through multiobjective decision making (MODM) [42][43][44][45] methods. Fuzzy MCDM methods are used to assess the alternatives through a single or a committee of decision-makers, where the values of alternatives, weights of criteria can be evaluated using linguistic values which can be represented by fuzzy numbers [46][47][48]. Several approaches have been developed to solve the fuzzy MCDM problems. A review and comparison of various of these methods are presented in [49][50][51] and [52]. A brief review of the category in fuzzy MCDM and some of its recent applications is presented in [53][54][55][56].
The proposed method is dealt with using Pythagorean Neutrosophic graphs which is a recently developed fuzzy set having the advantage of holding a bit more fuzziness when compared with the previously developed fuzzy sets. Although the constrain is restricted to two, this provides independence to two membership values. With all these additional advantages our proposed method is more useful in developing models or methods for real-life problems. The proposed method using Pythagorean Neutrosophic graphs for the decision model is more compatible and at the same time, holds little restrictions to select the suitable criterion. Thus the proposed method, is an advantage in the decisionmaking field, because of its usage and recent developments.
This paper is arranged as follows: Section 2 discusses the preliminary concepts which are necessary for the work presented in the manuscript. Section 3 proposes the ideas of edge regular, regular, partially edge regular, strongly regular, full edge regular, and bi-regular PNGs, and examines their properties. In Section 4, an illustrative example is given for the newly proposed MCDM method using PNGs and finally, we conclude Section 5. In this article, V denotes a crisp universe of generic elements, G stands for the crisp graph, G is the PN. The membership, non-membership, and indeterminacy are represented by M, NM, I.

Preliminaries
Fuzzy set theory is an effective mathematical concept that is used to deal with uncertain and vague values. The basic definition of the fuzzy subset and its extended fuzzy sets were given for the basic reference. In addition, a few more definitions of the fuzzy graphs were given for the base understanding of the paper. The section holds basic definitions and terminologies for the paper:     1] symbolize the M and NM grades of B, correspondingly.

Regularity of Pythagorean neutrosophic graphs
In this section, we describe the regularity ideas of PNGs. The concept of degree, total degree, regular, and totally regular were discussed in detail with their characterizations and properties.
The PNG in Figure 1     Definition 3.7. A PNG G having same degree for every edge is named as the edge regular PNG (ERPNG). G is called < g 1 , g 2 , g 3 > ER, when each edge has degree < g 1 , Definition 3.8. A PNG G is termed as totally edge regular PNG (TERPNG) when each edge has the same TD.
has M value exactly of the edge in it's respective crisp graph times. Thus, Proposition 3.11. G = (σ, µ) be a PNG on a k-RCG G. Then Proof. Let G be a PNG on k-RCG G. Then by theorem 3.10, Proposition 3.12. G = (σ, µ) be a PNG on a RCG G.
Theorem 3.14. If a PNG G is both ER and TER, then (µ 1 , µ 2 , µ 3 ) is a CF. Proof. G be a PNG on a k-RCG G. Consider (µ 1 , µ 2 , µ 3 ) is a CF, that is, µ 1 (h i h j ) = y 1 , µ 2 (h i h j ) = y 2 and µ 3 (h i h j ) = y 3 ∀ h i h j ∈ E, where y 1 , y 2 and y 3 are Cs. From the definition of vertex degree, we have Likewise, td I (h i h j )= y 2 (2k − 1) and td NM = y 3 (2k − 1) ∀ h i h j ∈ E can also be expressed. Thus G is a TERPNG. Conversely, consider G is both RPNG and TERPNG. We have to prove that < µ 1 , µ 2 , µ 3 > is a CF.
In the same way, we can illustrate that µ 2 (h i h j ) = 2g 2 − t 2 and Theorem 3.16. Let G = (σ, µ) be a PNG on a crisp graph G. If (µ 1 , µ 2 , µ 3 ) is a CF, then G is an ERPNG iff G is ER.
where y 1 , y 2 and y 3 are Cs. Assume that G is an ERPNG. We claim that G is an ER. On the other hand assume G is not an ER. i.e., , a contradiction. Therefore G is an ERPNG.
Proof. G be a < g 1 , where y 1 , y 2 and y 3 are Cs.
Proof. Suppose that < µ 1 , µ 2 , µ 3 > is a CF. Then where y 1 , y 2 and y 3 are Cs. G be a FRPNG, then Therefore G is an ERPNG. Thus G is a FERPNG.
Theorem 3.20. Let G = (σ, µ) be a t-TER and t -PERPNG. Then Therefore, Definition 3.21. A g =< g 1 , g 2 , g 3 > RPNG G on n vertices is said to be strongly regular PNG (SRPNG), if it holds the following characteristics: 1) Total of M, I, and NM values in the same neighbourhood of any 2 adjacent vertices of G have λ =< λ 1 , λ 2 , λ 3 > weight. 2) Total of M, I, and NM values in the same neighbourhood of any 2 non-adjacent vertices of G have χ =< χ 1 , χ 2 , χ 3 > weight.
then G is said to be a complete bipartite PNG (CBPNG).

MCDM method based on the Pythagorean neutrosophic graphs
PN sets have become an interesting topic in research due to their powerful dealing with incomplete, inconsistent information. PNG can illustrate the uncertainty in a real-life context, thus we propose the use of PNG in solving MCDM problems. This newly proposed model is named the PNG-based MCDM method.
At first to frame the algorithm or method, we describe the decision making problem. Consider that P = {p 1 , p 2 , p 3 , ...p m } is a collection of alternatives and B = {α 1 , α 2 , α 3 , ...α n } is a set of criteria, with weight vector w = (w 1 , w 2 , w 3 , ...w n ) T fulfilling w j ∈ [0, 1], n j=1 w j = 1. If the decision maker provides a PN value for the alternative p k (k = 1, 2, 3, ...., m) under the attribute α j ( j = 1, 2, .., n) and can be characterized by a PN number (PNN) d k j = m k j , id k j , nm k j (where m, id, nm represents the membership, indeterminacy and non-membership value) j = 1, 2, .., n; k = 1, 2, 3..., m. Imagine that D = [d k j ] m×n is the decision matrix, where d k j is given by PNN. If the decision maker provides a PN value for the alternative p k (k = 1, 2, ...m) under the criteria α j ( j = 1, 2...n), these values are illustrated as e k j = (m k j , id k j , nm k j ), ( j = 1, 2, ...n, k = 1, 2...m). If there exists a relation between two criteria α i = (m i , id i , nm i ) and α j = (m j , id j , nm j ), we represent the PN relation as β i j = (m i j , id i j , nm i j ), with the properties: m i j ≤ min(m i , m j ), id i j ≤ min(id i , id j ), nm i j ≤ max(nm i , nm j ) ∀ (i, j = 1, 2, ...m); otherwise β i j =< 0, 0, 1 >. Based on the established PNG structure, we suggest a technique for decision-maker to choose the best alternative with PN information.
The technique has been illustrated in the steps following: Step 1: Compute the influence co-efficient between the criteria α i and α j (i, j = 1, 2, ...n) in decision process by, where β i j = (m i j , id i j , nm i j ) is the PN edge between the verices α i and α j (i, j = 1, 2, ...n). we have χ i j = 1 and χ i j = χ ji for i = j.
Step 2: Obtain the complete criterion value of the alternative p k (k = 1, 2, ...m) by where e ki = (m ki , id ki , nm ki ) is a PNN.
Step 3: Compute the score value of the alternative p k (k = 1, 2, ...m) which is expressed by: Step 4: Rank all the alternatives p k (k = 1, 2, ...m) and choose the top one in concordance with S ( p k ).

An illustrative example
In this subsection, an illustration of PNG based MCDM problem with PN information is applied to show the application and efficiency of the suggested decision-making method.
An investing company wants to invest a sum of money in the best option. There is a panel of attributes with YouTube channels in which to invest the money: (1) p 1 is a movie review channel, (2) p 2 is an educational content channel, (3) p 3 is a food related channel (food review, cooking guidance) (4) p 4 is a technical content channel. The investment company must take a decision according to the criteria (1) α 1 is the subscribers (2) α 2 is the content worth (3) α 3 is the growth of the criteria is given by w = (.34, .33, .33). The four possible alternatives are to be calculated under these three criteria shown in the form of PN information by decision-maker. The information evaluation is consistent to α j ( j = 1, 2, 3) on the alternative p k (k = 1, 2, 3, 4) under the factors α j ( j = 1, 2, 3) and the resultant PN decision matrix is represented as D : The relation among factors α j is given by the complete graph G = (V, E) where V = {α 1 , α 2 , α 3 } and E = {α 1 α 2 , α 1 α 3 , α 2 α 3 }. We can get the influence coefficients to quantify the relationships among the criteria. Suppose that PN edges denoting the connection among the criteria are described in Figure 3 as: l 12 = (.4, .5, .5), l 13 = (.5, .6, .2), l 23 = (.5, .3, .2). G = (V, E) describes the PNG according to relationship among criteria for each alternative. The steps given below are followed to achieve the best alternative. Step 1: The influence coefficients between criteria were computed by using (4.1) and the values are χ 12 = .325, χ 13 = .41, χ 23 = .53 Step 2: The overall criterion value of the alternatives p i , i = 1, 2, 3, 4 are obtained by using (4.2) are as follows: Step 3: Calculating the score value of the alternatives p k (k = 1, 2, 3, 4), we get S( p 1 ) = .1317475, S( p 2 ) = .32011, S( p 3 ) = .6596175, S( p 4 ) = .4372125.
Step 4: Since S( p 3 ) > S( p 4 ) > S( p 2 ) > S( p 1 ), the ranking of four alternatives is Thus, the alternative p 3 is chosen among the alternatives.