RENEWED STRUCTURE OF NEUTROSOPHIC SOFT GRAPHS AND ITS APPLICATION IN DECISION-MAKING PROBLEM

. This study is designed with the renewed concept of neutrosophic soft graph (briefly ns-graph) which is a combination of graphs and neutrosophic soft sets. We re-define notions of ns-graphs and ns-subgraphs with the different perspective from the study in [4]. Also, we introduce some new operations on ns-graphs and detailed them with convenient examples. Moreover, we present an application of ns-graphs to determine of optimal object by using given data with the help of an algorithm. This algorithm we developed is new inventive-ness domain for problems which are involving uncertainty, and effectively finds the optimal result between the states where vagueness exists. We also provide a comparative analysis with the existing method given in [4].


Introduction
It is clear that the uncertainty arising from different areas cannot be explained and expressed with precise definitions.Different types of uncertainty are common in many fields such as economics, biology, physics, engineering, medicine and social sciences.Since uncertainty is a complex and broad concept that is not clearly defined, many fields dealing with uncertainty have failed to model this situation successfully with classical mathematical methods.In valuing a phenomenon in real life, we use intermediate values, that is, fuzzy values.For example, when evaluating the temperature of the air, we make ratings such as cold, slightly cold, warm, slightly hot and hot.Therefore, classical set theory falls short of expressing intermediate state values.This inadequate situation in classical set theory was first tried to be overcome with fuzzy set theory [21].A fuzzy set A is characterized with the help of a membership function µ A (x), a mapping from the universal set X to the unit interval [0,1], where x in the fuzzy set A has a certain degree of membership.Although a phenomenon can be represented with only one of 0 and 1 values in 1056 Y. C ¸EL İK classical set theory, it can take infinite values in fuzzy logic.Thus, a phenomenon can have uncertain values in the fuzzy approach.Fuzzy logic controllers have been applied many areas from electrical household appliances to auto electronics, from business machines we use daily to production engineering, from industrial control technologies to automation.Atanassov [5] put forward the intuitionistic fuzzy set theory which is a generalized type of the fuzzy set theory.In intuitionistic fuzzy sets, unlike fuzzy sets, the elements have degrees of non-membership.However, in these theories, the uncertainty of an element is not discussed, although the values of an element such as whether it is a member or not.Based on this, Smarandache [19] introduced the neutrosophic set theory which is an extended and special case of fuzzy set theory.The neutrosophic sets are stated by three functions.These are truth, indeterminacy and falsity membership function.The neutrosophic models produce more suitable solutions for the complex systems.In addition, an individual may not always be fully informed about a subject.In this case, the indeterminacy membership function comes into play and it provides a very large place for modeling events involving many uncertainties.Maji [14] defined the neutrosophic soft set concept and examined the properties of this concept.Broumi [7] worked on generalized netrosophic soft sets.Deli [11] gave the notion of interval-valued neutrosophic soft set and also applied this concept to the a decision-making problem.
Some great scientific theories grew out of answers to simple questions.Graf theory is one of them.Graf theory was first put forward by Euler [12].Graph theory which is an important mathematical tool for solving complicated problems in many different fields.Graphs are used to put forth a rellevance between elements in a given set, where every element can be expressed with the help of vertices and their relation edges.Since graph theory provides conveniences in modeling complicated systems, it has many number of applications.A simple graph is showed by G * = (ν, ε) where ν and ε represents sets of vertices and edges, respectively.After Euler's introduction of the graph concept, Rosenfeld [17] introduced the fuzzy graph theory.Bhattacharya [6] gave some properties of fuzzy graphs.Mordeson and Peng [16] have defined some operations on fuzzy graphs.Later, many researchers discussed the concept of fuzzy sets on the graph theory and defined different structures.Akram and Dudek [1] gave the concept of interval-valued fuzzy graphs and examined their related properties.Broumi et al. [9] gave the concept of interval valued pentapartitioned neutrosophic graphs.Broumi et al. [8] defined interval-valued fermatean neutrosophic graphs and presented some operations on this.Thumbakara and George [20] gave the concepts of soft graph and soft subgraph, and examined the properties of these structures.Akram and Nawaz [2] described some new algebraic operations on soft graphs.Mohinta and Samanta [15] defined the concept of fuzzy soft graph.Later, Akram and Nawaz [3] studied different types of fuzzy soft graphs.Zihni et al. [22] gave the concept of interval-valued fuzzy soft graph and examined its basic properties.C ¸elik [10] gave the concept of bipolar fuzzy soft graph and investigated some operations on this concept.Kandasamy et al. [13] gave the concept of neutrosophic graph and made various applications of neutrosophic graphs.Akram et al. [4] combined neutrosophic soft set concept and graph theory, then define notion of neutrosophic soft graph (ns-graph).They also applied the ns-graphs to the a decision-making problem.Shah and Hussain [18] gave new features on ns-graphs.
In the current study, the renewed concept of ns-graphs is defined and some new operations not previously defined such us extended union, restricted union, extended intersection, restricted intersection and complement are presented.Also illustrative examples related these operations are given.Hence an application of nsgraphs for a decision-making problem is examined with the method we developed.Moreover a comparative analysis between proposed method and existing method given in [4] are revealed.

Preliminaries
Definition 1. [19] Let X ̸ = ∅ be an universe.Then a neutrosophic set A on X is given by A = {⟨x, T A (x), I A (x), F A (x)⟩, x ∈ X}, where the functions T A , I A and F A are fuzzy sets on X under the conditions 0 ≤ T A (x) + I A (x) + F A (x) ≤ 3 for all x ∈ X.The family of all neutrosophic sets on X is denoted by N (X).Definition 2. [14] Let X be an initial universe.Let E be a set of parameters.Then, a pair (f, E) is called a neutrosophic soft set (briefly ns-set) over X representing a mapping by f : E → N (X).[14] Let (f, E 1 ) and (g, E 2 ) be two ns-sets over X, then (f, E 1 ) is said to be a ns-subset of (g, for all e ∈ E 1 and x ∈ X.
3. Renewed Structure of Ns-graphs with Some New Operations Definition 5. A ns-graph is an order 4-tuple  1.
, e 3 } be a set of parameters.Let consider ns-sets f and g over ν and ε, respectively, as given in Table 3.

Comparative Study and Discussion
In this section, for determining of optimal object, a comparative study based on the results of numerical computation is discussed.For this, based on the application discussed above, we present a comparative analysis between the our proposed method and the existing method in [4].The method given in [4] just takes into 1074 Y. C ¸EL İK consideration "AND" or "OR" operations to obtain resultant ns-graph which is used determine of optimal object.However if these two operations are applied separately, different results occur in the selection of the optimal object.So this leads to uncertainty in determining the most appropriate choice.Our proposed approach just takes into consideration both of ∧-intersection and ∨-union operations to obtain resultant ns-graphs, because we observe that these two operations should be interdependent in determining the possible choice.When the existing method given in [4] is compared with the current proposed method, ranking orders both of them are appeared as in Table 17.
Table 17.Comparison Model Ranking order Existing method given in [4] In existing method, the obtained ranking results are quite close where the first and fifth ranking order are changed but the second, third and fourth ranks are consistent.Nevertheless, obtaining two different rankings causes problems in decisionmaking process.In proposed method, the obtained ranking result is unique and more effective in determining the appropriate choice.Clearly, the proposed method consider the problem in all aspects and reveals a final result although existing method provides a set of alternatives as a final selection to consider the problem.

Conclusions
When compared with soft graph and fuzzy soft graph models, ns-graphs are more useful mathematical tools.Ns-graphs can be used in many areas with uncertainty.
We have introduced the concept of ns-graphs of a simple graph with some new notions such as union and intersection, and gave illustrative examples related to these notions.Also we have applied the concept of ns-graph to a decision-making problem, and then a case study has been given to show the application of the technique.Hence a comparative analysis is conducted to show the applicability and validity of the proposed approach.The proposed method can be used in dealing with decision making process involving uncertainty especially in solving the real scientific and engineering problems.For future research, another algorithm can be developed by incorporate the complement of ns-graphs.Therefore we will work for the extension of the this method in different neutrosophic structures and decisionmaking applications.We plan to extend this research work to (i) Vague ns-graphs, (ii) Intuitionistic ns-graphs, and (iii) Bipolar ns-graphs.

9 )Figure 7 .
Figure 7. Ns-graph G N G ′ N = (G * , f, g, E) e 3 }.If the membership values of vertices and edges of G N G ′ N are calculated, then the ns-graph G N G ′ N = (G * , f, g, E) is obtained as in the Table11.

Table 14 .
Score value of incidence matrice h(e)

Table 15 .
Score value of incidence matrice h ′ (e)

Table 16 .
Avarage score values of h(e) and h ′ (e) N = (G * , f, g, E). 4. Compute the resultant neutrosophic graphs h(e) and h ′ (e) with h(e) = ∨ k∈Λ h(e k ) and h ′ (e) = ∧ k∈Λ h(e k ), respectively, for all k ∈ Λ. 5. Construct incidence matrice forms of h(e) and h ′ (e).6. Calculate the score S k of v k for all k ∈ Λ. 7. Determine decision as v