Theory and Application of Interval-Valued Neutrosophic Line Graphs

Neutrosophic graphs are used to model inconsistent information and imprecise data about any real-life problem. It is regarded as a generalization of intuitionistic fuzzy graphs. Since interval-valued neutrosophic sets are more accurate, compatible, and fexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defned. Te interval-valued neutrosophic graph is a fundamental issue in graph theory that has wide applications in the real world. Also, problems may arise when partial ignorance exists in the datasets of membership [0, 1], and then, the concept of IVNG is crucial to represent the problems. Line graphs of neutrosophic graphs are signifcant due to their ability to represent and analyze uncertain or indeterminate information about edge relationships and complex networks in graphs. However, there is a research gap on the line graph of interval-valued neutrosophic graphs. In this paper, we introduce the theory of an interval-valued neutrosophic line graph (IVNLG) and its application. In line with that, some mathematical properties such as weak vertex isomorphism, weak edge isomorphism, efective edge, and other properties of IVNLGs are proposed. In addition, we defned the vertex degree of IVNLG with some properties, and by presenting several theorems and propositions, the relationship between fuzzy graph extensions and IVNLGs was explored. Finally, an overview of the algorithm used to solve the problems and the practical application of the introduced graphs were provided.


Introduction
Graph theory is a mathematical discipline that deals with mathematical representations of the links between objects.Not all systems described by science and technology can accommodate complex processes and occurrences.For situations like these, mathematical models have been created to handle different kinds of systems with uncertainty-containing components.In 1965, Zadeh presented fuzzy sets by giving membership grades to each object of the interval set [1].Based on Zadeh's fuzzy relations, Kaufman [2] proposed fuzzy graphs.Later on, Rosenfeld [3] discussed the fuzzy analogy of many graph-theoretic notions.After this, researchers started to introduce many classes of fuzzy graphs, and they have brought remarkable advances to impressive applications of fuzzy theory.
However, linguistic terms are very important in decision-making theory, such as data mining, multiattribute decision-making (MADM) problems, and a novel type of linguistic information form to facilitate decision-makers in evaluating online learning platforms in a comprehensive manner, and the determination of decision-makers' weights is a key step prior to the aggregation of individual assessment information into a collective result [4,5].
Te membership function was insufcient to explain exactly the complexity of object features, leading to the suggestion of a nonmembership function with fuzzy sets (FS).Te extension of FS, which is called intuitionistic fuzzy sets (IFS), was introduced [6][7][8].Later, the notion of m-polar interval-valued intuitionistic fuzzy graphs was introduced [9].Also, several types of arcs in the interval-valued intuitionistic (S, T)-fuzzy graphs and their properties were studied [10,11].Due to the dynamic nature of certain problems that cannot be addressed by FS and IFS, Smarandache [12] introduced neutrosophic sets (NS).He added another component, the indeterminacy degree, to the definition of IFS.Te notion of SVNSs, which has multiple applications in entropy measure, decision-making, index distance, and similarity measure, can be independently expressed as a truth-membership function (TMF), an indeterminacy membership function (IMF), and a nonmembership or falsity function (FMF) by adhering to the defnition of NS [13].In any case, from a philosophical perspective, the membership, indeterminate, and nonmembership values are independent with respect to one another, and the TMF, IMF, and FMF values are in the interval of [0, 1] presented in [14].For that reason, sometimes it happens that the membership, indeterminate, and nonmembership values cannot be measured as a point, but they can be measured as an interval.Considering this, later, an IVNS and its properties were introduced [15,16].In comparison to an SVNS, an IVNS provides a more accurate and fexible description of graphs.An IVNS is a generalization of SVNSs, and it has many applications in decisionmaking [17].
Besides the fact that single-valued neutrosophic graphs (SVNG) were proposed [18], graph theory is a basic idea in modern mathematics.Graphs are used as a mathematical tool to visually represent and evaluate social networks after all of this has been considered.Consequently, neutrosophic graphs with interval values were studied by Broumi et al. [19].However, Akram and Shahzadi [20] provided a different defnition of SVNG because this defnition goes against the concepts of complement and join characteristics.He also presented the concept of interval-valued neutrosophic competition graphs [21].An IVNG and some of its functions were discussed in light of the revised defnition of SVNSs.Tey also noticed that IVNG may be altered to take on a regular structure [22].Connectivity concepts are the key to graph clustering and networks, and they are the most important concept in the entire graph theory [23].
In a network, vertices hold signifcance due to their connections with other vertices, while edges in a line graph can be applied with vertices' attributes [24].Typically, the structure of a line graph L(G) is more complex than that of the corresponding graph G.Many other researchers studied diferent classes of L(G), such as classical line graphs [25], fuzzy line graphs [26], interval-valued fuzzy line graphs (IVFLGs) [27], intuitionistic fuzzy line graphs (IFLGs) [28], and the L(G) of IVIFG [29].
Subsequently, SVNGs were explored by researchers and used to tackle a variety of real-world modeling and optimization issues.Also, the defnition and mathematical properties of SVNLG were derived from the single-valued neutrosophic graph [30].Tey provided both necessary and sufcient criteria for SVNG and its corresponding SVNLG to be isomorphic.Also, neutrosophic vague line graphs were investigated [31].Isomorphic properties of those graphs were also initiated.Te reader should read articles [19,[32][33][34] to understand the fundamental ideas of the line graph and its properties.
Te interval-valued neutrosophic graph is an elementary graph theory problem with numerous real-world applications.Nevertheless, no other academics have yet to introduce the IVNLG theory.In this study, we introduced the theory and application of interval-valued neutrosophic line graphs and described some of their properties.One of the motives of this research was to apply the concepts introduced to real-life problems.Finally, a procedure to drive IVNLG from a connected simple NG and its application are presented.Tis work's framework is organized as follows: In Section 1, we provide a basic overview of fuzzy graphs (FGs), IFG, neutrosophic graphs (NGs), and the line graphs that correspond to each of these concepts.In Section 2, we cover the foundational mathematical ideas that will be applied to the research.A comprehensive defnition and appropriate examples of IVNLG are provided in Section 3. Some basic IVNLG properties are presented in Section 4, along with some propositions.In Section 5, a real-world application for a decision-making problem was designed using IVNLG.Finally, further research work related to the research paper is discussed in the conclusion.

Preliminaries
Here, we have used standard defnitions, terminologies, and results from the rest of the article.
Defnition 1 (see [35]).An ordered triple Defnition 2 (see [36]).Te graph G � (V, E) is an IFG if the following conditions are satisfed: is a node set of the graph P(E) and an edge of the graph Defnition 4. Let P(E) � (S, Λ) be the intersection graph of G � (V, E).Ten, L(G) of G can be derived by defnition of the intersection graph P(E).Tis implies L(G) � (Z, W) is a line (edge) graph where 2 Journal of Mathematics Defnition 5 (see [37]).Let A be a subset of a universal set X.
are TMF, IMF, and FMF, respectively, with the condition where the functions TMF, IMF, and FMF of A will be real standard or nonstandard subsets of ]0 − , 1 + [.Due to the fact that it is difcult to use the above definition of NS in real-life situations, Wang et al. introduced the idea of SVNS having TMF, IMF, and FMF values in the range of [0, 1], which will be used in scientifc and engineering applications [12].
Defnition 6 (see [20]).An SVN-graph is a pair of function G � (A, B) where A � (t 1 , i 1 , f 1 ) is the subset of V and B � (t 2 , i 2 , f 2 ) is the subset of E with the conditions: are TMF, IMF, and FMF of the vertex set of u ∈ V, respectively, such that 0 represent the TMF, IMF, and FMF of edge E, respectively, with the condition be SVNS on V and E, respectively.Ten, the intersection graph (S, Λ) � (A 2 , B 2 ) of SVNG G, where Defnition 8 (see [30]).Consider represent the SVNSs with Z and W, respectively, so that (y), t B2 (S x S y ) � t B1 (x) ∨ t B1 (y), for all S x S y ∈ W Defnition 9 (see [28]).Suppose there are two SVNGs and also, where Defnition 10 (see [22]).Let X ≠ ∅ set of points (objects) and A ⊆ X. Ten, we defne an IVNS of A as follows: where Troughout this article, we used the modifed defnition of IVNG which is introduced by Mohammed Akram and Nasir [22].
) are an IVN relation, which satisfes the following conditions: where 0 Defnition 13.Given an IVNG G � (A, B), then, for u i ∈ V, the degree u is denoted by d(u i ) and given by where Journal of Mathematics Defnition 14 (see [38]).An IVNG G is strong IVNG if and only if all of the following holds: Defnition 15.An IVNG G is complete IVNG if a graph G satisfes the following properties: Tus, the neighborhood of the vertex u i in IVNG is denoted by N(u i ), and it is defned by For instance, the vertex u i is a neighbor of u j , and u j is neighbor of u i .

Interval-Valued Neutrosophic Line Graph
We have here introduced an IVNLG for undirected IVNG and some mathematical properties of undirected IVNG with examples.We only considered IVNG without self-loops and parallel edges.
(a) A 2 and B 2 are IVNSs of S and Λ, respectively, (b) , for all S i S j ∈ Λ. Terefore, any IVNG of P(E) is called an IVNintersection graph.

Defnition 18. Consider the line graph
(ii) Te edge of an IVNLG of G is computed as , for all S e i S e j ∈ W. ( Example 1.Consider an IVNG G � (A 1 , B 1 ) as shown in Figure 1 where the vector set of a graph G is is the edge set on A 1 , as shown in Tables 1 and 2.
From above Figure 1, we can drive a line graph as follows: Ten, by the defnition of the line graph, L(G) � (A 2 , B 2 ) can be obtained by routine computation; the vertex set of IVNLG G is as follows: Using the defnition of the line graph, an edge set of IVNLG G is as follows: so that an IVNLG G is shown in Figure 2.

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6 Journal of Mathematics

E). Ten, L(K) is the subgraph of an IVNLG L(G).
Proof.Suppose that K is a subgraph of IVNG G. Ten, we have V ′ ⊆ V and E ′ ⊆ E. We know that V(L(K)) ⊆ E(G), and it is also the subset of V(L(G)).Moreover, the edge set of the line graph L(K) is a subset of the edge set of IVNLG G. Hence, L(K) ⊆ L(G).Table 1: Te vertices of IVNG G.
Table 2: Te edges of IVNG G.
Te weak line (edge) isomorphism of IVNG is a bijective homomorphism ψ: K 1 ⟶ K 2 if the following conditions hold: Defnition 23.If the mapping ψ: K 1 ⟶ K 2 is a bijective weak vertex and weak edge isomorphism, then we say that ψ is a weak isomorphism map of IVNGs from K 1 to K 2 .
Defnition 24 (see [17]).A path P in undirected If P is a path with n-vertices, then the length of P is n − 1.A single node u i may also be taken as a path with length ([0, 0], [0, 0], [0, 0]).Te edges of the path are successive pairs (u i−1 , u i ).If n ≥ 3 and u 0 � u n , then P is referred to as a cycle.

Defnition 25.
If there is at least one path between each pair of nodes in an IVNG G � (A, B), then IVNG G is said to be connected; otherwise, it is disconnected.

Proposition 26. Te IVNLG L(G) is connected if its original graph IVNG G is a connected graph.
where Proof.Suppose (i), (ii), and (iii) are true.Tat implies For each x ∈ E, we have

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Now, from (i), (ii), and (iii), we have We know that IVNS which is sufcient.Te converse of this statement is obvious from the defnition of IVNLG, and hence, the proof holds.

□
Defnition 34.Let G � (V, E) be an IVNG G. Ten, we have ) is the maximum degree of an IVNgraph G where ) is the minimum degree of an IVN-graph G where An IVNG is strong if every edge is an efective edge.

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Theorem 36.Let G be an IVNG and IVNLG be the corresponding line graph of G. Ten, every edge of IVNLG is an efective edge.
Proof.From the defnition of an IVNLG, the proof of this theorem is straightforward.

Application
Suppose that an investor invested in four diferent companies, namely, food, automobile, computer, and textile companies.Investors also had employees at each company, and all four employees knew each other.Teir friendship is for diferent purposes.Some of them are for the success of the organization.Some of them are for unknown reasons beyond the control of the manager of the company (say, political view, culture, ethnicity, language, and so on), and some employees make relationships for their own advantage only, and those who do not care about the company's mission and objectives.Te investor wants to analyze the friendship of his employees between the companies to identify which relationship is for success, unknown, and failure of the organization.Also, there is uncertainty and imprecise situation of the employees in each company to perform the organization's goals.Te investor conducted a survey and collected information from managers of each company about how the problem afects the performance of the organization due to the friendship of employees between the companies and within the company.Consider employee friendship within a car company, a computer company, a food company, and a textile company as a vertex set and employee friendship between companies as a set of edges.Te NS results from uncertainty, impreciseness, and inconsistency in the data, which is caused by the fact that the information gathered is dependent upon the manager of each company.Since IVNSs are more appropriate than SVNSs, we also use this concept.As TMF, IMF, and FMF values, it is evident that the employees' friendship is defned independently by IVNSs.An IVNG will be used to represent this analysis.
Te degree of friendship activities of employees in the company represents the membership values of a node.Similarly, the degree of the relationship between the nodes measures the edge membership value.As a result, there are three diferent kinds of edge interval membership values: truth, indeterminacy, and false.Such a type of network is an example of an IVNG.Terefore, since the investor wants to analyze the relationship between edges, which means friendship between each company, transforming the given graph into a line graph is a better way to solve the problem.
In order to construct an interval-valued neutrosophic line graph for the friendship relationships between employees of diferent companies, a few steps can be followed: Step 1: Defne the original graph that represents the employees and their friendship relationships.Each node in the graph represents an employee, and the edges represent the friendships between them.Assign weights or values to the edges representing the strength or closeness of the friendships.
Step 2: Determine the interval values: Assign interval values to each edge in the graph to represent the membership, nonmembership, and indeterminacy associated with the friendship relationship.Tese interval values can be based on subjective assessments, surveys, or expert opinions.Te intervals should capture the range of possibilities for the strength of the friendship, considering both the lower and upper bounds.
Step 3: Construct the interval-valued neutrosophic line graph by creating a new graph where the nodes represent the edges of the original graph.Te edges in the interval-valued neutrosophic line graph represent the adjacency or connections between the edges in the original graph.
Step 4: Capture the neutrosophic aspect by accounting for the indeterminacy, ambiguity, and incomplete knowledge associated with the friendship relationships.Tis can be done by allowing for the existence of uncertain or ambiguous information within each interval.
Step 5: Analyze and interpret the results from the constructed interval-valued neutrosophic line graph.Tis can involve exploring the ranges of possibilities for the strength of the friendship relationships, identifying any uncertain or ambiguous regions, and understanding the overall patterns or trends in the graph.above example, the vertex degree of S e6 is [0.4,0.8], [0.4,1.4], [1.2, 2.8], which means that the truth-membership degree of S e6 is minimum, the indeterminacy-membership degree of S e6 is maximum, and the nonmembership degree of S e6 is maximum when compared with other vertices.So, the investor should focus as much as possible on the employee's relationship (S e6 ), which is the computer company and the food company.Terefore, either disconnecting or managing employee relationships with the two companies is a better option to be competitive in investment by managing the employees' relationships within the company and across diferent organizations.

Conclusion
An interval-valued neutrosophic model is more complex than an IVIF or an IVF model.Numerous real-world systems with varying levels of precision, incompleteness, vagueness, and uncertainty can be modeled using this technique.As a result, the study concentrated on the IVNLG concept, which is crucial to real-world problems.
In this paper, our focus is to introduce both the theory and application of IVNG.Tese include defnition, vertex degree, edge degree, isomorphic properties, and daily life applications of IVNLGs.In this regard, we explained the maximum degree and minimum degree of a vertex of the IVNLGs and their role in the art of decision-making.Many types of line graphs have already been discussed from a diferent perspective by other researchers, for example, classical line graphs, fuzzy line graphs, intervalvalued fuzzy line graph (IVFLG), intuitionistic fuzzy line graphs (IFLG), and L(G) of IVIFG.Also, the line graph of single-valued neutrosophic graphs was introduced.Interval-valued neutrosophic line graphs are the generalization of interval-valued fuzzy line graphs and intervalvalued intuitionistic line graphs.In addition, weak vertex, weak line, and homomorphism properties are demonstrated.We also presented some theorems, propositions, and properties of the IVNLG.Finally, the algorithm that is used to calculate the degree of IVNLG as well as the application of IVNLG has also been discussed and illustrated by numerical examples.Based on this result, we  Journal of Mathematics can extend the introduced concept to several extensions of neutrosophic graphs since it is the most general form of graph today and was designed in order to capture our complex real world.We can also extend this new concept to direct neutrosophic graphs and other areas of graph theory.

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Journal of MathematicsProof.Given G is IVNG and L(G) is a connected IVNLG of G, we must demonstrate that precondition.Assume G is not connected IVNG.Ten, G has at least two nodes that are not connected by a path.If we choose one edge from the frst component, there are no edges that are adjacent to edges in other components of G. Te L(G) of G is then broken and contradicting, so that G is connected.Conversely, assume that G is not a disconnected IVNG.Ten, there is a path that connects each pair of nodes.Adjacent edges in G are thus neighboring nodes in L(G), according to the defnition of L(G).As a result, each pair of nodes in L(G) has a path that connects them.Hence, the proof holds.
□Proposition 27.Consider G * � (V, E) with underlying set V and L � (V, E) and G � (A 1 , B 1 ) is an IVNG.For an IVNLG L(G) � (A 2 , B 2 ) where A 2 � t A 2 , i A 2 , f A 2   and B 2 � t B 2 ,  i B 2 , f B 2 }are IVNS on Z and W, respectively.Ten, we denoted the vertex degree of L(G) by d(S x ) defned □ Theorem 3 .Let G be a connected IVNG path graph.Ten, an IVNLG L(G) is a connected path graph.Proof.Consider a path G is connected IVN-path graph with |V(G)| � k.Tis implies that |E(G)| � k − 1 and that G is a P k path graph.Since the vertex set of an IVNLG G is the *

Table 6 :
Edge set of IVNLG G.