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Neutrosophic soft graphs.

1 Introduction

Graph theory is a nice tool to depict information in a very nice way. Usually graphs are represented pictorially, algebraically in the form of relations or by matrices. Their representation depends on application for which a graph is being employed. Graph theory has its origins in a 1736 paper by the celebrated mathematician Leonhard Euler [13] known as the father of graph theory, when he settled a famous unsolved problem known as Ko'nigsburg Bridge problem. Subject of graph theory may be considered a part of combinatorial mathematics. The theory has greatly contributed to our understanding of programming, communication theory, switching circuits, architecture, operational research, civil engineering anthropology, economics linguistic and psychology. From the standpoint of applications it is safe to say that graph theory has become the most important part of combinatorial mathematics. A graph is also used to create a relationship between a given set of elements. Each element can be represented by a vertex and the relationship between them can be represented by an edge.

L.A. Zadeh [26] introduced the notion of fuzzy subset of a set in 1965 which is an extension of classical set theory. His work proved to be a mathematical tool for explaining the concept of uncertainty in real life problems. A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. In 1975 Azriel Rosenfeld [20] considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs which have many applications in modeling, Environmental science, Social science, Geography and Linguistics etc. which deals with problems in these areas that can be better studied using the concept of fuzzy graph structures. Many researchers contributed a lot and gave some more generalized forms of fuzzy graphs which have been studied in [8] and [10]. These contributions show a new dimension of graph theory.

Molodstov introduced the theory of soft sets [18] which is generally used to deal with uncertainty and vagueness. He introduced the concept as a mathematical tool free from difficulties and presented the fundamental results of the new theory and successfully applied it to several directions. During recent past soft set theory has gained popularity among researchers, scholars practitioners and academicians. The theory of neutrosophic set is introduced by Smarandache [21] which is useful for dealing real life problems having imprecise, indeterminacy and inconsistent data. The theory is generalization of classical sets and fuzzy sets and is applied in decision making problems, control theory, medicines, topology and in many more real life problems. Maji [17] first time proposed the definition of neutrosophic soft sets and discussed many operations such as union, intersection and complement etc of such sets. Some new theories and ideas about neutrosophic sets can be studied in [6], [7] and [12]. In the present paper neutrosophic soft sets are employed to study graphs and give rise to a new class of graphs called neutrosophic soft graphs. We have discussed different operations defined on neutrosophic soft graphs using examples to make the concept easier. The concept of strong neutrosophic soft graphs and the complement of strong neutrosophic soft graphs is also discussed. Neutrosophic soft graphs are pictorial representation in which each vertex and each edge is an element of neutrosophic soft sets. This paper has been arranged as the following;

In section 2, some basic concepts about graphs and neutrosophic soft sets are presented which will be employed in later sections. In section 3, concept of neutrosophic soft graphs is given and some of their fundamental properties have been studied. In section 4, the concept of strong neutrosophic soft graphs and its complement is studied. Conclusion are also given at the end of section 4.

2 PRELIMINARIES

In this section, we have given some definitions about graphs and neutrosophic soft sets. These will be helpful in later sections.

2.1 Definition [25]: A graph [G.sup.*] consists of set of finite objects V = {[v.sub.1], [v.sub.2], [v.sub.3], ...... [v.sub.n]} called vertices (also called points or nodes) and other set E = {[e.sub.1], [e.sub.2], [e.sub.3], ...... [e.sub.n]} whose elements are called edges (also called lines or arcs). Usually a graph is denoted as [G.sup.*] = (V, E). Let [G.sup.*] be a graph and {u, v} an edge of [G.sup.*]. Since {u, v} is 2-element set, we may write {v, u} instead of {u, v}. It is often more convenient to represent this edge by uv or vu. If e = uv is an edges of a graph [G.sup.*], then we say that u and v are adjacent in [G.sup.*] and that e joins u and v. A vertex which is not adjacent to any other node is called isolated vertex.

2.2 Definition [25]: An edge of a graph that joins a node to itself is called loop or self loop.

2.3 Definition [25]: In a multigraph no loops are allowed but more than one edge can join two vertices, these edges are called multiple edges or parallel edges and a graph is called multigraph.

2.4 Definition [25]: A graph which has neither loops nor multiple edges is called a simple graph.

2.5 Definition [25]: A sub graph [H.sup.*] of [G.sup.*] is a graph having all of its vertices and edges in [G.sup.*]. If [H.sup.*] is a sub graph of [G.sup.*], then [G.sup.*] is a super graph of [H.sup.*].

2.6 Definition [25]: Let [G.sup.*.sub.1] = ([V.sub.1], [E.sub.1]) and [G.sup.*.sub.2] = ([V.sub.2], [E.sub.2]) be two graphs. A function f: [V.sub.1] [right arrow] [V.sub.2] is called isomorphism if

i) f is one to one and onto.

ii) for all a, b [member of] [V.sub.1], {a, b} [member of] [E.sub.1] if and only if {f (a), f (b)} [member of] [E.sub.2] when such a function exists, [G.sup.*.sub.1] and [G.sup.*.sub.2] are called isomorphic graphs and is written as

[G.sup.*.sub.1] [congruent to] [G.sup.*.sub.2].

In other words, two graph [G.sup.*.sub.1] and [G.sup.*.sub.2] are said to be isomorphic to each other if there is a one to one correspondence between their vertices and between edges such that incidence relationship is preserved.

2.7 Definition [25]: The union of two simple graphs [G.sup.*.sub.1] = ([V.sub.1], [E.sub.1]) and [G.sup.*.sub.2] = ([V.sub.2], [E.sub.2]) is the simple graph with the vertex set [V.sub.1] [union] [V.sub.2] and edge set [E.sub.1] [union] [E.sub.2]. The union of [G.sup.*.sub.1] and [G.sup.*.sub.2] is denoted by [G.sup.*] = [G.sup.*.sub.1] [union] [G.sup.*.sub.2] = ([V.sub.1] [union] [V.sub.2], [E.sub.1] [union] [E.sub.2]).

2.8 Definition [25]: The join of two simple graphs [G.sup.*.sub.1] = ([V.sub.1], [E.sub.1]) and [G.sup.*.sub.2] = ([V.sub.2], [E.sub.2]) is the simple graph with the vertex set [V.sub.1] [union] [V.sub.2] and edge set [E.sub.1] [union] [E.sub.2] [union] E' where E is the set of all edges joining the nodes of [V.sub.1] and [V.sub.2] assume that [V.sub.1] [intersection] [V.sub.2] [not equal to] [phi]. The join of [G.sup.*.sub.1] and [G.sup.*.sub.2] is denoted by [G.sup.*] = [G.sup.*.sub.1] + [G.sup.*.sub.2] = ([V.sub.1] [union] [V.sub.2], [E.sub.1] [union] [E.sub.2] [union] E').

2.9 Definition [18]: Let U be an initial universe and E be the set of all possible parameters under consideration with respect to U. The power set of U is denoted by P (U) and A is a subset of E. Usually parameters are attributes, characteristics, or properties of objects in U.

A pair (F, A) is called a soft set over U, where F is a mapping F: A [right arrow] P (U). In other words, a soft set over U is a parameterized family of subsets of the universe U. For e [member of] A,F (e) may be considered as the set of [e.sup.-] approximate elements of the soft set (F, A).

2.10 Definition [21]: A neutrosophic set A on the universe of discourse X is defined as A = {<x, [T.sub.A](x), [I.sub.A](x), [F.sub.A](x)>, x [member of] X}, where

T, I, F: X [right arrow]] [bar.0], [1.sup.+][and [bar.0] [less than or equal to] [T.sub.A](x) + [I.sub.A](x) + [F.sub.A](x) [less than or equal to] [3.sup.+].

From philosophical point of view, the neutrosophic set takes the value from real standard or non-standard subsets of ][bar.0], [1.sub.+][. But in real life application in scientific and engineering problems it is difficult to use neutrosophic set with value from real standard or non-standard subset of ][bar.0], [1.sup.+][. Hence we consider the neutrosophic set which takes the value from the subset of [0,1].

2.11 Definition [17]: Let N(U) be the set of all neutrosophic sets on universal set U, E be the set of parameters that describes the elements of U and A [subset or equal to] E. A pair (F, A) is called a neutrosophic soft set NSS over U, where F is a mapping given by F: A [right arrow] N (U). A neutrosophic soft set is a mapping from parameters to N (U). It is a parameterized family of neutrosophic subsets of U. For e [member of] A, F (e) may be considered as the set of e-approximate elements of the neutrosophic soft set (F, A). The neutrosophic soft set (F, A) is parameterized family {F([e.sub.i]), i = 1,2,3, e [member of] A}.

2.12 Definition [17]: Let [E.sub.1], [E.sub.2] [member of] E and (F, [E.sub.1]), (G, [E.sub.2]) be two neutrosophic soft sets over U then (F, [E.sub.1]) is said to be a neutrosophic soft subset of (G, [E.sub.2]) if

(1) [E.sub.1] [subset or equal to] [E.sub.2]

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all e [member of] [E.sub.1], x [member of] U.

In this case, we write (F, [E.sub.1]) [subset or equal to] (G, [E.sub.2]).

2.13 Definition [17]: Two neutrosophic soft sets (F, [E.sub.1]) and (G, [E.sub.2]) are said to be neutrosophic soft equal if (F, [E.sub.1]) is a neutrosophic soft subset of (G, [E.sub.2]) and (G, [E.sub.2]) is a neutrosophic soft subset of In this case, we write (F, [E.sub.1]) = (G, [E.sub.2]).

2.14 Definition [14]: Let U be an initial universe, E be the set of parameters, and A [subset or equal to] E.

(a) (H, A) is called a relative whole neutrosophic soft set (with respect to the parameter set A), denoted by [[phi].sub.A], if [T.sub.H(e)](x) = 1, [I.sub.H(e)](x) = 1, [F.sub.H(e)](x) = 0, for all e [member of] A x [member of] U.

(b) (G, A) is called a relative null neutrosophic soft set (with respect to the parameter set A), denoted by [[phi].sub.A], if [T.sub.H(e)](x) = 0, [I.sub.H(e)](x)= 0, [F.sub.H(e)](x) = 1, for all e [member of] A, x [member of] U.

The relative whole neutrosophic soft set with respect to the set of parameters E is called the absolute neutrosophic soft set over U and simply denoted by [U.sub.E]. In a similar way, the relative null neutrosophic soft set with respect to E is called the null neutrosophic soft set over U and is denoted by [[phi].sub.E].

2.15 Definition [17]: The complement of a NSS (G, A) is denoted by [(G, A).sup.c] and is defined by [(G, A).sup.c] = ([G.sup.c], [logical not]A) where [G.sup.c]: [logical not]A [right arrow] N(U) is a mapping given by [G.sup.c] ([logical not]e) = neutrosophic soft complement with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2.16 Definition [14](1): Extended union of two NSS (H, A) and (G, B) over the common universe U is denoted by (H, A) [[union].sub.E] (G, B) and is define as (H,A) [[union].sub.E] (G,B) = (K,C), where C = A [union] B and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2.17 Definition [14]: The restricted union of two NSS (H,A) and (G,B) over the common universe U is denoted by (H, A) [[union].sub.R] (G,B) and is define as (H, A) [[union].sub.R] (G, B) = (K,C), where C = A [intersection] B and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2.18 Definition [14]: Extended intersection of two NSS (H,A) and (G, B) over the common universe U is denoted by (H, A) [[intersection].sub.E] (G, B) and is define as (H,A) [[intersection].sub.E] (G,B) = (K,C), where C = A [union] B and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2.19 Definition [14]: The restricted intersection of two NSS (H, A) and (G, B) over the common universe U is denoted by (H, A) [[intersection].sub.R] (G, B) and is define as (H,A) [[intersection].sub.R] (G,B) = (K,C), where C = A [intersection] B and the truth-membership, indeterminacy-membership and falsity-membership of (K, C) are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3 Neutrosophic soft graphs

3.1 Definition Let [G.sup.*] = (V, E) be a simple graph and A be the set of parameters. Let N (V) be the set of all neutrosophic sets in V. By a neutrosophic soft graph NSG, we mean a 4-tuple G = ([G.sup.*], A, f, g) where f: A [right arrow] N (V), g: A [right arrow] N (V x V) defined as f (e) = [f.sub.e] = {<x, [T.sub.fe](x), [I.sub.fe](x) [I.sub.fe](x)>, x [member of] V} and g (e) = [g.sub.e] = {<(x, y), [T.sub.fe](x, y), [I.sub.fe](x,y), [F.sub.fe](x,y)), (x, y)>, (x, y) [member of] V x V} are neutrosophic sets over V and V x V respectively, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for all (x,y) [member of] V x V and e [member of] A. We can also denote a NSG by G = ([G.sup.*], A, f, g) = {N (e): e [member of] A} which is a parameterized family of graphs N(e) we call them Neutrosophic graphs.

3.2 Example

Let [G.sup.*] =(V, E) be a simple graph with V = {[x.sub.1], [x.sub.2], [x.sub.3]}, A = {[e.sub.1], [e.sub.2], [e.sub.3]} be a set of parameters. A NSG is given in Table 1 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge]([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1, for all ([x.sub.i], [x.sub.j]) [member of] V x V\{([x.sub.1], [x.sub.2]), ([x.sub.2], [x.sub.3]), ([x.sub.3], [x.sub.1])} and for all e [member of] A.

N ([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 1 OMITTED]

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 2 OMITTED]

N ([e.sub.3]) Corresponding to [e.sub.3]

[FIGURE 3 OMITTED]

3.3 Definition A neutrosophic soft graph

G = ([G.sup.*], [A.sup.1], [f.sup.1], [g.sup.1]) is called a neutrosophic soft subgraph of G = ([G.sup.*], A, f, g) if

(i) [A.sup.1] [subset or equal to] A

(ii) [f.sup.1.sub.e] [subset or equal to] f, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iii) [g.sup.1.sub.e] [subset or equal to] g, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for all e [member of] [A.sup.1].

3.4 Example

Let [G.sup.*] = (V, E) be a simple graph with V = {[x.sub.1], [x.sub.2], [x.sub.3]} and set of parameters A = {[e.sub.1], [e.sub.2]}. A neutrosophic soft subgraph of example 3.2 is given in Table 2 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge] ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1, for all ([x.sub.i], [x.sub.j]) [member of] V x V\{([x.sub.1], [x.sub.2]), ([x.sub.2], [x.sub.3]), ([x.sub.3], [x.sub.1])} and for all e [member of] A.

N([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 4 OMITTED]

N([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 5 OMITTED]

3.5 Definition A neutrosophic soft subgraph G = ([G.sup.*], [A.sup.1], [f.sup.1], [g.sup.1]) is said to be spanning neutrosophic soft subgraph of G = ([G.sup.*], A, f, g) if [f.sup.1.sub.e](x) = f(x), for all x [member of] V, e [member of] [A.sup.1].

(Here two neutrosophic soft graphs have the same neutrosophic soft vertex set, But have opposite edge sets.

3.6 Definition The union of two neutrosophic soft graphs [G.sub.1] = ([G.sup.*.sub.1], [A.sub.1], [f.sup.1], [g.sup.1]) and [G.sub.2] = ([G.sup.*.sub.2], [A.sub.2], [f.sup.2], [g.sup.2]) is denoted by G = ([G.sup.*], A, f, g), with A = [A.sub.1] [union] [A.sub.2] where the truth membership, indeterminacy-membership and falsity-membership of union are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3.7 Example

Let [G.sup.*.sub.1] = ([V.sub.1], [E.sub.1]) be a simple graph with [V.sub.1], = {[x.sub.1], [x.sub.2], [x.sub.3]) and set of parameters [A.sub.1] = {[e.sub.1], [e.sub.2], [e.sub.3]). Let [G.sup.*.sub.2] = ([V.sub.2], [E.sub.2]) be a simple graph with [V.sub.2] = {[x.sub.2], [x.sub.3], [x.sub.5]) and set of parameters [A.sub.2] = {[e.sub.2], [e.sub.4]). A NSG [G.sub.1] = ([G.sup.*.sub.1], [A.sub.1], [f.sup.1], [g.sup.1]) is given in Table 3 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge], ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1 for all ([x.sub.i], [x.sub.j]) [member of] [V.sub.1] x [V.sub.1]\{([x.sub.1], [x.sub.4]), ([x.sub.3], [x.sub.4]), ([x.sub.1], [x.sub.3])) and for all

N ([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 6 OMITTED]

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 7 OMITTED]

N ([e.sub.3]) Corresponding to [e.sub.3]

[FIGURE 8 OMITTED]

A NSG [G.sub.2] = ([G.sup.*.sub.2], [A.sub.2], [f.sup.2], [g.sup.2]) is given in Table 4 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge] ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1 for all ([x.sub.i], [x.sub.j]) [member of] [V.sub.2] x [V.sub.2]\ {([x.sub.2], [x.sub.3]), ([x.sub.3], [x.sub.5]), ([x.sub.2], [x.sub.5])} and for all e [member of] [A.sub.2].

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 9 OMITTED]

N ([e.sub.4]) Corresponding to [e.sub.4]

[FIGURE 10 OMITTED]

The union G = ([G.sup.*], A, f, g) is given in Table 5 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge]([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1, for all ([x.sub.i], [x.sub.j]) [member of] V x V\{([x.sub.1], [x.sub.4]), ([x.sub.3], [x.sub.4]), ([x.sub.1], [x.sub.3]), ([x.sub.2], [x.sub.3]), ([x.sub.3], [x.sub.5]), ([x.sub.2], [x.sub.5])} and for all e [member of] A.

N([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 11 OMITTED]

N([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 12 OMITTED]

N([e.sub.3]) Corresponding to [e.sub.3]

[FIGURE 13 OMITTED]

N([e.sub.4]) Corresponding to [e.sub.4]

[FIGURE 14 OMITTED]

3.8 Proposition

The union [G.sup.*] = {V, A, f, g) of two neutrosophic soft graph [G.sub.1] = ([G.sup.*], [A.sub.1], [f.sup.1], [g.sup.1]) and [G.sub.2] = ([G.sup.*], [A.sub.2], [f.sup.2], [g.sup.2]) is a neutrosophic soft graph.

Proof

Case i) If e [member of] [A.sub.1]-[A.sub.2] and (x, y) [member of] ([V.sub.1] x [V.sub.1]) - ([V.sub.2] x [V.sub.2]), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so [T.sub.ge](x, y) [less than or equal to] min{[T.sub.fe](x), [T.sub.fe](y)}

Also [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so [I.sub.ge](x,y) [less than or equal to] min{[I.sub.fe](x), [I.sub.fe](y)}

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly If {e [member of] [A.sub.1] - [A.sub.2] and (x, y) [member of] ([V.sub.1] x [V.sub.1]) [intersection] ([V.sub.2] x [V.sub.2])}, or If {e [member of] [A.sub.1] [intersection] [A.sub.2] and (x, y) [member of] ([V.sub.1] x [V.sub.1]) - ([V.sub.2] x [V.sub.2])}, we can show the same as done above.

Case ii) If e [member of] [A.sub.1] [intersection] [A.sub.2] and (x, y) [member of] ([V.sub.1] x [V.sub.1]) [intersection] ([V.sub.2] x [V.sub.2]), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the union G = [G.sub.1] [union] [G.sub.2] is aneutrosophic soft graph.

3.9 Definition The intersection of two neutrosophic soft graphs [G.sub.1] = ([G.sup.*.sub.1], [A.sub.1], [f.sup.1], [g.sup.1]) and [G.sub.2] = ([G.sup.*.sub.2], [A.sub.2], [f.sup.2], [g.sup.2]) is denoted by G = ([G.sup.*], A, f, g) where A = [A.sub.1] [intersection] [A.sub.2], V = [V.sub.1] [intersection] [V.sub.2] and the truthmembership, indeterminacy-membership and falsity-membership of intersection are as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 3.10

3.10 Example

Let [G.sup.*.sub.1] = ([V.sub.1], [E.sub.1]) be a simple graph with with [V.sub.1] = {[x.sub.1], [x.sub.2], [x.sub.3]} md set of parameters [A.sub.1] = {[e.sub.1], [e.sub.2]}. A NSG [G.sub.1] = ([V.sub.1], [A.sub.1], [f.sup.1], [g.sup.1]) s given in Table 6 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge] ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1, for all ([x.sub.i], [x.sub.j]) [member of] [V.sub.1] x [V.sub.1]\{([x.sub.1], [x.sub.5]), ([x.sub.1], [x.sub.2]), ([x.sub.2], [x.sub.5])} and for all e [member of] [A.sub.1].

N ([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 15 OMITTED]

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 16 OMITTED]

Let [G.sup.*.sub.2] = ([V.sub.2], [E.sub.2]) be a simple graph with [V.sub.2] = {[x.sub.1], [x.sub.2], [x.sub.3]} and set of parameters [A.sub.2] = {[e.sub.2], [e.sub.3]} [A.sub.2] = ([e.sub.2], [e.sub.3]}. A NSG [G.sub.2] = ([V.sub.2], [A.sub.2], [f.sup.2], [g.sup.2]) is given in Table 7 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge] ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1, for all ([x.sub.i], [x.sub.j]) [member of] [V.sub.2] x [V.sub.2]{([x.sub.2], [x.sub.3]), ([x.sub.3], [x.sub.5]), ([x.sub.2], [[x.sub.5])} and for all e [member of] [A.sub.2].

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 17 OMITTED]

N ([e.sub.3]) Corresponding to [e.sub.3]

[FIGURE 18 OMITTED]

Let V = [V.sub.1] [intersection] [V.sub.2] = {[x.sub.2], [x.sub.5]}, A = [A.sub.1] [union] [A.sub.2] = {[e.sub.1], [e.sub.2], [e.sub.3]}

The intersection of two neutrosophic soft graphs [G.sub.1] = ([G.sup.*.sub.1], [A.sub.1], [f.sup.1], [g.sup.1]) and [G.sub.2] = ([G.sup.*.sub.2], [A.sub.2], [f.sup.2], [g.sup.2]) is given in Table 8.

N ([e.sub.1]) corresponding to [e.sub.1]

[FIGURE 19 OMITTED]

N ([e.sub.2]) corresponding to [e.sub.2]

[FIGURE 20 OMITTED]

N ([e.sub.3]) corresponding to [e.sub.3]

[FIGURE 21 OMITTED]

3.11 Proposition

The intersection G = ([G.sup.*], A, f, g) of two neutrosophic soft graphs [G.sub.1] = ([G.sup.*], [A.sub.1], [f.sup.1], [g.sup.1]) and [G.sub.2] = ([G.sup.*], [A.sub.2], [f.sup.2], [g.sup.2]) is a neutrosophic soft graph where, A = [A.sub.1] [union] [A.sub.2] and V = [V.sub.1] [intersection] [V.sub.2].

Proof

Case i) If e [member of] [A.sub.1] - [A.sub.2] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so [T.sub.ge](x, y) [less than or equal to] min{[T.sub.fe](x), [T.sub.fe](y)}

Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so [I.sub.ge](x, y) [less than or equal to] min{[I.sub.fe](x), [I.sub.fe](y)}

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly If e [member of] [A.sub.2] - [A.sub.1] we can show the same as done above.

Case ii) If e [member of] [A.sub.1] [intersection] [A.sub.2] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence the intersection G = [G.sub.1] [intersection] [G.sub.2] is a neutrosophic soft graph.

4 Strong Neutrosophic Soft Graph

4.1 Definition A neutrosophic soft graph G = ([G.sup.*], A, f, g), is called strong if [g.sub.e](x, y) = [f.sub.e](x) [intersection] [f.sub.e](y), for all x, y [member of] V, e [member of] A. That is if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

for all (x, y) [member of] E.

4.2 Example

Let V = {[x.sub.1], [x.sub.2], [x.sub.3]}, A = {[e.sub.1], [e.sub.2]}. A strong NSG G = ([G.sup.*], A, f, g) is given in Table 9 below and [T.sub.ge] ([x.sub.i], [x.sub.j]) = 0, [I.sub.ge] ([x.sub.i], [x.sub.j]) = 0 and [F.sub.ge] ([x.sub.i], [x.sub.j]) = 1 for all ([x.sub.i], [x.sub.j]) [member of] V x V\{([x.sub.1], [x.sub.2]), ([x.sub.2], [x.sub.3]), ([x.sub.1], [x.sub.3])} and for all e [member of] A.

N ([e.sub.1]) Corresponding to [e.sub.1]

[FIGURE 22 OMITTED]

N ([e.sub.2]) Corresponding to [e.sub.2]

[FIGURE 23 OMITTED]

4.3 Definition Let G = ([G.sup.*], A, f, g) be a strong neutrosophic soft graph that is [g.sub.e](x, y) = [f.sub.e](x) [intersection] [f.sub.e](y), for all for all x, y [member of] V, e [member of] A. The complement [bar.G] = ([[bar.G].sup.*], [bar.A], [bar.f], [bar.g]) of strong neutrosophic soft graph G = ([G.sup.*], A, f, g) is neutrosophic soft graph where

(i) [bar.A] = A

(ii) [T.sub.fe](x) = [bar.[T.sub.fe]](x), [bar.[I.sub.fe]](x) = [bar.[I.sub.fe]](x), [F.sub.fe](x) = [[bar.F].sub.fe](x) for all x [member of] V

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4.4 Example

For the strong neutrosophic soft graph in previous example, the complements are given below for [e.sub.1] and [e.sub.2]. Corresponding to [e.sub.1], the complement of

[FIGURE 24 OMITTED]

is given by

[FIGURE 25 OMITTED]

Corresponding to [e.sub.2], the complement of

[FIGURE 26 OMITTED]

is given by

[FIGURE 27 OMITTED]

Conclusion: Neutrosophic soft set theory is an approach to deal with uncertainty having enough parameters so that it is free from those difficulties which are associated with other contemporary theories dealing with study of uncertainty. A graph is a convenient way of representing information involving relationship between objects. In this paper we have combined both the theories and introduced and discussed neutrosophic soft graphs which are representatives of neutrosophic soft sets.

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Received: Feb. 20, 2016. Accepted: Mar. 30, 2016

Nasir Shah (1) and Asim Hussain (2)

(1) Department of Mathematics, Riphah International University, I-14, Islamabad, Pakistan. Email: memaths@yahoo.com

(2) Department of Mathematics, Barani Institute of Management Sciences (BIMS), Rawalpindi, Pakistan. Email: sh.asim.hussain@gmail.com
Table 1

f                 [x.sub.1]                [x.sub.2]

[e.sub.1]       (0.4,0.5,0.6)            (0.4,0.5,0.7)
[e.sub.2]       (0.3,0.4,0.5)            (0.1,0.3,0.4)
[e.sub.3]       (0.2,0.3,0.5)            (0.1,0.2,0.4)
g           ([x.sub.1], [x.sub.2])   ([x.sub.2], [x.sub.3])
[e.sub.1]       (0.2,0.3,0.8)               (0,0,1)
[e.sub.2]       (0.1,0.3,0.6)               (0,0,1)
[e.sub.3]       (0.1,0.1,0.9)            (0.1,0.2,0.7)

f                 [x.sub.3]

[e.sub.1]          (0,0,1)
[e.sub.2]       (0.1,0.3,0.6)
[e.sub.3]       (0.1,0.5,0.7)
g           ([x.sub.1], [x.sub.3])
[e.sub.1]          (0,0,1)
[e.sub.2]       (0.1,0.3,0.8)
[e.sub.3]       (0.1,0.3,0.8)

Table 2.

[f.sup.1]   [x.sub.1]       [x.sub.2]       [x.sub.3]

[e.sub.1]   (0.3,0.2,0.5)   (0.3,0.2,0.6)   (0,0,1)
[e.sub.2]   (0.1,0.1,0.5)   (0.1,0.2,0.4)   (0.1,0.2,0.6)

[g.sup.1]   ([x.sub.1],     ([x.sub.2],     ([x.sub.1],
            [x.sub.2])      [x.sub.3])      [x.sub.3])

[e.sub.1]   (0.2,0.2,0.7)   (0,0,1)         (0,0,1)
[e.sub.2]   (0.1,0.1,0.6)   (0,0,1)         (0.1,0.2,0.8)

Table 3

[f.sup.1]   [x.sub.1]       [x.sub.3]       [x.sub.4]

[e.sub.1]   (0.1,0.2,0.3)   (0.2,0.3,0.4)   (0.2,0.5,0.7)
[e.sub.2]   (0.1,0.3,0.7)   (0.4,0.6,0.7)   (0.1,0.2,0.3)
[e.sub.3]   (0.5,0.6,0.7)   (0.6,0.8,0.9)   (0.3,0.4,0.6)

[g.sup.1]   ([x.sub.1],     ([x.sub.3],     ([x.sub.1],
            [x.sub.4])      [x.sub.4])      [x.sub.3])

[e.sub.1]   (0.1,0.2,0.7)   (0.1,0.3,0.8)   (0.1,0.2,0.5)
[e.sub.2]   (0.1,0.2,0.7)   (0.1,0.1,0.9)   (0.1,0.2,0.8)
[e.sub.3]   (0.1,0.3,0.8)   (0.2,0.3,0.9)   (0,0,1)

Table 4

[f.sup.2]     [x.sup.2]       [x.sup.3]       [x.sup.5]

[e.sub.1]   (0.1,0.2,0.4)   (0.2,0.3,0.4)   (0.4,0.6,0.7)
[e.sub.2]   (0.3,0.6,0.8)   (0.5,0.7,0.9)   (0.3,0.4,0.5)

[g.sup.2]    ([x.sub.2],     ([x.sub.3],     ([x.sub.2],
             [x.sub.3])       [x.sub.5)       [x.sub.5)

[e.sub.1]   (0.1,0.2,0.8)   (0.2,0.3,0.9)      (0,0,1)
[e.sub.2]   (0.1,0.1,0.9)   (0.2,0.2,0.9)   (0.2,0.3,0.8)

Table 5

f             [x.sub.1]        [x.sub.2]         [x.sub.3]

[e.sub.1]   (0.1,0.2,0.3)       (0,0,1)        (0.2,0.5,0.7)
[e.sub.2]   (0.1,0.3,0.7)    (0.1,0.2,0.3)    (0.2, 0.4, 0.4)
[e.sub.3]   (0.5,0.6,0.7)      (0, 0, 1)       (0.6,0.8,0.9)
[e.sub.5]      (0,0,1)      (0.3, 0.6, 0.8)   (0.5, 0.7, 0.9)

g            ([x.sub.1],      ([x.sub.3],       ([x.sub.1],
             [x.sub.4])       [x.sub.4])        [x.sub.3])

[e.sub.1]   (0.1,0.2,0.7)    (0.1,0.3,0.8)     (0.1,0.2,0.8)
[e.sub.2]   (0.1,0.2,0.7)    (0.1,0.1,0.9)     (0.1,0.2,0.8)
[e.sub.3]   (0.1,0.3,0.8)    (0.2,0.3,0.9)        (0,0,f)
[e.sub.4]      (0,0,1)          (0,0,1)           (0,0,f)

f             [x.sub.4]        [x.sub.5]

[e.sub.1]   (0.2,0.3,0.4)       (0,0,1)
[e.sub.2]   (0,.1,0,2,0,3)   (0.4,0.6,0.7)
[e.sub.3]   (0.3,0.4,0.6)       (0,0,1)
[e.sub.5]      (0,0,1)       (0.3,0.4,0.5)

g            ([x.sub.2],      ([x.sub.3],     ([x.sub.2],
              [x.sub.3])      [x.sub.5])      [x.sub.5])

[e.sub.1]      (0,0,1)          (0,0,1)         (0,0,1)
[e.sub.2]   (0,.1,0,2,0,8)   (0.2,0.3,0.9)      (0,0,1)
[e.sub.3]      (0,0,1)          (0,0,1)         (0,0,1)
[e.sub.4]   (0.1,0.1,0.9)    (0.2,0.2,0.9)   (0.2,0.3,0.8)

Table 6

[f.sub.1]         [x.sub.1]               [x.sub.2]

[e.sub.1]       (0.1,0.2,0.3)           (0.2,0.4,0.5)
[e.sub.2]       (0.2,0.3,0.7)           (0.4,0.6,0.7)
[g.sub.1]   ([x.sub.1],[x.sub.5])   ([x.sub.1],[x.sub.5])
[e.sub.1]       (0.1,0.1,0.8)           (0.1,0.3,0.8)
[e.sub.2]       (0.2,0.3,0.7)           (0.3,0.4,0.8)

[f.sub.1]         [x.sub.5]

[e.sub.1]       (0.,,0.5,0.7)
[e.sub.2]       (0.3,0.4,0.6)
[g.sub.1]   ([x.sub.1],[x.sub.2])
[e.sub.1]       (0.1,0.1,0.6)
[e.sub.2]       (0.2,0.3,0.7)

Table 7.

[f.sup.2]         [x.sub.2]               [x.sub.3]

[e.sub.2]       (0.3,0.5,0.6)           (0.2,0.4,0.6)
[e.sub.3]       (0.2,0.4,0.5)           (0.1,0.2,0.6)
[g.sub.2]   ([x.sub.2],[x.sub.3])   ([x.sub.3],[x.sub.5])
[e.sub.2]       (0.1,0.3,0.7)           (0.2,0.4,0.9)
[e.sub.3]       (0.1,0.2,0.8)           (0.1,0.2,0.9)

[f.sup.2]         [x.sub.5]

[e.sub.2]       (0.4,0.5,0.9)
[e.sub.3]       (0.1,0.5,0.7)
[g.sub.2]   ([x.sub.2],[x.sub.5])
[e.sub.2]       (0.2,0.4,0.9)
[e.sub.3]       (0.1,0.4,0.8)

Table 8.

f           [x.sub.2]       [x.sub.5]

[e.sub.1]   (0.2,0.4,0.5)   (0.1,0.5,0.7)
[e.sub.2]   (0.3,0.5,0.7)   (0.3,0.4,0.9)
[e.sub.3]   (0.2,0.4,0.5)   (0.1,0.5,0.7)

g           ([x.sub.2],[x.sub.5])

[e.sub.1]   (0.1,0.3,0.8)
[e.sub.2]   (0.2,0.4,0.9)
[e.sub.3]   (0.1,0.4,0.8)

Table 9.

f                 [x.sub.1]               [x.sub.2]

[e.sub.1]       (0.1,0.2,0.4)           (0.2,0.3,0.5)
[e.sub.2]       (0.3,0.6,0.8)           (0.4,0.5,0.9)
g           ([x.sub.1],[x.sub.2])   ([x.sub.2],[x.sub.3])
[e.sub.1]       (0.1,0.2,0.5)           (0.2,0.3,0.7)
[e.sub.2]       (0.3,0.5,0.9)           (0.3,0.4,0.9)

f                 [x.sub.5]

[e.sub.1]       (0.3,0.4,0.7)
[e.sub.2]       (0.3,0.4,0.5)
g           ([x.sub.1],[x.sub.3])
[e.sub.1]          (0,0,1)
[e.sub.2]       (0.3,0.4,0.8)
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Author:Shah, Nasir; Hussain, Asim
Publication:Neutrosophic Sets and Systems
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Date:Mar 1, 2016
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