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The notion of neutrosophic algebra, ideal of neutrosophic algebra, kernel and neutrosophic quotient algebra have been proposed in this paper. We characterize some properties of neutrosophic algebra and proved that every quotient neutrosophic algebra is quotient algebra. Also we proved that every neutrosophic algebra is algebra and direct product of neutrosophic algebras over a neutrosophic field is algebra

This paper introduced the concept of neutrosophic multi topological space. The definitions of the interior, closure, exterior, and boundary of neutrosophic multiset is defined and also, its propositions will be studied with examples.

This paper demonstrates a concept of split domination in neutrosophic graphs.Minimal split domination, lower and upper split dominations in neutrosophic graphs are discussed. Theorems are derived for minimal split domination in neutrosophic graphs with suitable examples.

A single-valued neutrosophic set is one of the advanced fuzzy sets that is capable of handling complex real-world information satisfactorily. A development of single-valued neutrosophic set and fuzzy bipolar set, called a bipolar neutrosophic set, was introduced. Distance measures between fuzzy sets and advanced fuzzy sets are important tools in diagnostics and prediction problems. Sometimes they are defined without considering the condition of the inclusion relation on sets. In decision-making applications, this condition should be required (here it is called the inference of the measure). Moreover, in many cases, a distance measure capable of discriminating between two nearly identical objects is considered an effective measure. Motivated by these observations, in this paper, a new distance measure is proposed in a bipolar neutrosophic environment. Furthermore, an entropy measure is also developed by the similarity between two sets of mutual negation. Finally, an application to medical diagnosis is presented to illustrate the effective applicability of the proposed distance measure, where entropy values are used to characterize noises of different attributes.

The objective of this paper is to study algebraic properties of neutrosophic matrices, where a necessary and sufficient condition for the invertibility of a square neutrosophic matrix is presented by defining the neutrosophic determinant. On the other hand, this work introduces the concept of neutrosophic Eigen values and vectors with an easy algorithm to compute them. Also, this article finds a necessary and sufficient condition for the diagonalization of a neutrosophic matrix.

The idea behind the neutrosophic set is we can connect the concept by dynamics of opposite interacts and its neutral that are uncertain and get common parts. Automata theory is beneficial to solve computational complexity problem and also it is an influential mathematical modeling tool in computer science. Inspired by the concepts of neutrosophic sets and automata theory, here, we are introducing and discussing the algebraic concept of neutrosophic finite automata based on the paper [10]. Generally, composite machines can be achieved by the output of the one machine that will be used as input for another machines. This paper introduced the concept of composite automata under the environment of the neutrosophic set and also examined the box function between the composite neutrosophic finite automata. &nbsp

Distance measure is a numerical measurement of the distance between any two objects. The aim of this paper is to propose a new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids with graphical representation. In addition, the metric properties of the proposed measure are examined in detail. A decision making problem also has been solved using the proposed distance measure for a software selection process. comparative analysis has been done with the existing methods to show the potential of the proposed distance measure and various forms of trapezoidal fuzzy neutrosophic number have been listed out to show the uniqueness of the proposed graphical representation. Further, advantages of the proposed distance measure have been given.

In paper, neutrosophic soft points with the concept of one point greater than the other and their properties, generalized neutrosophic soft open set known as soft b-open set, neutrosophic soft separation axioms theoretically with support of suitable examples with respect to soft points, neutrosophic soft -space engagement with generalized neutrosophic soft closed set, neutrosophic soft -space engagement with generalized neutrosophic soft open set are addressed. In continuation, neutrosophic soft -space behave as neutrosophic soft -space with the plantation of some extra condition on soft -space, neutrosophic soft -space and related theorems, neutrosophic soft -space, monotonous behavior of neutrosophic soft function with connection of different neutrosophic soft separation axioms, monotonous behavior of neutrosophic soft function with connection of different neutrosophic soft close sets are reflected. Secondly, long touched has been given to neutrosophic soft countability connection with bases and sub-bases, neutrosophic soft product spaces and its engagement through different generalized neutrosophic soft open set and close sets, neutrosophic soft coordinate spaces and its engagement through different generalized neutrosophic soft open set and close sets, Finally, neutrosophic soft countability and its relationship with Bolzano Weirstrass Property through engagement of compactness, neutrosophic soft strongly spaces and its related theorems, neutrosophic soft sequences and its relation with neutrosophic soft compactness, neutrosophic soft Lindelof space and related theorems are supposed to address.

Unipolar is less fundamental than bipolar cognition based on truth, and composure is a restraint for truth-based worlds. Bipolarity is the most powerful phenomenon that survives when truth disappeared in a black hole due to Hawking radiation or particular / anti particular emission. The purpose of this research study is to dene few four operations, including residue product, rejection, maximal product and symmetric difference of bipolar single-valued neutrosophic graph (BSVNG) and to explore some of their related properties with examples. Bipolar single-valued neutrosophic graph (BSVNG) is the generalization of the single-valued neutrosophic graph (SVNG), intuitionistic fuzzy graph, bipolar intuitionistic fuzzy graph, bipolar fuzzy graph and fuzzy graph. BSVNG plays a signicant role in the study of neural networks, daily energy issues, energy systems, and coding. Moreover, we will determine related properties like the degree of a vertex in a BSVNG or total degree of a vertex in a BSVNG. We provide examples of the vertex degree in BSVNG and the total vertex degree in BSVNG. In order to make this useful, we develop an algorithm for our useful method in steps.

In this paper, the concept of t-neutrosophic cubic set is introduced and investigated the t-neutrosophic cubic set through subalgebra, ideal and closed ideal of BF-algebra. Homomorphic properties of t-neutrosophic cubic subalgebra and ideal are also investigated with some related properties.

Many results have been obtained on isolatedgraphs and complete graphs. In this paper, a necessaryand sufficient condition will be proved for a single valuedneutrosophic graph to be an isolated single valuedneutrosophic graph.

Many extension and generalization of fuzzy sets have been studiedand introduced in the literature. Hesitancy fuzzy digraph is ageneralization of intuitionistic fuzzy set and fuzzy graph. In thispaper, we redefine some basic operations of hesitancy fuzzy graphand it is referred as hesitancy fuzzy digraph (in short HFDG). Wediscuss some arithmetic operations and relations among HFDG. Wefurther proposed a method to solve a shortest path problem throughscore function.

In this paper, we introduced a new neutrosophic graphs called bipolar complex neutrosophic graphs of type1 (BCNG1) and presented a matrix representation for it and studied some properties of this new concept. The concept of BCNG1 is an extension of generalized fuzzy graphs of type 1 (GFG1), generalized single valued neutrosophic graphs of type 1 (GSVNG1), Generalized bipolar neutrosophic graphs of type 1(GBNG1) and complex neutrosophic graph of type 1(CNG1).

Neutrosophic set and neutrosophic logic theory are renowned theories to deal with complex, not clearly explained and uncertain real life problems, in which classical fuzzy sets/models may fail to model properly. This paper introduces an algorithm for finding minimum spanning tree (MST) of an undirected neutrosophic weighted connected graph (abbr. UNWCG) where the arc/edge lengths are represented by a single valued neutrosophic numbers. To build the MST of UNWCG, a new algorithm basedon matrix approach has been introduced. The proposed algorithm is compared to other existing methods and finally a numerical example is provided.

This paper presents a study of neutrosophic shortestpath with interval valued neutrosophic number on a network. Aproposed algorithm also gives the shortest path length usingranking function from source node to destination node. Hereeach arc length is assigned to interval valued neutrosophicnumber. Finally, a numerical example has been provided forillustrating the proposed approach.

In this research paper, a new approach is proposed for computing the shortest path length from source node to destination node in a neutrosophic environment. The edges of the network are assigned by trapezoidal fuzzy neutrosophic numbers. A numerical example is provided to show the performance of the proposed approach.

In this article, we combine the concept of bipolar neutrosophic set and graph theory. We introduce the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs and investigate some of their related properties.&nbsp

In this paper, we first give the cartesian product of two neutrosophic multi sets(NMS). Then, we define relations on neutrosophic multi sets to extend the intuitionistic fuzzy multi relations to neutrosophic multi relations. The relations allows to compose two neutrosophic sets. Also, various properties like reflexivity, symmetry and transitivity are studied.

In this paper, we first present the concept of Q- intuitionistic fuzzy soft sets which combine Qintuitionistic fuzzy sets and soft sets. Basic properties are introduced along with illustrative examples. We also define its basic operations, namely equality, subset, complement, union, intersection, AND and OR, and study some related properties with supporting proofs. This concept is a generalization of Q-fuzzy soft sets

An advantage of dealing indeterminacy is possible only with Neutrosophic Sets. Graph theory plays a vital role in the field of networking. If uncertainty exist in the set of vertices and edge then that can be dealt by fuzzy graphs in any application and using Neutrosophic Graph uncertainty of the problems can be completely dealt with the concept of indeterminacy. In this paper, Dombi Interval Valued Neutrosophic Graph has been proposed and Cartesian product and composition of the proposed graphs have been derived.

In this paper, some types of edge irregular single valued neutrosophic graphs such as neighbourly edge totally irregular single valued neutrosophic graphs, strongly edge irregular single valued neutrosophic graphs and strongly edge totally irregular single valued neutrosophic graphs are introduced. A comparative study between neighbourly edge irregular single valued neutrosophic graphs and neighbourly edge totally irregular single valued neutrosophic graphs is done. Likewise some properties of them are studied. Finally, we have given some interesting results about edge irregular single valued neutrosophic graphs that are very useful in computer science and networks.

Neutrosophic set is a generalization of classical set, fuzzy set, and intuitionistic fuzzy set by employing a degree of truth (T), a degree of indeterminacy (I), and a degree of falsehood (F) associated with an element of the dataset. One of the most essential problems is studying set-theoretic operators in order to be applied to practical applications. Developing Matlab toolboxes for computing the operational matrices in neutrosophic environments is essential to gain more widely-used of neutrosophic algorithms. In this paper, we propose some computing procedures in Matlab for neutrosophic operational matrices, especially i) computing the single-valued neutrosophic matrix; ii) determining complement of a single-valued neutrosophic matrix; iii) computing max-min-min and min-max-max of two single-valued neutrosophic matrices; v) computing power of a single-valued neutrosophic matrix; vi) computing additional operation and subtraction of two single-valued neutrosophic matrices; and ix) computing transpose of a single-valued neutrosophic matrix. Numerical examples are given to illustrate their applicability.

Neutrosophic set theory provides a new tool to handle the uncertainties in shortest path problem (SPP). This paper introduces the SPP from a source node to a destination node on a neutrosophic graph in which a positive neutrosophic number is assigned to each edge as its edge cost. We define this problem as neutrosophic shortest path problem (NSSPP). A simple algorithm is also introduced to solve the NSSPP. The proposed algorithm finds the neutrosophic shortest path (NSSP) and its corresponding neutrosophic shortest path length (NSSPL) between source node and destination node. Our proposed algorithm is also capable to find crisp shortest path length (CrSPL) of the corresponding neutrosophic shortest path length (NSSPL) which helps the decision maker to choose the shortest path easily. We also compare our proposed algorithm with some existing methods to show efficiency of our proposed algorithm. Finally, some numerical experiments are given to show the effectiveness and robustness of the new model. Numerical and graphical results demonstrate that the novel methods are superior to the existing method.