Interval-valued Neutrosophic Graph Research Articles

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 flexible than single neutrosophic sets, interval-valued neutrosophic graphs (IVNGs) were defined. The 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 significant 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, effective edge, and other properties of IVNLGs are proposed. In addition, we defined 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.

Particle swarm optimization for the shortest path problem

Solving the shortest path problem is very difficult in situations such as emergency rescue after a typhoon: road-damage caused by a typhoon causes the weight of the rescue path to be uncertain and impossible to represent using single, precise numbers. In such uncertain environments, neutrosophic numbers can express the edge distance more effectively: membership in a neutrosophic set has different degrees of truth, indeterminacy, and falsity. This paper proposes a shortest path solution method for interval-valued neutrosophic graphs using the particle swarm optimization algorithm. Furthermore, by comparing the proposed algorithm with the Dijkstra, Bellman, and ant colony algorithms, potential shortcomings and advantages of the proposed method are deeply explored, and its effectiveness is verified. Sensitivity analysis performed using a 2020 typhoon as a case study is presented, as well as an investigation on the efficiency of the algorithm under different parameter settings to determine the most reasonable settings. Particle swarm optimization is a promising method for dealing with neutrosophic graphs and thus with uncertain real-world situations.

Pragmatic results in Taiwan education system based IVFG & IVNG

An interval-valued fuzzy graph (IVFG) and degree of vertices have been applied for performance evaluation in an educational system. The approach is mainly developed based on real membership values of vertices of an IVFG. In this manuscript, some definitions of generalized fuzzy graphs and neutrosophic graphs structures are improved. First, we have concentrated to improve the existing definitions for union and join of two IVFG’s, complete IVFG with supporting examples. Then, their modified version is developed. Secondly, the modified version of interval-valued neutrosophic graph (IVNG) is given. Third, an algorithm and a flowchart of the proposed method are described. Fourth, the generalized form of complete and strong IVNG is given with examples. Finally, a real-life application using interval-valued fuzzy graph in education system in Taiwan is exhibited.

Introduction of some new results on interval-valued neutrosophic graphs

In this paper, inspired by the concept of generalized single-valued neutrosophic graphs(GSVNG) of the first type, we define yet another generalization of neutrosophic graph called the generalized interval-valued neutrosophic graph of 1 type (GIVNG1) in addition to our previous work on complex neutrosophic graph (CNG1) in [47]. We will also show a matrix representation for this new generalization. Many of the fundamental properties and characteristics of this new concept is also studied. Like the concept CNG1 in [47], the concept of GIVNG1 is another extension of generalized fuzzy graphs 1 (GFG1) and GSVNG1.

Energy and Spectrum Analysis of Interval Valued Neutrosophic Graph using MATLAB

In recent time graphical analytics of uncertainty and indeterminacy has become major concern for data analytics researchers. In this direction, the mathematical algebra of neutrosophic graph is extended to interval-valued neutrosophic graph. However, building the interval-valued neutrosophic graphs, its spectrum and energy computation is addressed as another issues by research community of neutrosophic environment. To resolve this issue the current paper proposed some related mathematical notations to compute the spectrum and energy of interval-valued neutrosophic graph using the MATAB.

Some Root Level Modifications in Interval Valued Fuzzy Graphs and Their Generalizations Including Neutrosophic Graphs

Fuzzy graphs (FGs) and their generalizations have played an essential role in dealing with real-life problems involving uncertainties. The goal of this article is to show some serious flaws in the existing definitions of several root-level generalized FG structures with the help of some counterexamples. To achieve this, first, we aim to improve the existing definition for interval-valued FG, interval-valued intuitionistic FG and their complements, as these existing definitions are not well-defined; i.e., one can obtain some senseless intervals using the existing definitions. The limitations of the existing definitions and the validity of the new definitions are supported with some examples. It is also observed that the notion of a single-valued neutrosophic graph (SVNG) is not well-defined either. The consequences of the existing definition of SVNG are discussed with the help of examples. A new definition of SVNG is developed, and its improvement is demonstrated with some examples. The definition of an interval-valued neutrosophic graph is also modified due to the shortcomings in the current definition, and the validity of the new definition is proved. An application of proposed work is illustrated through a decision-making problem under the framework of SVNG, and its performance is compared with existing work.

Representation of Graph Structure Based on I-V Neutrosophic Sets

In this research article, we apply the concept of interval-valued neutrosophic sets to graph structures. We present the concept of interval-valued neutrosophic graph structures. We describe certain operations on interval-valued neutrosophic graph structures and elaborate them with appropriate examples. Further,we investigate some relevant properties of these operators. Moreover,we propose some open problems on interval-valued neutrosophic line graph structures.

Interval-Valued Neutrosophic Graph Representation of Concept Lattice and Its ( $$\alpha ,\beta ,\gamma $$ α , β , γ )-Decomposition

Recently, three-way representation of fuzzy concept lattice is introduced to characterize the uncertainty and vagueness in the given data set based on its acceptation, rejection and uncertain regions, by a well-defined single-valued neutrosophic set. This paper put forward effort to deal with partial ignorance in the given three-way fuzzy context using the properties of interval-valued neutrosophic set and its graphical properties. The objective is to discover some of the important hidden patterns (i.e. formal concepts) in the interval-valued neutrosophic context, and visualize their super–sub-concept hierarchical-order reflection in the concept lattice. To achieve this goal, a method is proposed to generate the interval-valued neutrosophic formal concepts based on properties of interval-valued neutrosophic set, interval-valued neutrosophic graph and concept lattice. Another method is proposed to refine some of the important patterns (i.e. interval-valued neutrosophic formal concepts) at user-defined ( $$\alpha , \beta , \gamma $$ )-cut for their truth (i.e. $$\alpha $$ -cut), indeterminacy (i.e. $$\beta $$ -cut) and falsity (i.e. $$\gamma $$ -cut) membership value, independently. In addition to that, both of the proposed methods are demonstrated with an illustrative example.

Interval-valued neutrosophic competition graphs

We first introduce the concept of interval-valued neutrosophic competition graphs. We then discuss certain types, including k-competition interval-valued neutrosophic graphs, p-competition interval-valued neutrosophic graphs and m-step interval-valued neutrosophic competition graphs. Moreover, we present the concept of m-step interval-valued neutrosophic neighbouhood graphs.

Interval-valued neutrosophic competition graphs

We first introduce the concept of interval-valued neutrosophic competition graphs. We then discuss certain types, including k-competition interval-valued neutrosophic graphs, p-competition interval-valued neutrosophic graphs and m-step interval-valued neutrosophic competition graphs. Moreover, we present the concept of m-step interval-valued neutrosophic neighbouhood graphs.

Representation of Graph Structure Based on I-V Neutrosophic Sets

In this research article, we apply the concept of interval-valued neutrosophic sets to graph structures. We present the concept of interval-valued neutrosophic graph structures. We describe certain operations on interval-valued neutrosophic graph structures and elaborate them with appropriate examples. Further, we investigate some relevant properties of these operators. Moreover,we propose some open problems on interval-valued neutrosophic line graph structures.

Concepts of Interval-Valued Neutrosophic Graphs

Broumi et al. [15] proposedthe concept of interval-valued neutrosophic graphs. In this research article, we first show that there are some flaws in Broumi et al. [15] ’s definition which violates condition of an interval. We then modify the definition of an interval-valued neutrosophic graph. We present some operations on interval-valued neutrosophic graphs. We discuss the concepts of self-complementary and self weak complementary interval-valued neutrosophic complete graphs. We also describe regularity of interval-valued neutrosophic graphs.