Distinguishable and Inverses of Neutrosophic Finite Automata

This chapter focuses on neutrosophic finite automata with output function. Some new notions on neutrosophic finite automata are established and studied, such as distinguishable, rational states, semi-inverses, and inverses. Interestingly, every state in finite automata is said to be rational when its inputs are ultimately periodic sequence that yields an ultimately periodic sequence of outputs. This concludes that any given state is rational when its corresponding sequence of states is distinguishable. Furthermore, this study is to prove that the semi-inverses of two neutrosophic finite automata are indistinguishable. Finally, the result shows that any neutrosophic finite automata and its inverse are distinguished, and then their reverse relation is also distinguished.

Finding the Shortest Path With Neutrosophic Theory

In this chapter, the authors study a kind of network where the edge weights are characterized by single-valued triangular neutrosophic numbers. First, rigorous definitions of nodes, edges, paths, and cycles of such a network were proposed, which are then defined in algebraic terms. Then, characterization on the length of paths in such a network were presented. This is followed by the presentation of an algorithm for finding the shortest path length between two given nodes on the network. The proposed algorithm gives the shortest path length from source node to destination node based on a ranking method that takes both the length of edges and the number of nodes into account. Finally, a numerical example based on a real-life scenario is also presented to illustrate the efficiency and usefulness of the proposed approach.

A Study of Neutrosophic Shortest Path Problem

Shortest path problem (SPP) is an important and well-known combinatorial optimization problem in graph theory. Uncertainty exists almost in every real-life application of SPP. The neutrosophic set is one of the popular tools to represent and handle uncertainty in information due to imprecise, incomplete, inconsistent, and indeterminate circumstances. This chapter introduces a mathematical model of SPP in neutrosophic environment. This problem is called as neutrosophic shortest path problem (NSPP). The utility of neutrosophic set as arc lengths and its real-life applications are described in this chapter. Further, the chapter also includes the different operators to handle multi-criteria decision-making problem. This chapter describes three different approaches for solving the neutrosophic shortest path problem. Finally, the numerical examples are illustrated to understand the above discussed algorithms.

Dominations in Neutrosophic Graphs

The aim of this chapter is to impart the importance of domination in various real-life situations when indeterminacy occurs. Domination in graph theory plays an important role in modeling and optimization of computer and telecommunication networks, transportation networks, ad hoc networks and scheduling problems, molecular physics, etc. Also, there are many applications of domination in fuzzy and intuitionistic fuzzy sets for solving problems in vague situations. Domination in neutrosophic graph is introduced in this chapter for handling the situations in case of indeterminacy. Dominating set, minimal dominating set, independent dominating set, and domination number in neutrosophic graph are determined. Some definitions, characterization of independent dominating sets, and theorems of neutrosophic graph are also developed in this chapter.

Neutrosophic Soft Digraph

Neutrosophic soft sets are an important tool to deal with the uncertainty-based real and scientific problems. In this chapter, the idea of neutrosophic soft (NS) digraph has been developed. These digraphs are mainly the graphical representation of neutrosophic soft sets. A graphical study of various set theoretic operations such as union, intersection, complement, cross product, etc. are shown here. Also, some properties of NS digraphs along with theoretical concepts are shown here. In the last part of the chapter, a decision-making problem has been solved with the help of NS digraphs. Also, an algorithm is provided to solve the decision-making problems using NS digraph. Finally, a comparative study with proposed future work along this direction has been provided.

Introduction to Plithogenic Subgroup

This chapter gives some essential scopes to study some plithogenic algebraic structures. Here the notion of plithogenic subgroup has been introduced and explored. It has been shown that subgroups defined earlier in the crisp, fuzzy, intuitionistic fuzzy, as well as neutrosophic environments, can also be represented as plithogenic fuzzy subgroups. Furthermore, introducing function in plithogenic setting, some homomorphic characteristics of plithogenic fuzzy subgroup have been studied. Also, the notions of plithogenic intuitionistic fuzzy subgroup, plithogenic neutrosophic subgroup have been introduced and their homomorphic characteristics have been analyzed.

New Algorithms for Hamiltonian Cycle Under Interval Neutrosophic Environment

A cycle passing through all the vertices exactly once in a graph is a Hamiltonian cycle (HC). In the field of network system, HC plays a vital role as it covers all the vertices in the system. If uncertainty exists on the vertices and edges, then that can be solved by considering fuzzy Hamiltonian cycle. Further, if indeterminacy also exist, then that issue can be dealt efficiently by having neutrosophic Hamiltonian cycle. In computer science applications, objects may not be a crisp one as it has uncertainty and indeterminacy in nature. Hence, new algorithms have been designed to find interval neutrosophic Hamiltonian cycle using adjacency matrix and the minimum degree of a vertex. This chapter also applied the proposed concept in a network system.

Application of Floyd's Algorithm in Interval Valued Neutrosophic Setting

An algorithm with complete and incremental access is called a Floyd algorithm (FA). It determines shortest path for all the pairs in the network. Though there are many algorithms have been designed for shortest path problems (SPPs), due to the completeness of Floyd's algorithm, it has been improved by considering interval valued neutrosophic numbers as the edge weights to solve neutrosophic SPP (NSPP). Further, the problem is extended to triangular and trapezoidal neutrosophic environments. Also, comparative analysis has been done with the existing method.

Two Centroid Point for SVTN-Numbers and SVTrN-Numbers

In this chapter, some basic definitions and operations on the concepts of fuzzy set, fuzzy number, intuitionistic fuzzy set, single-valued neutrosophic set, single-valued neutrosophic number (SVN-number) are presented. Secondly, two centroid point are called 1. and 2. centroid point for single-valued trapezoidal neutrosophic number (SVTN-number) and single-valued triangular neutrosophic number (SVTrN-number) are presented. Then, some desired properties of 1. and 2. centroid point of SVTN-numbers and SVTrN-numbers studied. Also, based on concept of 1. and 2. centroid point of SVTrN-numbers, a new single-valued neutrosophic multiple-attribute decision-making method is proposed. Moreover, a numerical example is introduced to illustrate the availability and practicability of the proposed method. Finally, since centroid points of normalized SNTN-numbers or SNTrN-numbers are fuzzy values, all definitions and properties of fuzzy graph theory can applied to SNTN-numbers or SNTrN-numbers. For example, definition of fuzzy graph theory based on centroid points of normalized SVTN-numbers and SVTrN-numbers is given.

Minimal Spanning Tree in Cylindrical Single-Valued Neutrosophic Arena

In this chapter, the concept of cylindrical single-valued neutrosophic number whenever two of the membership functions, which serve a crucial role for uncertainty conventional problem, are dependent to each other is developed. It also introduces a new score and accuracy function for this special cylindrical single valued neutrosophic number, which are useful for crispification. Further, a minimal spanning tree execution technique is proposed when the numbers are in cylindrical single-valued neutrosophic nature. This noble idea will help researchers to solve daily problems in the vagueness arena.