COMPLEX FERMATEAN NEUTROSOPHIC GRAPH AND APPLICATION TO DECISION MAKING

: A new growing area of the neutrosophic set theory called complex neutrosophic sets (CNS) provides useful tools for dealing with uncertainty in complex valued physical variables that are observed in the actual world. A CNS takes values for the truth, indeterminacy and falsity membership functions in the complex plane's unit circle. In this research, a novel concept of complex fermatean neutrosophic graph (CFNG) is established and various basic graphical ideas such as the order, size, degree and total degree of a vertex of it are introduced. Also, set theoretical operations such as complement, union, join, ring-sum and cartesian product of CFNG are studied. Further, the concept of a regular graph under a complex fermatean neutrosophic environment is introduced. Finally, we make use of the proposed CFNG in solving a multi-criteria decision-making problem in which the graphical structure of attributes is uncertain. This study also demonstrates the application of a CFNG in the educational system to evaluate a lecturer's research productivity.


Introduction
The idea of fuzzy sets (FS) was initially proposed by (Zadeh, 1965).It is very good at coping with the uncertainty that exists in a real-world decision-making.FS considers an element's membership degree (  ) alone and does not allow us to independently select non-membership degree (  ).In this sense, fuzzy sets are still somewhat classical.However, this fact may be considered as a disadvantage.Thus, (Atanassov, 1986) proposed intuitionistic fuzzy sets [IFS] which facilitate the decision maker (DM) to assign  and  independently with the condition 01   +  .By relaxing the restriction of IFS to Yager (2013Yager ( , 2014) ) proposed Pythagorean fuzzy sets [PFS].Senapati andYager (2019, 2019a) proposed Fermatean fuzzy set (FFS) by further relaxing the restriction of PFS  to handle a wider range of uncertainty.A comparison between the space of FS, IFS, PFS, FFS, and PS is shown in Table 1., 0.7, 0.6, 0.9 , , 0.6, 0.8, 0.7 D a b = Includes  One can add that there also other solutions of this kind.For example, Ibrahim et al. (2021) introduced (3,2)-fuzzy sets while Shanmathi and Nirmala (2022) studied (4,2)-fuzzy sets.In both cases, appropriate topological spaces were studied too.As for the neutrosophic sets (NS), they were initiated by (Smarandache (1999).They incorporate the indeterminacy membership degree (  ) along with the truth membership degree (  ) and falsity membership degree (  ).Details will be described later in the paper.Another aspect of our work is related to complex numbers and functions.Many physical quantities in practical problems such as wave function in quantum mechanics are complex-valued.Buckley (1989) first initiated the formal definition of complex fuzzy numbers (CFN) and their types.Later on, Ramot et al. (2002) presented the concept of complex fuzzy sets (CFS) by extending FS in the real field to the complex field.The membership degree of CFS is of the form  where 1 i =− whose amplitude term  lies in [0,1] and phase term  lies in [0,2 ]  .Many researchers have focused their attention on CFS over the past few years.Yazdanbakhsh and Dick (2018) reviewed CFS and highlighted potential future research topics in it.Many researchers investigated and worked on CFS and its extensions.Alkouri and Salleh (2012) generalized the concept of CFS to IFS and proposed a complex intuitionistic fuzzy set (CIFS) whose membership and nonmembership functions are complex valued.Ullah et al. (2019) presented the concept of complex Pythagorean fuzzy set (CPyFS).Chinnadurai et al. (2021) presented complex Fermatean fuzzy set (CFFS) in rectangular coordinates and its applications in decision-making problems.Ali and Smarandache (2017) proposed complex neutrosophic sets (CNS).Extensions of CFS under various fuzzy environment are shown in Table 2.
CPyFS (Chinnadurai et al., 2021) CFFS (Ali and Smarandache, 2017) CNS Our paper deals with graph theory too (but in a context of uncertainty).Therefore, we may say that fuzzy graphs (FG) are developed to model uncertainties in graphical network and their extensions are explored by many researchers (Saumya and Hegde, 2022).Thirunavukarasu et al. (2016) extended FG to complex fuzzy graphs (CFG) and proposed energy of CFG.Yaqoob et al. (2018) studied about complex neutrosophic graphs (CNG).Also, Yaqoob et al. (2019) suggested the complex intuitionistic fuzzy graphs (CIFG) using some basic operations.Akram and Naz (2019) proposed complex Pythagorean fuzzy graphs (CPFG).Shoaib et al. (2022) introduced the concept of picture fuzzy graphs and their properties.Also, Shoaib et al. (2022a) initiated the notion of complex spherical fuzzy graph.Pythagorean neutrosophic fuzzy graphs (PFNG) was suggested by Ajay and Chellamani (2020).Antony and Jansi (2021) proposed the fermatean neutrosophic sets (FNS) and Broumi et al. (2022) explored the fermatean neutrosophic graphs (FNG) with their applications.Motivated by these works we initiated the concept, complex fermatean neutrosophic fuzzy sets (CFNS) and complex fermatean neutrosophic fuzzy graph (CFNG).Butt et al. (2022) established a complex dombi fuzzy graph and their properties.In the single valued neutrosophic environment, regular graphs are explored by Samina et al. (2017).Uncertain decision-making problems can be easily organized and modeled using CFNG theory.
An overview of the motivation for this paper is provided below: 1.A complex Fermatean neutrosophic set (CFNS) is capable of handling the situation well when faced with imprecise and intuitive knowledge in an uncertain decision-making process.2. Due to the CFNS's phase term, there is no information loss, and are incredibly effective at making decisions and have a wider range of applications.The primary contributions of this work are as follows: 1. First and foremost, we initiated the new idea of CFNGs.2. We studied the properties of CFNGs.

 
T , I , F : 0 , 0 S u are truth, indeterminacy and falsity membership values respectively and are real standard or non-standard subsets of   0 , 0 −+ .It is worth to mention that the definition above is very general.In particular, it allows to combine neutrosophic sets with some ideas taken from the world of non-standard analysis.However, the vast majority of researchers limit their interest to single-valued neutrosophic sets.In this case T, I and F are just real numbers from [0, 1].This approach is more practical.Also interval-valued neutrosophic sets are studied.Definition 2.2 (Ajay and Chellamani, 2020): A Pythagorean neutrosophic set (PNS) S defined on X is represented as SS uu are dependent components and ( ) S defined on X is represented as Definition 2.4 (Ali and Smarandache, 2017): A complex neutrosophic set (CNS) S is described on X is represented as Definition 2.5 (Yaqoob et al, 2018;Khan et al., 2022): A mapping ( ) Definition 2.6 (Yaqoob et al., 2018): A neutrosophic graph (NG) on X is a pair ( ) where P and Q are NS defined on vertex set V and edge set Definition 2.7 (Ajay and Chellamani, 2020): A Pythagorean neutrosophic graph (PyNG) on X is a pair ( ) where P and Q are PyNS defined on vertex set Definition 2.8 (Antony and Jansi, 2021): A Fermatean neutrosophic graph (FNG) on X is a pair ( ) where P and Q are FNS defined on vertex set V and edge set E V V respectively such that Definition 2.9 (Yaqoob et al., 2018): A complex neutrosophic graph (CNG) on X is a pair ( ) where P and Q are CNS defined on vertex set V and edge set

Complex Fermatean neutrosophic graphs
This section explores the concept of the Complex Fermatean Neutrosophic Set (CFNS) and the Complex Fermatean Neutrosophic Graph (CFNG) as well as their fundamental operations.

Proposed CFNS and some important definitions
Definition 3.1: A complex Pythagorean neutrosophic set (CPNS) S on X is given by ,, In the next definition we show how to perform operations of union and intersection on CFNNs.
Definition 3.6 : A complex fermatean neutrosophic graph (CFNG) defined on X is a pair ( ) where P is an CFNS on vertex set V and Q is a CFNS on edge set Figure 1.The vertex set P is a CFNS defined on V given in Table 4 and , ( ) , ( ) , ( ) | then the order and size of a CFNG is defined by ,, Example 2: The order and the size of the CFNG G represented in Figure 1 is given by ( ) = and ( )

Basic operations of complex fermatean fuzzy graphs
This section introduces some of the fundamental CFNG operations and explores some of its associated properties.
Definition 3.8: The complement of a CFNG ( ) , ( , ) 0 max , ( , ) 0 , 1 Example 3: The complement of CFNG G is represented in Figure 2 whose vertex set P is shown in 1 1 , G P Q be two CFNGs whose vertex sets and edge sets are provided in Table 8, Table 9, Table 10 and Table 11 and are represented in Figure 3 and Figure 4. respectively.( ) , , , ( ) is the CFNG, Definition 3.11: Let ( ) ( ) be two CFNSs, then the join Q is the set of all edges joining the vertices of 1 P and 2 , Definition 3.13: For CFNG G , the total degree of a vertex uV  is denoted by

Regular and totally regular CFNGs
Regular and totally regular CFNGs are introduced in this section along with examples.Definition 3.17: A CFNG ( ) ,, u u u -regular.
Example 5: Consider a CFNG ( ) , , , as in Figure 6 whose vertex and edge set are given in Table 14 and Table 15 respectively.
Definition 3.18: A CFNG ( ) G is called a totally regular CFNG if the total degree of its each vertex is same.If each vertex has total degree ( ) G is called totally regular of degree ( ) ,, m m m or ( ) ,, m m m -totally regular.Example 6: Consider a CFNG ( ) as in Figure 7 whose vertex and edge set are given in Table 16 and Table 17 respectively.
Since each edge is considered twice, so ) ,, m m m totally regular graph CFNG.Then n n n -regular and ( ) ,, n n n -regular CFNG.Then the size of G is ( ) ,, m m m -totally regular CFNG.Using theorem 4, we have ( ) ( ) ( )

Decision making approach under CFN environment
In this section, based on the proposed CFNGs a decision-making approach is introduced.Further a real-life application on education system is discussed.
,, Then the complex fermatean neutrosophic ordered weighted averaging (CFNOWA) operator is a mapping , , , denotes the permutation of ( )   Then the complex fermatean neutrosophic ordered weighted geometric (CFNOWG) operator is a mapping , , , denotes the permutation of ( )

Decision making approach
We developed a decision making approach under CFN environment using the proposed CFNWA, CFNOWA, CFNWG, CFNOWG operators.Let Step 2: The alternatives are aggregated using the proposed operators CFNWA, CFNOWA, CFNWG, CFNOWG.
Step 3: The aggregated value's score function is calculated.
Step 4: The best alternative is selected based on ranking given in step 3.

Illustrative example
To provide evidence that the suggested arithmetic and geometric aggregation operator of CFNS, an illustrative example is discussed below.Teachers play a major role in imparting quality education to students in institutions.In this section, a decision-making approach to evaluate lecturer's research productivity at the end of year (Nguyen et al., 2020) is discussed.Assume that there are 3 lecturers to be evaluated.A committee of 4 decision makers are considered to make their assessment individually, according to their preferences of criteria and the ratings of alternative s.After a discussion with committee members five criteria are selected as in Table 18.Using the score function of CFNN, we arrive at the ranking order and stated in Table 25.
Table 25.Ranking order Aggregation operator Ranking Proposed CFNWA Clearly, we can see that the CFNWA and CFNOWA provides same ranking and CFNWG and CFNOWG provides same ranking.

Conclusion
To model uncertain decision-making problem, graph theory plays a very important role.The novel idea of CFNG is introduced and investigated different properties of it in this paper.We have defined the order, size, degree and total degree of CFNG with appropriate examples.Also, the primary operation such as complement, union, join and ring sum of CFNG are proposed with appropriate examples and some of their important theorems are also described.It is shown with

Definition 2. 3 (
Antony and Jansi, 2021): A Fermatean neutrosophic set (FNS) Figure 5. CFNG 12 GG  Definition 3.12: For CFNG G , the degree of a vertex uV  is denoted by ( )Du and is defined by  of two CFNGs 1G and 2 G , then two cases are to be considered.If Complex fermatean neutrosophic graph and application to decision making 489 any edge incident at is either in 12 QQ − or 21 .
Figure 6.Regular CFNG G Then the degrees of vertices 1 2 3 4 , , , v v v v are determined as:

Table 6 .
Table 6 and edge set Q is shown in Table 7 and is denoted by CFNG G .Vertices of CFNG G iii eee Table 7. Edges of CFNG G Figure 2. CFNG G Theorem 1: For a CFNG G , we have .GG Proof: Let G be a CFNG.Using definition 3.7 we have

Table 11 .
Edges of CFNG 2 is shown in Figure5whose vertex set and edge set given in Table12and Table13respectively.

Table 14 .
Vertices of regular CFNG G

Table 15 .
Edges of regular CFNG

Table 16 .
Vertices of totally regular CFNG

Table 17 .
Edges of totally regular CFNG G Edge CFNS , we propose the idea of score function and accuracy function in order to compare two CFNN.

Table 18 .
Criteria for research productivity CriteriaType To select the most desirable teacher, we utilize the above step-wise procedure using CFNWA, CFNOWA, CFNWG, CFNOWG.Consider the CFNN decision matrix as stated in Table19, Table20, Table21 respectively.

Table 21 .
CFNN decision matrix for 3Now we aggregate the criteria using the proposed CFNWA, CFNOWA, CFNWG, CFNOWG and the values are stated in Table22, Table23, Table 24.

Table 22 .
Aggregated values of 1

Table 23 .
Aggregated values of 2

Table 24 .
Aggregated values of the CFNN information Aggregation operator