Accessible single-valued neutrosophic graphs

This paper derived single-valued neutrosophic graphs from single-valued neutrosophic hypergraphs via strong equivalence relation. We show that any weak single-valued neutrosophic graph is a derived single-valued neutrosophic graph and any linear weak single-valued neutrosophic tree is an extendable linear single-valued neutrosophic tree.

trosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic set and neutrosophic logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth (T ), a degree of indeterminacy (I ) and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ] − 0, 1 + [.
Most of the problems in engineering, medical science, economics, environments etc. have various uncertainties. In 1995, Smarandache talked for the first time about neutrosophy and in 1999 and 2005 [11,14] he initiated the theory of neutrosophic set as a new mathematical tool for handling problems involving imprecise, indeterminacy, and inconsistent data. Alkhazaleh et al. generalized the concept of fuzzy soft set to neutrosophic soft set and they gave some applications of this concept in decision making and medical diagnosis [4].
A graph is a convenient way of representing information involving relationship between objects. The objects are represented by vertices and the relations by edges. When there is vagueness in the description of the objects or in their relationships or in both, normally that we need to design a fuzzy graph model. The extension of fuzzy graph theory [12,16] have been developed by several researchers including intuitionistic fuzzy graphs [1,13] considered the vertex sets and edge sets as intuitionistic fuzzy sets.
Smarandache [15] have defined four main categories of neutrosophic graphs, two of which are based on literal indeterminacy (I ), which are called; I -edge neutrosophic graph and I -vertex neutrosophic graph, these concepts are studied deeply and have gained popularity among the researchers due to their applications via real world problems [7,17]. The other two graphs are based on (t, i, f ) components and are called; The (t, i, f )-Edge neutrosophic graph and the (t, i, f )-vertex neutrosophic graph, these concepts are not developed at all. Later on, Broumi et al. [5] introduced a third neutrosophic graph model. This model allowed the attachment of truth-membership (t), indeterminacy-membership (i) and falsity-membership degrees ( f ) both to vertices and edges, and investigated some of their properties.
Fuzzy hypergraph was introduced by the Kaufmann [10]. Lee-kwang et al. generalized the concept of fuzzy hypergraph and redefined it to be useful for fuzzy partition of a system. Akram and Dudek [2] investigated some properties of intuitionistic fuzzy hypergraph and gave applications of intuitionistic fuzzy hypergraph.
Regarding these points, the aim of this paper is to generalize the notion of singlevalued neutrosophic graphs by considering the notion of strong equivalence relation and to define the concept of extendable single-valued neutrosophic graphs. It is a normal question about the relationships between extendable single-valued neutrosophic graphs and extendable single-valued neutrosophic hypergraphs. From here comes the main motivation for this and in this regard, we have considered the quotient of single-valued neutrosophic hypergraphs via equivalence relations. Also, we want to establish the relationship between (α, β, γ )-level single-valued neutrosophic graphs and (α, β, γ )-level single-valued neutrosophic hypergraphs. Moreover, by using strong equivalence relation, we have defined a well-defined operation on singlevalued neutrosophic hypergraphs that the quotient of any single-valued neutrosophic hypergraphs via this relation is a single-valued neutrosophic graph.
We use single-valued neutrosophic hypergraphs to represent of the complex systems as networks, social, biological, ecological and technological systems where the use of complex networks gives very limited to information about the structure of the system. By introducing the concept of the complex hyper-network, the use of complex hypernetworks appears to be a necessary for exploring these systems and representation their relationships. We have introduced several valuable measures as truth-membership, indeterminacy and falsity-membership values for studying complex hyper-networks, such as node and hypergraph centralities as well as clustering coefficients for both the hyper-networks and the networks.

Preliminaries
In this section, we recall some definitions and results that are indispensable to our research paper.
The elements x 1 , x 2 , . . . , x n of G are called vertices, and the sets E 1 , E 2 , . . . , E m are the edges (hyperedges) of the hypergraph. For any 1 ≤ k ≤ m if |E k | ≥ 2, then E k is represented by a solid line surrounding its vertices, if |E k | = 1 by a cycle on the element (loop). If for all 1 ≤ k ≤ m|E k | = 2, the hypergraph becomes an ordinary (undirected) graph.
Then a binary relation ρ on G is defined as follows: for every integer n ≥ 1, ρ n is defined as follows: Obviously the relation ρ = n≥1 ρ n is an equivalence relation on G. We denote the set of all equivalence classes of ρ by G/ρ.
and |E n+1 | = n; (ii) H is called a discrete complete hypergraph, if for any respectively the truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element x ∈ X to the set A such that 1] denote the degree of membership, degree of indeterminacy and non-membership of the element v i ∈ V ; respectively, and for every 1 1] are called degree of the truth-membership, the indeterminacy-membership and the falsity-membership of the edge (v i , v j ) ∈ E respectively, such that for any Also A is called the single-valued neutrosophic vertex set of V and B is called the singlevalued neutrosophic edge set of E.
Definition 2.7 [3] (i) A single-valued neutrosophic hypergraph (SVN-HG) is defined to be a pair is called the family of single-valued neutrosophic hyperedges of H and V is the crisp vertex set of H .
) be a hypergraph, 1 ≤ i, j ≤ n and k ∈ N. Then H is called a partitioned hypergraph, if P = {E 1 , E 2 , . . . , E n } is a partition of G. We will denote the set of partitioned hypergraphs with |P| = k on G that |E i | = |E j |, by P Proof Since R is an equivalence relation on H , we get that (ii) Any finite set can be a (partitioned) single-valued neutrosophic hypergraph.
(iii) Any finite set can be a complete single-valued neutrosophic hypergraph.
Example 3.4 Let V = {a 1 , a 2 , . . . , a n }. Consider the complete graph K n and define . It is clear that (V, A, B) is a complete single-valued neutrosophic graph and supp(A) = V .

Corollary 3.5 Any finite set can be a complete weak single-valued neutrosophic graph.
Proof It is obtained from Corollary 3.2.

Lemma 3.6 Let X be a finite set and A
is a single-valued neutrosophic hypergraph. Since R is an equivalence relation on V , for any x = y ∈ V we get that R(x) ∩ R(y) = ∅ and so it is a partitioned single-valued neutrosophic hypergraph.

single-valued neutrosophic hypergraph. If R is an equivalence relation on H , then H R
Clearly H R is obtained in Fig. 2a and H/R is obtained in Fig. 2b.

single-valued neutrosophic hypergraph and R be an equivalence relation on H . Then
Hence and so we have the hypergraph in Fig. 4b. Fig. 4a.
In a similar way we obtain   and H = (V , {E x } x∈V ) be a single-valued neutrosophic hypergraph. We say that the single-valued neutrosophic graph G is derived from the single-valued neutrosophic hypergraph H if G is isomorphic to a nontrivial quotient of H.
If V = V we will say that it is an extended single-valued neutrosophic graph.
) be a single-valued neutrosophic hypergraph. Then there exists an operation " * on H/ρ such that (H/ρ, * ) is a single-valued neutrosophic graph.
For any ρ(x) = ρ((x, T E i (x), I E i (x), F E i (x))) and ρ(y) = ρ((y, T E i (y), I E i (y), F E i (y)) ∈ H/ρ, define an operation " * on H/ρ by where for any x, y ∈ G, (ρ(x), ρ(y)) is represented as an ordinary (simple) edge and ∅ = ρ(x) means that there is not edge. It is easy to see that , * is a single-valued neutrosophic graph.
where is obtained in Fig. 8.
) as follows: ) is a singlevalued neutrosophic hypergraph. Clearly for any 1 ≤ i ≤ n, ρ(a i ) = E i and since E i ∩ E i = ∅, we get that H/ρ = {ρ(a i ) | 1 ≤ i ≤ n} and so obtain If ρ(x) = ρ(y), then |E x | = |E y | and so E x = E y . Thus ϕ(ρ(x)) = ϕ(ρ(y)). Since for any 1 ≤ i, i ≤ n, in other words, if ρ(a i ) and ρ(a i ) in G/ρ are adjacent, then ϕ(ρ(a i )) and ϕ(ρ(a i )) in G are adjacent. So ϕ is a homomorphism. It is easy to see that ϕ is bijection and so is an isomorphism. It follows that by Theorem 4.2, any weak single-valued neutrosophic graph is a derived single-valued neutrosophic graph. By definition of single-valued neutrosophic hypergraph, the converse of theorem is obtained immediately.    ((a j )).
In a similar way can see that      a 2 , . . . , a n } and for any 1 Fig. 16.
Since for any 1 ≤ i ≤ n, E m i = {(a 1 , T A (a 1 ), I A (a 1 ), F A (a 1 )), . . . , (a i , T A (a i ), I A (a i ), F A (a i ))}, we get that for any 1 ≤ i ≤ n, ρ((a i , T A (a i ), I A (a i ), F A (a i ))) = {(a i , T A (a i ), I A (a i ), F A (a i ))} and so for any 1 ≤ i = j ≤ n, ρ(a i ) * ρ(a j ) = ρ(a i ), ρ(a j ). Since (K n , A, B) is a complete single-valued neutrosophic graph, for any 1 ≤ i, j ≤ n we get ((a j )).
In a similar way can see that ((a j )) and A, B) and since V = n, have (K n , A, B) is an extended complete single-valued neutrosophic graph. Then Fig. 18.

Corollary 4.13 Let (G, A, B) be a weak single-valued neutrosophic graph. (G, A, B) is an extended single-valued neutrosophic graph if and only if is a (G, A, B) be a complete weak single-valued neutrosophic graph.
Theorem 4.14 Let m, n ∈ N. Then complete weak single-valued neutrosophic bigraph (K m,n , A, B) is an extendable complete single-valued neutrosophic bigraph. a 2 , . . . , a m , a 1 , a 2 , . . . , a n } and (K m,n , V ∪ V , A, B) be a complete single-valued neutrosophic bigraph.
n j=1 E j = ∅ and for any 1 ≤ i ≤ m, 1 ≤ j ≤ n, E i ∩ E j = ∅. By definition for any 1 ≤ i ≤ m, 1 ≤ j ≤ n, E m i = E i , E m j = E j , ρ(a i ) = E i and ρ(a j ) = E j . Hence for any 1 ≤ i ≤ m and 1 ≤ j ≤ n we obtain It is easy to see that for any 1 ≤ i ≤ m, 1 ≤ j ≤ n we get that, . It follows that H/ρ is a complete single-valued neutrosophic bigraph.

An applications of accessible single-valued neutrosophic (hyper)graphs in complex networks
In this section, we describe some applications of accessible single-valued neutrosophic graphs.
The study of complex networks play a main role in the important area of multidisciplinary research involving physics, chemistry, biology, social sciences, and information sciences. These systems are commonly represented by means of simple or directed graphs that consist of sets of nodes representing the objects under investigation, e.g., people or groups of people, molecular entities, computers, etc., joined together in pairs by links if the corresponding nodes are related by some kind of relationship. These networks include the internet, the world wide web, social networks, information networks, neural networks, food webs, and protein-protein interaction networks. In some cases the use of simple or directed graphs to represent complex networks does not provide a complete description of the real-world systems under investigation. For instance, in a collaboration network represented as a simple graph, we only know whether scientists have collaborated or not, but we can not know whether three or more authors linked together in the network were coauthors of the same paper or not. A possible solution to this problem is to represent the collaboration network as a bipartite graph in which a disjoint set of nodes represents papers and another disjoint set represents authors. However, in this case the homogeneity in the definition of nodes is lost, because we have certain nodes that represent papers and others that represent authors. In the study of connectivity, clustering and other topological properties, this distinction between two classes of nodes with completely different interpretations may lead to artifacts in the data.
A natural way of representing these systems is to use the hypergraphs. In the hypergraphs, hyper-edges can relate groups of more than two nodes. Thus, we can represent the collaboration network as a hypergraph in which nodes represent authors and hyperedges represent the groups of authors that have published papers together. Despite the fact that complex weighted networks have been covered in some detail in the physical literature, there are no reports on the use of hypergraphs to represent complex systems. Consequently, we will formally introduce the hypergraph concept as a generalization for representing complex networks and will call them complex hyper-networks. The hypergraph concept includes, as particular cases, a wide variety of other mathematical structures that are appropriate for the study of complex networks. Since still these representations are unsuccessful to deal with all the competitions of the world, for that purpose SVN-HG are introduced. Now, we discuss applications of SVN-HG to study the competition along with algorithms. The SVN-G have many utilizations in different areas, where by using the especial equivalence relations, we connect SVN-G and SVN-HG. We will first show some examples of complex systems for which hypergraph representation is necessary.
Example 5.1 In social networks nodes represent people or groups of people, normally called actors, that are connected by pairs according to some pattern of contact or interactions between them. Such patterns can be of friendship, collaboration, business relationships, etc. There are some cases in which hypergraph representations of the social network are indispensable. Let X = {a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 } be a society and a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 be names of its people. These people create some groups as E 1 = {a 1 , a 2 , a 3 }, E 2 = {a 4 , a 3 } and E 1 = {a 4 , a 5 , a 6 , a 7 }. Let, the degree of contribution in the business relationships of a 1 is 10/100, degree of indeterminacy of contribution is 0/100 and degree of false-contribution is 15/100, i.e. the truth-membership, indeterminacy-membership and falsity-membership values of the vertex human is (0.1, 0, 0.15). The likeness, indeterminacy and dislikeness of contribution in the business relationships this society is shown in the Table 1.
Consider the social complex network H = ({a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 }, {E 1 , E 2 , E 3 }) in Fig. 23. Since   So we obtain the single-valued neutrosophic graph in Fig. 24. By Fig. 24, for society X , we have 3 representatives ρ(a 1 ), ρ(a 2 ) and ρ(a 3 ) where the likeness, indeterminacy and dislikeness of contribution in the business relationships of group of this society is shown in the Table 2. Example 5.2 Trophic relations in ecological systems are normally represented through the use of food webs, which are oriented graphs (digraphs) whose nodes represent species and links represent trophic relations between species. Another way of representing food webs is by means of competition graphs, which have the same set of nodes as the food web but in which two nodes are connected if, and only if, the corresponding species compete for the same prey in the food web. In the competition graph we can only know if two linked species have some common prey, but we can not know the composition of the whole group of species that compete for common prey. In order to solve this problem a competition hypergraph has been proposed in which nodes represent species in the food web and hyper-edges represent groups of species that compete for common prey. It has been shown that in many cases competition hyper-networks yield a more detailed description of the predation relations among the species in the food web than competition graphs. Let X = {a, a , b, b , c, d, d , e, f, g} be a ecological  system and a, a , b, b , c, d, d , e, f, g be species. These species create some groups species as E 1 = {a, a }, E 2 = {a, b , b}, E 3 = {b, c, d , d} and E 4 = {d, g, f, e, a }. Let, the degree of contribution in the business relationships of a is 60/100, degree of indeterminacy of contribution is 50/100 and degree of false-contribution is 40/100, i.e. the truth-membership, indeterminacy-membership and falsity-membership values of the vertex human is (0.6, 0.5, 0.4). The likeness, indeterminacy and dislikeness of contribution in the business relationships this society is shown in the Table 3. Consider the food competition hyper-network is illustrated in Fig. 25. Since So we obtain the single-valued neutrosophic graph in Fig. 26. By Fig. 26, for society X , we have 4 representatives ρ(a), ρ(b), ρ(c) and ρ(e) where the likeness, indeterminacy and dislikeness of trophic relations between species of group of this society is shown in the Table 4.

Conclusion
The current paper has considered the notion of single-valued neutrosophic hypergraph, single-valued neutrosophic graph and by introducing weak single-valued neutrosophic graph, we have established a relation between them. Also: (i) Any weak single-valued neutrosophic graph is a derived single-valued neutrosophic graph. (ii) Every linear weak single-valued neutrosophic tree (T l m , A, B) is an extendable linear single-valued neutrosophic tree. (iii) All complete weak single-valued neutrosophic graphs (K n , A, B) are extended complete single-valued neutrosophic graphs. (iv) Any complete weak single-valued neutrosophic bigraph (K m,n , A, B) is an extendable complete single-valued neutrosophic bigraph. (v) The concept of intuitionistic neutrosophic sets provides an additional possibility to represent imprecise, uncertainty, inconsistent and incomplete information which exists in real situations. In this research paper, we have described the concept of single-valued neutrosophic graphs. We have also presented applications of single-valued neutrosophic hypergraphs and single-valued neutrosophic graphs in food webs and social networks.
We hope that these results are helpful for further studies in graph theory. In our future studies, we hope to obtain more results regarding graphs, hypergraphs and their applications.