Notions of Rough Neutrosophic Digraphs

mathematics Article

Notions of Rough Neutrosophic Digraphs Nabeela Ishfaq 1 , Sidra Sayed 1 , Muhammad Akram 1, * and Florentin Smarandache 2 1 2

*

Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan; nabeelaishfaq123@gmail.com (N.I.); sidratulmuntha228@yahoo.com (S.S.) Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; fsmarandache@gmail.com Correspondence: m.akram@pucit.edu.pk

Received: 7 December 2017; Accepted: 23 January 2018; Published: 29 January 2018

Abstract: Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we apply the concept of rough NS theory to graphs and present a new kind of graph structure, rough neutrosophic digraphs. We present certain operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also present an application of a rough neutrosophic digraph in decision-making.

Keywords: rough neutrosophic digraphs; lexicographic product; tensor product; decision-making

1. Introduction There are many real-life problems that are beyond a single expert. These are because of the need to involve a wide domain of knowledge. As a generalization of the intuitionistic fuzzy set, a neutrosophic set (NS) has been introduced by Smarandache [1]. This is a useful tool to deal with uncertainty in several social and G natural aspects. Neutrosophy provides a foundation for a whole family of new mathematical theories with the generalization of both classical and fuzzy counterparts.

In a NS, an element has three associated defining functions, the truth membership function (µ), the indeterminate membership function (σ) and the false membership function (λ), defined on a universe of discourse X. These three functions are completely independent. To apply NSs in real-life problems more conveniently, Smarandache [2] and Wang et al. [3] defined single-valued NSs. Ye [4] studied the correlation coefficient and improved the correlation coefficient of NSs, as well as determined that in NSs the cosine similarity measure is a special case of the correlation coefficient.

Rough set theory was proposed by Pawlak [5] in 1982. Rough set theory is useful to study the intelligence systems containing incomplete, uncertain or inexact information. The lower and upper approximation operators of rough sets are used for managing hidden information in a system.

Therefore, many hybrid models have been built, such as soft rough sets, rough fuzzy sets, fuzzy rough sets, soft fuzzy rough sets, neutrosophic rough sets, and rough NSs for handling uncertainty and incomplete information effectively. Dubois and Prade [6] introduced the notions of rough fuzzy sets and fuzzy rough sets. Liu and Chen [7] have studied different decision-making methods.

Broumi et al. [8] introduced the concept of rough NSs. Yang et al. [9] proposed single-valued neutrosophic rough sets by combining single-valued NSs and rough sets, and they established an algorithm for decision-making problems based on single-valued neutrosophic rough sets on two universes. Mordeson and Peng [10] presented operations on fuzzy graphs. Akram et al. [11–14] considered several new concepts of neutrosophic graphs with applications. Zafer and Akram [15] dealt with rough fuzzy digraphs with applications. Recently, Sayed et al. [16] considered rough

Mathematics 2018, 6, 18; doi:10.3390/math6020018

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neutrosophic digraphs. They discussed some fundamental properties of rough neutrosophic digraphs.

In this research paper, we investigate further new operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also consider an application of a rough neutrosophic digraph in decision-making.

2. Rough Neutrosophic Digraphs For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [17–21].

Definition 1. [3] Let Z be a nonempty universe. A NS N on Z is defined as follows: N = {< x : µ N ( x ), σN ( x ), λ N ( x ) >, x ∈ Z } where the functions µ, σ, λ :Z → [0, 1] represent the degree of membership, the degree of indeterminacy and the degree of falsity.

Definition 2. [5] Let Z be a nonempty universe and R be an equivalence relation on Z. A pair ( Z, R) is called an approximation space. Let N ∗ be a subset of Z; the lower and upper approximations of N ∗ in the approximation space ( Z, R) denoted by RN ∗ and RN ∗ are defined as follows: RN ∗ = { x ∈ Z |[ x ] R ⊆ N ∗ } RN ∗ = { x ∈ Z |[ x ] R ⊆ N ∗ } where [ x ] R denotes the equivalence class of R containing x. A pair ( RN ∗ , RN ∗ ) is called a rough set.

Definition 3. [8] Let Z be a nonempty universe and R be an equivalence relation on Z. Let N be a NS on Z. The lower and upper approximations of N in the approximation space ( Z, R), denoted by RN and RN, respectively, are defined as follows: RN = {< x, µ R( N ) ( x ), σR( N ) ( x ), λ R( N ) ( x ) >: y ∈ [ x ] R , x ∈ Z } RN = {< x, µ R( N ) ( x ), σR( N ) ( x ), λ R( N ) ( x ) >: y ∈ [ x ] R , x ∈ Z } where µ R( N ) ( x ) = σR( N ) ( x ) = λ R( N ) ( x ) =

V

y∈[ x ] R

V

y∈[ x ] R

W

y∈[ x ] R

µ N ( y ),

µ R( N ) ( x ) =

σN (y),

σR( N ) ( x ) =

λ N ( y ),

λ R( N ) ( x ) =

W

y∈[ x ] R

W

y∈[ x ] R

V

y∈[ x ] R

µ N (y) σN (y) λ N (y)

A pair ( RN, RN ) is called a rough NS. Definition 4. A rough neutrosophic digraph on a nonempty set V ∗ is an 4-ordered tuple G = ( R, RV, S, SE) such that the following hold: (a) (b) (c) (d) (e)

R is an equivalence relation on V ∗ ; S is an equivalence relation on E∗ ⊆ V ∗ × V ∗ ; RV = ( RV, RV ) is a rough NS on V ∗ ; SE = (SE, SE) is a rough neutrosophic relation on V ∗ ; ( RV, SE) is a rough neutrosophic digraph where G = ( RV, SE) and G = ( RV, SE) are lower and upper approximate neutrosophic digraphs of G such that µSE ( x, y) ≤ min{µ RV ( x ), µ RV (y)} σSE ( x, y) ≤ min{σRV ( x ), σRV (y)} λSE ( x, y) ≤ max{λ RV ( x ), λ RV (y)}

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and

≤ min{µ RV ( x ), µ RV (y)}

µSE ( x, y)

≤ min{σRV ( x ), σRV (y)}

σSE ( x, y)

≤ max{λ RV ( x ), λ RV (y)} ∀ x, y ∈ V ∗

λSE ( x, y)

Example 1. Let V ∗ = { a, b, c, d} be a set and R be an equivalence relation on V ∗ defined as follows:   1 1 0 0  1 1 0 0    R=   0 0 1 1  0 0 1 1

Let V1 = {( a, 0.2, 0.4, 0.9), (b, 0.1, 0.3, 0.5), (c, 0.2, 0.3, 0.6), (d, 0.5, 0.6, 0.7)} be a NS on V ∗ . The lower and upper approximations of V1 are given by RV1 = {( a, 0.1, 0.3, 0.9), (b, 0.1, 0.3, 0.9), (c, 0.2, 0.3, 0.7), (d, 0.2, 0.3, 0.7)} RV1 = {( a, 0.2, 0.4, 0.5), (b, 0.2, 0.4, 0.5), (c, 0.5, 0.6, 0.6), (d, 0.5, 0.6, 0.6)} Let E∗ = {( a, b), (b, c), (b, d), (c, d)} ⊆ V ∗ × V ∗ and as follows:  1 0 0 0  0 1 1 0  S=  0 1 1 0 0 0 0 1

S be an equivalence relation on E∗ , defined     

Let E1 = {(( a, b), 0.1, 0.2, 0.4), ((b, c), 0.1, 0.3, 0.6), ((b, d), 0.1, 0.2, 0.6), ((c, d), 0.2, 0.1, 0.5)} be a NS on E∗ and SE1 = (SE1 , SE1 ) be a rough neutrosophic relation, where SE1 and SE1 are given as follows: SE1 = {(( a, b), 0.1, 0.2, 0.4), ((b, c), 0.1, 0.2, 0.6), ((b, d), 0.1, 0.2, 0.6), ((c, d), 0.2, 0.1, 0.5)} SE1 = {(( a, b), 0.1, 0.2, 0.4), ((b, c), 0.1, 0.3, 0.6), ((b, d), 0.1, 0.3, 0.6), ((c, d), 0.2, 0.1, 0.5)} Thus, G1 = ( RV1 , SE1 ) and G1 = ( RV1 , SE1 ) are neutrosophic digraphs as shown in Figure 1. b(0.1, 0.3, 0.9) b

a(0.2, 0.4, 0.5) (0.1, 0.2, 0.4) ) 0.6 .3, 0 1, (0.

(0.1, 0.2, 0.6)

) 0.6 .2, 0 1, (0. b

(0.2, 0.1, 0.5) d(0.2, 0.3, 0.7)

b

c(0.2, 0.3, 0.7)

G1 = (RV1 , SE1 )

b(0.2, 0.4, 0.5) b

b

(0.1, 0.2, 0.4)

b

(0.1, 0.3, 0.6)

a(0.1, 0.3, 0.9) b

(0.2, 0.1, 0.5) c(0.5, 0.6, 0.6) d(0.5, 0.6, 0.6) b

G1 = (RV1 , SE1 )

1. Rough neutrosophic digraph G G11 = G1 , G, 1G ). 1 ) Figure Figure 1: Rough neutrosophic digraph =((G 1

Example 2. Let V ∗ = { a, b, c} be a crisp set and R be an equivalence relation on V ∗ defined as follows: Example 2.2. Let V ∗ = {a, b, c} be a crisp set and R an equivalence relation on V ∗ defined as:    1 0 0 1 0 0  R= 0 1 1  R =  0 1 1 .

0 1 1 0 1 1 Let V2 = {( a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.5), (c, 0.2, 0.4, 0.3)} be a NS on V ∗ and Let V2 = {(a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.5), (c, 0.2, 0.4, 0.3)} be a neutrosophic set on V ∗ and RV2 = ( RV2 , RV2 ) be a rough NS, where RV2 and RV2 are given as follows: RV2 = (RV2 , RV2 ) a rough neutrosophic set, where RV2 and RV2 are given as:

RV2 = {(a, 0.1, 0.7, 0.8), (b, 0.2, 0.4, 0.5), (c, 0.2, 0.4, 0.5)}, RV2 = {(a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.3), (c, 0.9, 0.6, 0.3)}.

Let E ∗ = {(a, b), (b, c)} ⊆ V ∗ × V ∗ and S an equivalence relation on E ∗ defined as:

 1 0 0 R =  0 1 1 .

0 1 1 

Let V = {(a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.5), (c, 0.2, 0.4, 0.3)} be a neutrosophic set on V ∗ and 4 of 19 RV2 = (RV2 , RV2 ) a rough neutrosophic set, where RV2 and RV2 are given as:

2 2018, 6, 18 Mathematics

RV2 = {(a, 0.1, 0.7, 0.8), (b, 0.2, 0.4, 0.5), (c, 0.2, 0.4, 0.5)}, RV = {( a, 0.1, 0.7, 0.8), (b, 0.2, 0.4, 0.5), (c, 0.2, 0.4, 0.5)} RV2 2= {(a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.3), (c, 0.9, 0.6, 0.3)}.

RV2 = {( a, 0.1, 0.7, 0.8), (b, 0.9, 0.6, 0.3), (c, 0.9, 0.6, 0.3)} ∗ Let E = {(a, b), (b, c)} ⊆ V ∗ × V ∗ and S an equivalence relation on E ∗ ∗defined as: Let E∗ = {( a, b), (b, c)} ⊆ V ∗ × V ∗ and S be an equivalence relation on E defined as follows:

"1 0 # S= 1 0 .

S= 0 1 0 1 Let E2 = {((a, b), 0.1, 0.4, 0.7), ((b, c), 0.2, 0.3, 0.2)} be a neutrosophic set on E ∗ , then by definitionLetweE2have = {(( a, b), 0.1, 0.4, 0.7), ((b, c), 0.2, 0.3, 0.2)} be a NS on E∗ ; then by definition we have SE SE2 2=={((a, {(( a,b), b)0.1, , 0.1,0.4, 0.4,0.7), 0.7)((b, , ((b,c), c)0.2, , 0.2,0.3, 0.3,0.2)}, 0.2)} SE = {((a, b), 0.1, 0.4, 0.7), ((b, c), 0.2, 0.3, SE2 2 = {(( a, b), 0.1, 0.4, 0.7), ((b, c), 0.2, 0.3,0.2)}.

0.2)} (RV neutrosophicdigraphs digraphs shown in Fig.2.2.

Thus, G2G= = 2 , SE 2 )2 and 2 ,2 SE Thus, ( RV ) andGG22==(RV ( RV , SE22)) are are neutrosophic as as shown in Figure 2 2 , SE b(0.2, 0.4, 0.5)

b(0.9, 0.6, 0.3) b

b

a(0.1, 0.7, 0.8)

(0.

1, 0.4 ,0 .7) b

b

c(0.2, 0.4, 0.5)

.2) ,0 0.3 .2, (0

.2) ,0 0.3 .2, (0

(0.

1, 0.4 ,0 .7)

b

b

a(0.1, 0.7, 0.8)

G2 = (RV2 , SE2 )

c(0.9, 0.6, 0.3)

G2 = (RV2 , SE2 )

Figure Roughneutrosophic neutrosophic digraph digraph GG2 2==( G Figure 2: 2.

Rough (G , 2G).2 ) 2 ,2G

Definition 5. Let = (G G11 , = G1(G ) and G2 = ( G2 ,2G= be two rough neutrosophic digraphs on a set V ∗ . Then 2 ) (G Definition 2.5.G1Let 1 , G1 ) and G 2 , G2 ) be two rough neutrosophic digraphs theonlexicographic of Glexicographic neutrosophic digraph G1 G , G 1 G 2 ), 2 = ( G1 G2digraph 1 and G2 is a rough a set V ∗ . product Then the product of G1 and G2 isGa=rough neutrosophic where G1 G2 = ( RV1 RV2 , SE1 SE2 ) and G1 G2 = ( RV1 RV2 , SE1 SE2 ) are neutrosophic digraphs, respectively, such that

(1)

µ RV1 RV2 ( x1 , x2 ) = min{µ RV1 ( x1 ), µ4RV2 ( x2 )} σRV1 RV2 ( x1 , x2 ) = min{σRV1 ( x1 ), µ RV2 ( x2 )}

λ RV1 RV2 ( x1 , x2 ) = max{λ RV1 ( x1 ), µ RV2 ( x2 )} ∀ ( x1 , x2 ) ∈ RV1 n RV2

µSE1 SE2 (( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µSE2 ( x2 , y2 )} σSE1 SE2 (( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σSE2 ( x2 , y2 )}

λSE1 SE2 (( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λSE2 ( x2 , y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) ∈ SE2

µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = min{µSE1 ( x1 , y1 ), µSE2 ( x2 , y2 )} σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = min{σSE1 ( x1 , y1 ), σSE2 ( x2 , y2 )}

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = max{λSE1 ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2

(2)

µ RV1 RV2 ( x1 , x2 ) = min{µ RV1 ( x1 ), µ RV2 ( x2 )} σRV1 RV2 ( x1 , x2 ) = min{σRV1 ( x1 ), µ RV2 ( x2 )}

λ RV1 RV2 ( x1 , x2 ) = max{λ RV1 ( x1 ), µ RV2 ( x2 )} ∀ ( x1 , x2 ) ∈ RV1 n RV2 µSE

(( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µSE2 ( x2 , y2 )}

σSE

(( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σSE2 ( x2 , y2 )}

λSE

(( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λSE2 ( x2 , y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) ∈ SE2

1 SE2 1 SE2 1 SE2

µSE

(( x1 , x2 ), (y1 , y2 )) = min{µSE1 ( x1 , y1 ), µSE2 ( x2 , y2 )}

σSE

(( x1 , x2 ), (y1 , y2 )) = min{σSE1 ( x1 , y1 ), σSE2 ( x2 , y2 )}

λSE

(( x1 , x2 ), (y1 , y2 )) = max{λSE1 ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2

1 SE2 1 SE2 1 SE2

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Example 3. Consider the two rough neutrosophic digraphs G1 and G2 as shown in Figures 1 and 2.

The lexicographic product of G1 and G2 is G = G1 G2 = ( G1 G2 , G1 G2 ), where G1 G2 = ( RV1 RV2 , SE1 SE2 ) and G1 G2 = ( RV1 RV2 , SE1 SE2 ) are neutrosophic digraphs as shown in Figures 3 and 4. ((a, a), 0.1, 0.3, 0.9) ((a, a), 0.1, 0.3, 0.9)

((a, c), 0.1, 0.3, 0.9) ((a, b), 0.1, 0.3, 0.9) ((a, c), 0.1, 0.3, 0.9) ((a, b), 0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.1, 0.3, 0.9) ( 0 (0.

(0..1, 0 (0.1, 0. .

1 2 ,0 ,0 1, 2, .2, .4) 0.2 0.7 0.4 ,0 ) .7) ) ((b, b), 0.1, 0.3, 0.9) ((b, c), 0.1, 0.3, 0.9) ((b, a), 0.1, 0.3, 0.9) ((b, b), 0.1, 0.3, 0.9) ((b, c), 0.1, 0.3, 0.9) ((b, a), 0.1, 0.3, 0.9) (0.1, 0.3, 0.9) 0.3, 0.9) (0. (0.1, (0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.

(0.1, 0 1, .2, (0.1, 0 0.2 0.6 1, .2, ,0 ) 0.2 0.7 .6) ,0 ) .7) ((c, a), 0.1, 0.3, 0.8) ((c, b), 0.2, 0.3, 0.7) ((c, c), 0.2, 0.3, 0.7) ((c, a), 0.1, 0.3, 0.8) ((c, b), 0.2, 0.3, 0.7) ((c, c), 0.2, 0.3, 0.7) (0.1, 0.3, 0.7) (0.1, 0.3, 0.7) (0.

(0.

(0.1, 0.3, 0.7) (0.1, 0.3, 0.7) (0.1, 0 1, .1, (0.2, 0.

0.1 0.7 2, 0 1, 0 ,0 ) .1, .5) .7) 0.5 ) b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

) .7).7 , 02, 0 0.,20. .1, .1 (0 (0

((d, a), 0.1, 0.3, 0.8) ((d, a), 0.1, 0.3, 0.8) b

b

b

b

) ) .6 .6 ,0 ,0 .2 .2 ,0 ,0 .1 .1 (0 (0

b

((d, b), 0.2, 0.3, 0.7) ((d, c), 0.2, 0.3, 0.7) ((d, b), 0.2, 0.3, 0.7) ((d, c), 0.2, 0.3, 0.7) (0.1, 0.3, 0.7) (0.1, 0.3, 0.7) b

(0.1, 0.3, 0.7) (0.1, 0.3, 0.7) b

b

Figure 3. Neutrosophic digraph G1 G2 = ( RV1 RV2 , SE1 SE2 ).

Figure 3: G1 ⊙ G2 = (RV1 ⊙ RV2 , SE1 ⊙ SE2 ) Figure 3: G1 ⊙ G2 = (RV1 ⊙ RV2 , SE1 ⊙ SE2 )

((a, b), 0.2, 0.4, 0.5) ((a, c), 0.2, 0.4, 0.5) ((a, b), 0.2, 0.4, 0.5) (0.2, 0.3, 0.5) ((a, c), 0.2, 0.4, 0.5) (0.1, 0.4, 0.7) (0.2, 0.3, 0.5) (0. (0.1, 0.4, 0.7) (0.

(0.1, 0.

(0.1, 0 1, 2, 1, .2, 0.2 0.7 0.2 0.4 ,0 ) ,0 ) .7) .4)

((a, a), 0.1, 0.4, 0.8) ((a, a), 0.1, 0.4, 0.8) b

b

b

b

b

((b, a), 0.1, 0.4, 0.8) ((b, a), 0.1, 0.4, 0.8)(0.1, 0.4, 0.7) (0.1, 0.4, 0.7) (0.

(0.1, 0 1, .3, 0.3 0.7 ,0 ) .7) b

b

b

((b, b), 0.2, 0.4, 0.5) ((b, b), 0.2, 0.4, (0.2,0.5) 0.3, 0.5) 0.3, 0.5) ((0.2, 0 (0..1, 0 1, .3, 0.3 0.6 ,0 ) .6)

((b, c), 0.2, 0.4, 0.5) ((b, c), 0.2, 0.4, 0.5)

b

b

b

b

((c, b), 0.5, 0.6, 0.6) ((c, a), 0.1, 0.6, 0.8) ((c, c), 0.5, 0.6, 0.6) ((c, b), 0.5, 0.6, 0.6) ((c, a), 0.1, 0.6, 0.8) (0.1, 0.4, 0.7) (0.2, 0.3, ((c, 0.6)c), 0.5, 0.6, 0.6) (0.

(0.1, 0.4, 0.7) (0.2, 0.3, 0.6) (0.

(0.1, 0 1, .1, (0.2, 0.

0.1 0.7 2, 0 1, 0 ,0 ) .1, .5) .7) 0.5 ) ((d, a), 0.1, 0.6, 0.8) ((d, c), 0.5, 0.6, 0.6) ((d, b), 0.5, 0.6, 0.6) ((d, a), 0.1, 0.6, 0.8) ((d, c), 0.5, 0.6, 0.6) ((d, b), 0.5, 0.6, 0.6) (0.1, 0.4, 0.7) (0.2, 0.3, 0.6) (0.1, 0.4, 0.7) (0.2, 0.3, 0.6) b

b

b

b

b

b

) ) .6 .6 ,0 ,0 .3 .3 ,0 ,0 .1 .1 (0 (0

) .7).7 , 03, 0 0.,30. .1, .1 (0 (0

b

b

b

b

b

b

4: G1 ⊙digraph G2 = (RV RV ⊙,SE ) SE ).

1 ⊙= 2 , SE FigureFigure 4. Neutrosophic ( RV

1RV SE12

2 RV 1 G⊙ Figure 4: G ⊙ G = G (RV , 1SE ⊙2 SE ) 2 1

2

1

2

1

2

Theorem 1.σThe lexicographic product of two neutrosophic is a rough neutrosophic digraph. (x, y2 )) = σrough ∧ σSE2 (x2digraphs , y2 ) SE1 ⊙SE RV1 (x) 2 ((x, x2 ),

σSE1 ⊙SE2 ((x, x2 ), (x, y2 )) = σRV1 (x) ∧ σSE2 (x2 , y2 ) ≤ σRV1 (x) ∧ (σRV2 (x2 ) ∧ σRV2 (y2 )) (σ2RV (x2two ) ∧ σRV Proof. Let G1 = ( G1 , G1 ) and G≤ = 1 (x) ( G∧, G ) 2be rough 2 (y2 ))neutrosophic digraphs.

2 σRV = (σRV1 (x)2∧ σRV2 (x2 )) ∧ (σRV1 (x) ∧ σRV2 (y2 )) Let G = G1 G2 = ( G1 G2 , = G1(σ

RVG12(x) ) be lexicographic product of 2 )) G1 and G2 , ∧ σthe ∧ σRV2 (y RV2 (x 2 )) ∧ (σRV1 (x) σ2RV x ) ∧ σ (x, y ) ⊙RV2G(x, 2 RV ⊙RV 2 1and 1 2 where G1 G2 = ( RV1 RV2 , SE1 = SE )

G = ( RV

RV , SE

SE ) 2 2 . To prove 1 x )2 ∧ σ 1 1 = σRV1 ⊙RV2 (x, 2 RV1 ⊙RV2 (x, y2 ) that G = G1σ

rough digraph, is enough show that (x, SE1y

SE2 and SE1 SE2 x2neutrosophic ), (x, y2 )) ≤ min{σ (x, x2to ), σ 2 is a ((x, RVit⊙RV RV ⊙RV 2 )}, SEG⊙SE σSE11 ⊙SE22 ((x, x2 ), (x, y2 )) ≤ min{σRV11 ⊙RV22 (x, x2 ), σRV11 ⊙RV22 (x, y2 )}, are neutrosophic relations on RV

RV and RV

RV , respectively.

First, we show that SE1 SE2 2 λRV (x) 2 (x2 , y2 ) 1 ∨ λSE λSE1 ⊙SE2 ((x, x2 ), (x,1 y2 )) = λSE1 ⊙SE2 ((x, x2 ), (x, y2 )) = λRV11 (x) ∨ λSE22 (x2 , y2 ) ≤ λRV1 (x) ∨ (λRV2 (x2 ) ∨ λRV2 (y2 )) ≤ λRV1 (x) ∨ (λRV2 (x2 ) ∨ λRV2 (y2 )) = (λRV1 (x) ∨ λRV2 (x2 )) ∨ (λRV1 (x) ∨ λRV2 (y2 )) = (λRV1 (x) ∨ λRV2 (x2 )) ∨ (λRV1 (x) ∨ λRV2 (y2 )) 6 6

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is a neutrosophic relation on RV1 RV2 .

If x ∈ RV1 , ( x2 , y2 ) ∈ SE2 , then µSE1 SE2 (( x, x2 ), ( x, y2 )) = µ RV1 ( x ) ∧ µSE2 ( x2 , y2 )

≤ µ RV1 ( x ) ∧ (µ RV2 ( x2 ) ∧ µ RV2 (y2 ))

= (µ RV1 ( x ) ∧ µ RV2 ( x2 )) ∧ (µ RV1 ( x ) ∧ µ RV2 (y2 )) = µ RV1 RV2 ( x, x2 ) ∧ µ RV1 RV2 ( x, y2 )

µSE1 SE2 (( x, x2 ), ( x, y2 )) ≤ min{µ RV1 RV2 ( x, x2 ), µ RV1 RV2 ( x, y2 )} σSE1 SE2 (( x, x2 ), ( x, y2 )) = σRV1 ( x ) ∧ σSE2 ( x2 , y2 )

≤ σRV1 ( x ) ∧ (σRV2 ( x2 ) ∧ σRV2 (y2 ))

= (σRV1 ( x ) ∧ σRV2 ( x2 )) ∧ (σRV1 ( x ) ∧ σRV2 (y2 )) = σRV1 RV2 ( x, x2 ) ∧ σRV1 RV2 ( x, y2 )

σSE1 SE2 (( x, x2 ), ( x, y2 )) ≤ min{σRV1 RV2 ( x, x2 ), σRV1 RV2 ( x, y2 )}

λSE1 SE2 (( x, x2 ), ( x, y2 )) = λ RV1 ( x ) ∨ λSE2 ( x2 , y2 )

≤ λ RV1 ( x ) ∨ (λ RV2 ( x2 ) ∨ λ RV2 (y2 ))

= (λ RV1 ( x ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 ( x ) ∨ λ RV2 (y2 ))

= λ RV1 RV2 ( x, x2 ) ∨ λ RV1 RV2 ( x, y2 )

λSE1 SE2 (( x, x2 ), ( x, y2 )) ≤ max{λ RV1 RV2 ( x, x2 ), λ RV1 RV2 ( x, y2 )} If ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2 , then µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = µSE1 ( x1 , y1 ) ∧ µSE2 ( x2 , y2 )

≤ (µ RV1 ( x1 ) ∧ µ RV1 (y1 )) ∧ (µ RV2 ( x2 ) ∧ µ RV2 (y2 ))

= (µ RV1 ( x1 ) ∧ µ RV2 ( x2 )) ∧ (µ RV1 (y1 ) ∧ µ RV2 (y2 )) = µ RV1 RV2 ( x1 , x2 ) ∧ µ RV1 RV2 (y1 , y2 )

µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ min{µ RV1 RV2 ( x1 , x2 ), µ RV1 RV2 (y1 , y2 )} σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = σSE1 ( x1 , y1 ) ∧ σSE2 ( x2 , y2 )

≤ (σRV1 ( x1 ) ∧ σRV1 (y1 )) ∧ (σRV2 ( x2 ) ∧ σRV2 (y2 )) = (σRV1 ( x1 ) ∧ σRV2 ( x2 )) ∧ (σRV1 (y1 ) ∧ σRV2 (y2 )) = σRV1 RV2 ( x1 , x2 ) ∧ σRV1 RV2 (y1 , y2 )

σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ min{σRV1 RV2 ( x1 , x2 ), σRV1 RV2 (y1 , y2 )}

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = λSE1 ( x1 , y1 ) ∨ λSE2 ( x2 , y2 )

≤ (λ RV1 ( x1 ) ∨ λ RV1 (y1 )) ∨ (λ RV2 ( x2 ) ∨ λ RV2 (y2 )) = (λ RV1 ( x1 ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 (y1 ) ∨ λ RV2 (y2 )) = λ RV1 RV2 ( x1 , x2 ) ∨ λ RV1 RV2 (y1 , y2 )

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ max{λ RV1 RV2 ( x1 , x2 ), λ RV1 RV2 (y1 , y2 )} Thus, from the above, it is clear that SE1 SE2 is a neutrosophic relation on RV1 RV2 .

Similarly, we can show that SE1 SE2 is a neutrosophic relation on RV1 RV2 . Hence, G = ( G1 G2 , G1 G2 ) is a rough neutrosophic digraph.

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Definition 6. The strong product of two rough neutrosophic digraphs G1 and G2 is a rough neutrosophic digraph G = G1 G2 = ( G1 G2 , G1 G2 ), where G1 G2 = ( RV1 RV2 , SE1 SE2 ) and G1 G2 = ( RV1 RV2 , SE1 SE2 ) are neutrosophic digraphs, respectively, such that

(1)

µ RV1 RV2 ( x, y) = min{µ RV1 ( x ), µ RV2 (y)} σRV1 RV2 ( x, y) = min{σRV1 ( x ), σRV2 (y)}

λ RV1 RV2 ( x, y) = max{λ RV1 ( x ), λ RV2 (y)} ∀ ( x, y) ∈ RV1 × RV2

µSE1 SE2 (( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µSE2 ( x2 , y2 )} σSE1 SE2 (( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σSE2 ( x2 , y2 )}

λSE1 SE2 (( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λSE2 ( x2 , y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) ∈ SE2 µSE1 SE2 (( x1 , y), (y1 , y)) = min{µSE1 ( x1 , y1 ), µ RV2 (y)} σSE1 SE2 (( x1 , y), (y1 , y)) = min{σSE1 ( x1 , y1 ), σRV2 (y)}

λSE1 SE2 (( x1 , y), (y1 , y)) = max{λSE1 ( x1 , y1 ), λ RV2 (y)} ∀ ( x1 , y1 ) ∈ SE1 , y ∈ RV2

µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = min{µSE1 ( x1 , y1 ), µSE2 ( x2 , y2 )} σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = min{σSE1 ( x1 , y1 ), σSE2 ( x2 , y2 )}

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = max{λSE1 ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2

(2)

µ RV1 RV2 ( x, y) = min{µ RV1 ( x ), µ RV2 (y)} σRV1 RV2 ( x, y) = min{σRV1 ( x ), σRV2 (y)} λ RV1 RV2 ( x, y) = max{λ RV1 ( x ), λ RV2 (y)} ∀ ( x, y) ∈ RV1 × RV2

µSE

(( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µSE2 ( x2 , y2 )}

σSE

(( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σSE2 ( x2 , y2 )}

λSE

(( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λSE2 ( x2 , y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) ∈ SE2

1 SE2 1 SE2 1 SE2

µSE

(( x1 , y), (y1 , y)) = min{µSE1 ( x1 , y1 ), µ RV2 (y)}

σSE

(( x1 , y), (y1 , y)) = min{σSE1 ( x1 , y1 ), σRV2 (y)}

λSE

(( x1 , y), (y1 , y)) = max{λSE1 ( x1 , y1 ), λ RV2 (y)} ∀ ( x1 , y1 ) ∈ SE1 , y ∈ RV2

1 SE2 1 SE2 1 SE2

µSE

(( x1 , x2 ), (y1 , y2 )) = min{µSE1 ( x1 , y1 ), µSE2 ( x2 , y2 )}

σSE

(( x1 , x2 ), (y1 , y2 )) = min{σSE1 ( x1 , y1 ), σSE2 ( x2 , y2 )}

λSE

(( x1 , x2 ), (y1 , y2 )) = max{λSE1 ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2

1 SE2 1 SE2 1 SE2

Example 4. Consider the two rough neutrosophic digraphs G1 and G2 as shown in Figures 1 and 2. The strong product of G1 and G2 is G = G1 G2 = ( G1 G2 , G1 G2 ), where G1 G2 = ( RV1 RV2 , SE1 SE2 ) and G1 G2 = ( Rv1 RV2 , SE1 SE2 ) are neutrosophic digraphs as shown in Figures 5 and 6.

SE1 ⊠SE2

1

2

1

2

SE1

1

1

SE2

2

2

1

1

1

2

2

2

Example 2.4. Consider the two rough neutrosophic digraphs G1 and G2 as shown in Fig.1 and Fig.2. The strong product of G1 and G2 is G = G1 ⊠ G2 = (G1 ⊠ G2 , G1 ⊠ G2 ), where Mathematics 2018, 6, 18 8 of 19 G1 ⊠ G 2 = (RV1 ⊠ RV2 , SE1 ⊠ SE2 ) and G1 ⊠ G2 = (Rv1 ⊠ RV2 , SE1 ⊠ SE2 ) are neutrosophic digraphs as shown in Fig. 5 and Fig. 6. ((a, c), 0.1, 0.3, 0.9) ((a, a), 0.1, 0.3, 0.9) ((a, b), 0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.1, 0.3, 0.9) b

b

((b, a), 0.1, 0.3, 0.9) (0.1, 0.3, 0.9)

((b, b), 0.1, 0.3, 0.9) (0.1, 0.3, 0.9) (0.

1, 0.2 ,0 .6)

((c, b), 0.2, 0.3, 0.7) ((c, a), 0.1, 0.3, 0.8) (0.1, 0.3, 0.7) (0.

1, (0.

0.1 2, 0 ,0 .1, .7) 0.5 ) b

b

b

(0.2, 0.1, 0.5)

(0.1, 0.3, 0.7) b

((c, c), 0.2, 0.3, 0.7) (0.1, 0.3, 0.7)

.6) ,0 0.2 .1, (0

(0.1, 0 .2, 0.6 ) (0.2, 0.1, 0.5) .7) ,0 0.2 .1, (0

(0.1, 0.1, 0.8)

b

b

(0.1, 0.2, 0.6)

(0.1, 0.2, 0.6)

(0.

1, 0.2 ,0 .7)

((d, a), 0.1, 0.3, 0.8)

((b, c), 0.1, 0.3, 0.9)

(0.1, 0.2, 0.6)

(0.1, 0.2, 0.8)

(0.

1, 0.2 ,0 .4) b

b

(0.1, 0.2, 0.8)

(0.1, 0.2, 0.5)

(0.

1, 0.2 ,0 .7)

(0.1, 0.2, 0.5)

(0.1, 0.2, 0.8)

b

(0.1, 0.3, 0.7)

((d, b), 0.2, 0.3, 0.7)

b

((d, c), 0.2, 0.3, 0.7)

Figure 5: Rough neutrosophic digraph Figure 5. Neutrosophic digraph G1 G G12 .⊠ G2 ((a, b), 0.2, 0.4, 0.5)

((a, a), 0.1, 0.4, 0.8)

((a, c), 0.2, 0.4, 0.5)

Theorem 2.2. The strong product of two rough neutrosophic digraphs is a rough neutrosophic (0.1, 0.4, 0.7) (0.2, 0.3, 0.5) (0. digraph.

1, ( 0 b

b

(0.1, 0.2, 0.4)

(0.1, 0.2, 0.4)

(0.1, 0.2, 0.8)

b

0.1 ,0 .2, 0.4 neutrosophic digraphs.

(G2 , G2 ) be two rough Proof. Let G1 = (G1 , G1 ) and ) G1 ⊠ G2 = (G1 ⊠ G2 , G1 ⊠ G2 ) be the strong product of G1 and G2 , where G1 ⊠ G2 0.4, 0.8) b), 0.2, 0.4, 0.5) ((b, c), 0.2, 0.4, 0.5) anda),G0.1, (RV RV ((b, , SE ). To RV2 , SE1 ⊠ SE2 ) ((b, 1 ⊠ G2 =(0.1, 1⊠ 1 ⊠ SE (0.2,20.3, 0.5) prove that G = G1 ⊠ G2 0.4, 0.7) 2 (0.

1, (0.

0.3 1, ,0 0.3 .7) 8 ,0 .6)

.2, 0.7 G2 )=

b

b

b

(0.1, 0.3, 0.8)

(0.1, 0.1, 0.8)

(0.2, 0.1, 0.5) ) .6 ,0 .3 ,0 .1 (0

b

b

(0.1, 0.3, 0.6)

(0.1, 0.4, 0.7)

((d, a), 0.1, 0.6, 0.8)

(0.2, 0.1, 0.5)

(0.1, 0.3, 0.8)

b

.7) ,0 0.3 .1, (0

b

(0.1, 0.3, 0.6)

(0.1, 0.3, 0.6)

(0.1, 0.3, 0.6)

(0.1, 0.4, 0.7) ((c, b), 0.5, 0.6, 0.6) (0.2, 0.3, 0.6)((c, c), 0.5, 0.6, 0.6) (0.

1, ( 0.2 0.1 , 0. ,0 .7) 1, 0 .5)

((c, a), 0.1, 0.6, 0.8) b

Let G = = (RV1 ⊠ is a rough

(0.2, 0.3, 0.6) ((d, c), 0.5, 0.6, 0.6) b

((d, b), 0.5, 0.6, 0.6)

Figure 6. Neutrosophic digraph G1 G G12 .⊠ G2 Figure 6: Rough neutrosophic digraph

Theorem 2. The strong product of two rough neutrosophic a rough digraph. neutrosophic digraph it is enough to show that SEdigraphs and SE1 neutrosophic ⊠ SE2 are neutrosophic 1 ⊠ SE2 is relations on RV1 ⊠ RV2 and RV1 ⊠ RV2 , respectively. First, we show that SE1 ⊠ SE2 is a Let G1 = (relation G1 , G1 ) on andRV G21 ⊠ = RV ( G2 ., G2 ) be two rough neutrosophic digraphs. Let G = G1 G2 = Proof. neutrosophic , (x , y ) ∈ SE , then If x RV ( G1 G2∈, G

G ) be the strong product of G1 and G2 , where G1 G2 = ( RV1 RV2 , SE1 SE2 ) 2 2 2 1 1 and G1 G2 = ( RV1 RV2 , SE1 SE2 ). To prove that G = G1 G2 is a rough neutrosophic digraph, µSE1 ⊠SE2 ((x, x2 ), (x, y2 )) = µRV1 (x) ∧ µSE2 (x2 , y2 ) it is enough to show that SE1 SE2 and SE1 SE2 are neutrosophic relations on RV1 RV2 and ≤ µRV ∧ (µRV (x2 ) ∧ µRV2 (y2 ))relation on RV RV .

RV1 RV2 , respectively. First, we show that SE1 (x) 2 1 SE2 is 2a neutrosophic 1 = (µRV1 (x) ∧ µRV2 (x2 )) ∧ (µRV1 (x) ∧ µRV2 (y2 )) = µRV1 ⊠RV2 (x, x2 ) ∧ µRV1 ⊠RV2 (x, y2 ) If x ∈ RV1 , ( x2 , y2 ) ∈ SE2 , then µSE1 ⊠SE2 ((x, x2 ), (x, y2 )) ≤ min{µRV1 ⊠RV2 (x, x2 ), µRV1 ⊠RV2 (x, y2 )}, µσSE x, x2x)2,),( x, = µ RV1 ((x) x ) ∧∧µσSE ( x22, ,yy22)) SE2 ((((x, 2 2 (x (x,y2y)) 1 2 )) = σRV SE SE 1 ⊠SE 2 1 ≤≤µσRV ( x ) ∧ (µ RV22((x x22)) ∧ y22)) ∧ µσRV )) RV RV22((y 1 1 (x) ∧ (σRV ==((σ µ RV x ) ∧∧µσRV x22)) µRV1 ((x) x ) ∧ σµRV (y22)) )) )) ∧ ∧ ((σ RV22((x RV11((x) RV22 (y

==µσRV x, xx22))∧∧µσRV x, yy22)) RV RV11 ⊠RV 1 ⊠RV RV2 2((x, RV22((x, 1 ((x, x ), (x, y )) ≤ min{σ (x, x ), σ )}, σ 2 ≤ min{ µ RV RV1 ⊠RV2( x, x22), µ RV RV1 ⊠RV2((x, SE 1 ⊠SE(( 2 x, x2 )2, ( x, y2 )) µSE x, yy22)} 1 RV2 1 RV2 1 SE2 λSE1 ⊠SE2 ((x, x2 ), (x, y2 )) = λRV1 (x) ∨ λSE2 (x2 , y2 ) ≤ λRV1 (x) ∨ (λRV2 (x2 ) ∨ λRV2 (y2 ))

= (λRV1 (x) ∨ λRV2 (x2 )) ∨ (λRV1 (x) ∨ λRV2 (y2 ))

= λRV1 ⊠RV2 (x, x2 ) ∨ λRV1 ⊠RV2 (x, y2 )

λSE1 ⊠SE2 ((x, x2 ), (x, y2 )) ≤ max{λRV1 ⊠RV2 (x, x2 ), λRV1 ⊠RV2 (x, y2 )}.

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σSE1 SE2 (( x, x2 ), ( x, y2 )) = σRV1 ( x ) ∧ σSE2 ( x2 , y2 )

≤ σRV1 ( x ) ∧ (σRV2 ( x2 ) ∧ σRV2 (y2 ))

= (σRV1 ( x ) ∧ σRV2 ( x2 )) ∧ (σRV1 ( x ) ∧ σRV2 (y2 )) = σRV1 RV2 ( x, x2 ) ∧ σRV1 RV2 ( x, y2 )

σSE1 SE2 (( x, x2 ), ( x, y2 )) ≤ min{σRV1 RV2 ( x, x2 ), σRV1 RV2 ( x, y2 )}

λSE1 SE2 (( x, x2 ), ( x, y2 )) = λ RV1 ( x ) ∨ λSE2 ( x2 , y2 )

≤ λ RV1 ( x ) ∨ (λ RV2 ( x2 ) ∨ λ RV2 (y2 ))

= (λ RV1 ( x ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 ( x ) ∨ λ RV2 (y2 )) = λ RV1 RV2 ( x, x2 ) ∨ λ RV1 RV2 ( x, y2 )

λSE1 SE2 (( x, x2 ), ( x, y2 )) ≤ max{λ RV1 RV2 ( x, x2 ), λ RV1 RV2 ( x, y2 )} If x1 y1 ∈ SE1 , z ∈ RV2 , then µSE1 SE2 ( x1 , z)(y1 , z) = µSE1 ( x1 , y1 ) ∧ µ RV2 (z)

≤ (µ RV1 ( x1 ) ∧ µ RV1 (y1 )) ∧ µ RV2 (z)

= (µ RV1 ( x1 ) ∧ µ RV2 (z)) ∧ (µ RV1 (y1 ) ∧ µ RV2 (z)) = µ RV1 RV2 ( x1 , z) ∧ µ RV1 RV2 (y1 , z)

µSE1 SE2 ( x1 , z)(y1 , z) ≤ min{µ RV1 RV2 ( x1 , z), µ RV1 RV2 (y1 , z)} σSE1 SE2 ( x1 , z)(y1 , z) = σSE1 ( x1 , y1 ) ∧ σRV2 (z)

≤ (σRV1 ( x1 ) ∧ σRV1 (y1 )) ∧ σRV2 (z)

= (σRV1 ( x1 ) ∧ σRV2 (z)) ∧ (σRV1 (y1 ) ∧ σRV2 (z))

= σRV1 RV2 ( x1 , z) ∧ σRV1 RV2 (y1 , z)

σSE1 SE2 ( x1 , z)(y1 , z) ≤ min{σRV1 RV2 ( x1 , z), σRV1 RV2 (y1 , z)}

λSE1 SE2 ( x1 , z)(y1 , z) = λSE1 ( x1 , y1 ) ∨ λ RV2 (z)

≤ (λ RV1 ( x1 ) ∨ λ RV1 (y1 )) ∨ λ RV2 (z)

= (λ RV1 ( x1 ) ∨ λ RV2 (z)) ∨ (λ RV1 (y1 ) ∨ λ RV2 (z))

= λ RV1 RV2 ( x1 , z) ∨ λ RV1 RV2 (y1 , z)

λSE1 SE2 ( x1 , z)(y1 , z) ≤ max{λ RV1 RV2 ( x1 , z), λ RV1 RV2 (y1 , z)} If ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2 , then µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = µSE1 ( x1 , y1 ) ∧ µSE2 ( x2 , y2 )

≤ (µ RV1 ( x1 ) ∧ µ RV1 (y1 )) ∧ (µ RV2 ( x2 ) ∧ µ RV2 (y2 )) = (µ RV1 ( x1 ) ∧ µ RV2 ( x2 )) ∧ (µ RV1 (y1 ) ∧ µ RV2 (y2 )) = µ RV1 RV2 ( x1 , x2 ) ∧ µ RV1 RV2 (y1 , y2 )

µSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ min{µ RV1 RV2 ( x1 , x2 ), µ RV1 RV2 (y1 , y2 )} σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = σSE1 ( x1 , y1 ) ∧ σSE2 ( x2 , y2 )

≤ (σRV1 ( x1 ) ∧ σRV1 (y1 )) ∧ (σRV2 ( x2 ) ∧ σRV2 (y2 )) = (σRV1 ( x1 ) ∧ σRV2 ( x2 )) ∧ (σRV1 (y1 ) ∧ σRV2 (y2 )) = σRV1 RV2 ( x1 , x2 ) ∧ σRV1 RV2 (y1 , y2 )

σSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ min{σRV1 RV2 ( x1 , x2 ), σRV1 RV2 (y1 , y2 )}

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) = λSE1 ( x1 , y1 ) ∨ λSE2 ( x2 , y2 )

≤ (λ RV1 ( x1 ) ∨ λ RV1 (y1 )) ∨ (λ RV2 ( x2 ) ∨ λ RV2 (y2 ))

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= (λ RV1 ( x1 ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 (y1 ) ∨ λ RV2 (y2 )) = λ RV1 RV2 ( x1 , x2 ) ∨ λ RV1 RV2 (y1 , y2 )

λSE1 SE2 (( x1 , x2 ), (y1 , y2 )) ≤ max{λ RV1 RV2 ( x1 , x2 ), λ RV1 RV2 (y1 , y2 )} Thus, from the above, it is clear that SE1 SE2 is a neutrosophic relation on RV1 RV2 .

Similarly, we can show that SE1 SE2 is a neutrosophic relation on RV1 RV2 . Hence, G = ( G1 G2 , G1 G2 ) is a rough neutrosophic digraph.

Definition 7. Let G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) be two rough neutrosophic digraphs on a set V ∗ .

Then the rejection of G1 and G2 is a rough neutrosophic digraph G = G1 | G2 = ( G1 | G2 , G1 | G2 ), where G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ) and G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ) are neutrosophic digraphs, respectively, such that µ RV1 | RV2 ( x1 , x2 ) = min{µ RV1 ( x1 ), µ RV2 ( x2 )}

(1)

σRV1 | RV2 ( x1 , x2 ) = min{σRV1 ( x1 ), µ RV2 ( x2 )}

λ RV1 | RV2 ( x1 , x2 ) = max{λ RV1 ( x1 ), µ RV2 ( x2 )} ∀ ( x1 , x2 ) ∈ RV1 n RV2

µSE1 |SE2 (( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µ RV2 ( x2 ), µ RV2 (y2 )} σSE1 |SE2 (( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σRV2 ( x2 ), σRV2 (y2 )}

λSE1 |SE2 (( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λ RV2 ( x2 ), λ RV2 (y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) 6∈ SE2 µSE1 |SE2 (( x1 , z), (y1 , z)) = min{µ RV1 ( x1 ), µ RV1 (y1 ), µ RV2 (z)} σSE1 |SE2 (( x1 , z), (y1 , z)) = min{σRV1 ( x1 ), σRV1 (y1 ), σRV2 (z)}

λSE1 |SE2 (( x1 , z), (y1 , z)) = max{λ RV1 ( x1 ), λ RV1 (y1 ), λ RV2 (z)} ∀ ( x1 , y1 ) 6∈ SE1 , z ∈ RV2

µSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = min{µ RV1 ( x1 ), µ RV1 (y1 ), µ RV2 ( x2 ), µ RV2 (y2 )} σSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = min{σRV1 ( x1 ), σRV1 (y1 ), σRV2 ( x2 ), σRV2 (y2 )},

λSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = max{λ RV1 ( x1 ), λ RV1 (y1 ), λ RV2 ( x2 ), λ RV2 (y2 )} ∀ ( x1 , y1 ) 6∈ SE1 , ( x2 , y2 ) 6∈ SE2 µ RV1 | RV2 ( x1 , x2 ) = min{µ RV1 ( x1 ), µ RV2 ( x2 )}

(2)

σRV1 | RV2 ( x1 , x2 ) = min{σRV1 ( x1 ), µ RV2 ( x2 )}

λ RV1 | RV2 ( x1 , x2 ) = max{λ RV1 ( x1 ), µ RV2 ( x2 )} ∀ ( x1 , x2 ) ∈ RV1 n RV2 µSE

(( x, x2 ), ( x, y2 )) = min{µ RV1 ( x ), µ RV2 ( x2 ), µ RV2 (y2 )}

σSE

(( x, x2 ), ( x, y2 )) = min{σRV1 ( x ), σRV2 ( x2 ), σRV2 (y2 )}

λSE

(( x, x2 ), ( x, y2 )) = max{λ RV1 ( x ), λ RV2 ( x2 ), λ RV2 (y2 )} ∀ x ∈ RV1 , ( x2 , y2 ) 6∈ SE2

1 | SE2 1 | SE2 1 | SE2

µSE

(( x1 , z), (y1 , z)) = min{µ RV1 ( x1 ), µ RV1 (y1 ), µ RV2 (z)}

σSE

(( x1 , z), (y1 , z)) = min{σRV1 ( x1 ), σRV1 (y1 ), σRV2 (z)}

λSE

(( x1 , z), (y1 , z)) = max{λ RV1 ( x1 ), λ RV1 (y1 ), λ RV2 (z)} ∀( x1 , y1 ) 6∈ SE1 , z ∈ RV2

1 | SE2 1 | SE2 1 | SE2

µSE

(( x1 , x2 ), (y1 , y2 )) = min{µ RV1 ( x1 ), µ RV1 (y1 ), µ RV2 ( x2 ), µ RV2 (y2 )}

σSE

(( x1 , x2 ), (y1 , y2 )) = min{σRV1 ( x1 ), σRV1 (y1 ), σRV2 ( x2 ), σRV2 (y2 )}

λSE

(( x1 , x2 ), (y1 , y2 )) = max{λ RV1 ( x1 ), λ RV1 (y1 ), λ RV2 ( x2 ), λ RV2 (y2 )} ∀ ( x1 , y1 ) 6∈ SE1 , ( x2 , y2 ) 6∈ SE2

1 | SE2 1 | SE2 1 | SE2

Example 5. Consider the two rough neutrosophic digraphs G1 and G2 as shown in Figures 7 and 8. The rejection of G1 and G2 is G = G1 | G2 = ( G1 | G2 , G1 | G2 ), where G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ) and G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ) are neutrosophic digraphs, as shown in Figures 9 and 10.

b b

) SE1Rough GFigure 1 = (RV1 ,7: G1 = (RV1 , SE1 )

(0.1, 0.1, 0.6)

) 0.1, 0.8) (RV = 0.1, G2(0.1, 2 , SE2c(0.2, 0.6)

b(0.9, 0.6, 0.1) b(0.9, 0.6, 0.1) b b

(0 (0 (0 .2 .2 .2 ,0 ,0 ,0 .4 .4 .4 ,0 ,0 ,0 .6 .6 .6 ) ) ) (0 (0 (0 .2 .2 .2 ,0 ,0 ,0 .4 .4 .4 ,0 ,0 ,0 .5 .5 .5 ) ) )

, 0.1) (0.3, 0.2 b

b

d(0.5, 0.5, 0.6) d(0.5, 0.5, 0.6)

c(0.2, 0.1, 0.8)

b

c(0.5, 0.5, 0.6) b b

0.1, 0.3) (0.2, ) = (RV G2(0.2, 2 , SE2 c(0.5, 0.5, 0.6) 0.1, 0.3) c(0.5, 0.5, 0.6)

SE)2 ) =(G G2= ) 8. Rough SE2Rough ,8: GFigure neutrosophic digraph G2 =G(2G ). ,2 ,G 2, G 2(RV 2 = (RV2Figure neutrosophic digraph 2 2) 22 , SE G2 = (RV G2 = (RV2 , SE2 ) ((a, b), 0.2, 0.4, 0.6) ((a, c), 0.2, 0.1, 0.8) Figure 8: Rough neutrosophic digraph G G20.1, ) 0.8) ((a, 2 d),, 0.2, 2 = (G (0.2, 0.1, 0.8) digraph (0.2, 0.4, (0.2, 0.1,G 0.8) Figure 8: Rough0.6)neutrosophic 2 = (G2 , G2 ) b

b

b

((a, a), 0.2, 0.4, 0.6) ((a, a), 0.2, 0.4, 0.6)

b

((b, a), 0.2, 0.4, 0.6)

b

((b, a), 0.2, 0.4, 0.6) ((b, a), 0.2, 0.4, 0.6)

b

b

((c, a), 0.2, 0.4, 0.9)

b

((c, a), 0.2, 0.4, 0.9) ((c, a), 0.2, 0.4, 0.9)

b

b

(d, a), 0.2, 0.4, 0.9)

b

b

(d, a), 0.2, 0.4, 0.9) Figure (d, a), 0.2, 0.4, 0.9) b

b

(0.2, 0.1, (0.2, (0.2, (0.2, 0.9) 0.1, 0.1, (0.2, (0.2, 0.9) 0.9) 0.1, 0.9) 0.1, (0.2, 0.9) 0.1, 0.9) 0.1, (0.2, (0.2, 0.8) 0.1, 0.8) 0.1, 0.8)

((a, a), 0.2, 0.4, 0.6)

(0 ( (0 2, 0.8) ((a, b), 0.2, 0.4,0.0.6) .2, ((a, c), 0.2, .0.1, 2, 0 0. ((a, d), 0.2, 0.1, 0.8) 0.4 ((a, b), 0.2, 0.4, 0.6) ((a, c), 0.2, 0.1,.10.8) 1 ,0.1, (0.2, 0.1, (0.2, 0.4, (0.2, , 0 0.8) 0. 0.8) , 0 0.6) ((a, d), 0.2, 0.1, 0.8) 8) 0.8) . .

8 6 (0.2, 0.1, )0.8) (0.2, 0.4, 0.6) (0.2, 0.1, ) ( (0 0 (0 .2 (0 .2, 00.2, 0.1, 0.8) , (0 .2, 0 ((b,(0b), 2, .1, .2 00.2, .1 0.4, 0.6) ((b, .c), .2, .4 ((b, d), 0.2, 0.1, 0.8) , , 0 0.8) 0. 0.1, , 0 0.6) 0. 0.1, (0.2, (0.2, 1, 0.8) 0.8) (0.2,0.40.4, .8 .

1 6 , ) 0. ,0 ) 0 8 . .6) (0 ) (0 ) (0 0.2, 80.4, .2 0.2, 0.1, .2 ((b, b), 0.6) ((b, c), 0.8) , .2 ,0 , 0 0.4, 0.6) ((b, c), 00.2, .1 0.1, 0.8) ((b, d), 0.2, 0.1, 0.8) .4 ((b, b), 0.2, , 0 0.8) .1 0.8) (0.2, 0.1, (0.2, 0.1, , 0 0.6) (0.2, 0.4, ((b, d), 0.2, 0.1, 0.8) , . .

9 0 (0.2, 0.1, .0.8) 0.1,)0.8) 0.4,9)0.6) 9) ((0.2, ((0.2, 0.

0. ( 2, 0.

(0 2, 0 2 0.1, 0.9) ((c, (b), 0.4, 0.9) ((c,(0c), .2 00.2, .2 .4 0. ,0.2, ((c, d), 0.2, 0.1, 0.9) , 0 .1, 0 .1 0.9) , 0 0.4, 2, 00.1, , 0 0.9) (0.2, (0.2, (0.2, , .10.1, .9 0.9) .4 .9 0 0 , . ) ,0 ) 1, .9) 0 . .9 9) 0.

(0 (0 (0 ) 9) .2 .2 0.2, 0.1, .2 0.2, 0.4, 0.9) ((c, d), 0.2, 0.1, 0.9) ((c, b), 0.9) ((c, c), ,0 ,0 ,0 .4 . . ((c, 1 1 c), 0.9) ((c, d), 0.2, 0.1, 0.9) ((c,(0.2, b), 0.2, , 0 0.9) (0.2, 0.4, 0.1, 0.9)0.9) (0.2,0.2, 0.1, 0.9) , 0 0.1, , 0 0.4, . . .

9 9 9 (0.2, 0.1, )0.9) (0.2, 0.1, )0.9) (0(0.2, 0.4,)0.9) (0 (0 .2 .2 ((0.2, (0 .2, 0 0.1, 0.9) ((0.2, , 0 0.1, 0.9) (0.2, 0. , 0. 0.4, 0.9) 0 .2 .1 .2 .1 ((d, d), 0.2, 0.1, 0.9) 2, 4, , ,0 ,0 ,0 0. 0. (d, b), 0.2, 0.4, 0.9) .1 .9 ((d, c), 0.2, 0.1,0.10.9) 4, 9) , 0 .9) ,0 ) 0. .9 .9 9) ) ) (0.2, 0.1, 0.9) (0.2, 0.4, 0.9) (0.2, 0.1, 0.9) 9.(0.2, Neutrosophic digraph G | |SEd), | G = ( RV 2 ).0.2, 0.1, 0.9) 1 (0.2, 0.1, 0.9)2 , SE1 ((d, 0.4,(d, 0.9) 1 c), 2 0.2, 0.1, 0.9)RV ((d, b), 0.2, 0.4,(0.2, 0.9) 0.1, 0.9) 1 2 1 2((d, d), 0.2, 0.1, 0.9) 1 0.4,20.9) ((d, c), 0.2, 0.1, 0.9) (d, b), 0.2,

0.9) 0.4, (0.2, 0.9) 0.4, 0.9) 0.4, 0.9) 0.4, 0.6) 0.4, (0.2, (0.2, (0.2, (0.2, (0.2, 0.4, 0.9) 0.4,(0.2, 0.9) 0.6) 0.4, 0.6) 0.4, (0.2, (0.2,

d(0.2, 0.1, 0.8) d(0.2, 0.1, 0.8)

(0 (0 (0 .2 .2 .2 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .4 .4 .4 ) ) ) (0 (0 (0 .2 .2 .2 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .3 .3 .3 ) ) )

(0 (0 (0 .1 .1 .1 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .5 .5 .5 ) ) ) (0 (0 (0 .1 .1 .1 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .4 .4 .4 ) ) )

(0.1, 0.1, (0.1, (0.1, 0.1, 0.4) 0.4) 0.1, 0.4)

(0 (0 (0 .1, .1 .1, 0.1, 0.

0.1 , 01, 0 ,0 .6) .6 .6) )

b

b b

(0.5, 0.4, 0.1) (0.5, 0.4, 0.1)

0.1, 0.3) (0.2, , 0.1) d(0.5, 0.5, 0.6) (0.3, 0.2 .1) 0 , .2 0 b (0.3,

c(0.2, 0.1, 0.8) b

b b

.1) .1) .1) ,0 ,0 0 ) , 3 . .3) 3) 0.4 0.40.4 ,0 , 0 , 0. .5, 5, 0.1 0.1 .1 (0 (0.

2, 2, 2, 0 (0.

(0. (0.

0.4) 0.4) 0.4) , 0.1, , 0.1, (0.2, 0.1, (0.2 (0.2

b

b(0.9, 0.6, 0.1) .5, (0

b b

b b

c(0.5, 0.6, 0.8)

1

a(0.9, 0.6, 0.1) a(0.9, 0.6, 0.1)

b(0.5, 0.4, 0.3) b(0.5, 0.4, 0.3)

(0.1, 0.1, 0.6)

d(0.2, 0.1, 0.8)

(0.2, 0.5, 0.5)

) ) ) .3 .3 .3 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .2 .2 .2 (0 (0 (0 ) ) .4 .4 .4) ,0 ,0 0 .1 .1 .1, ,0 ,0 0 .2) ) .2 .2, ) (0 (0 , 0.4 , 0(0.,40.4 , 0.1 0.2, .02.,10.1 (0.2 ( (0

b

(0.3, 0.2, 0.1) (0.3, 0.2, 0.1)

.1) .1) .1) ,0 ,0 ,0 .6) .6) 6) 0.2 0.20.2 ,0 , 0 , 0. .3, 3, 0.1 0.1 .1 (0 (0.

1, 1, 1, 0 (0.

(0. (0.

0.5) 0.5) 0.5) , 0.1, , 0.1, (0.1, 0.1, (0.1 (0.1

b b

0.5, 10.5) ) , SE1c(0.5, = (RV G1 (0.2, 0.6, 0.8)

1

.3, (0

a(0.5, 0.4, 0.3) a(0.5, 0.4, 0.3)

b b

b

SE)1 ) (RV,1 ,G G= 1 =(G neutrosophic digraph G1G 1 1) 1 , SE = (RV

b b

b

c(0.5, 0.6, 0.8)

, G1 ) FigureFigure 7: Rough neutrosophic digraph 0.1) = a(0.9, 7. Rough digraph G10.6, =G , G(G 0.3) 1 ). 1 b(0.5, 0.4,neutrosophic 0.1) Figure 7: Rough neutrosophic digraph G(11G1= (G 0.4, (0.3, 0.2, 0.1) (0.5, 1) 1, G

) ) ) .4 .4 .4 ,0 ,0 ,0 .1 .1 .1 ,0 ,0 ,0 .1 .1 1 .

(0 (0 (0 ) ) 5) .5 .5 0. ,0 ,0 1, .1 .1 0. ,0 ,0 1, .1 .1 0. ) ) ) (0 (0 , 0.5 , 0(.,50.5 , 0.1 0.1, .01.,10.1 (0.1 ( (0

(0.(30.3 ( , 0.,20.2 0.3, 0.2 , 0.,10.1 , 0.1 ) ) )

a(0.5, 0.4, 0.3)

, 0.1) (0.3, 0.2 b

d(0.5, 0.6, 0.8) d(0.5, 0.6, 0.8)

(0 (0 (0 (0.(50.5 .2, .2 ( .2, , 0.,40.4 0.5, 0.4 0.1, 0.

0.1 , 0.,10.1 , 0.1 , 01, 0 ,0 ) ) ) .3) .3 .3) ) (0.2, 0.1, (0.2, (0.2, 0.3) 0.1, 0.3) 0.1, 0.3)

d(0.2, 0.5, 0.9) d(0.2, 0.5, 0.9)

(0.1, 0.1, 0.7) ) 0.5, 0.9) (RV = 0.1, G1(0.1, 1 , SE1c(0.2, 0.7) c(0.2, 0.5, 0.9)

(0.2, 0.1, (0.2, (0.2, (0.2, 0.8) 0.1, 0.1, (0.2, (0.2, (0.2, 0.8) 0.1, 0.8) 0.9) 0.1, 0.1, (0.2, (0.2, 0.9) 0.1, 0.9) 0.9) 0.1, 0.9) 0.1, 0.9)

b

b

) 0.4, 0 (0.1, .4, 0) 0 , .1 0 (

(0.2, , 0.1)0.5, 0.5) d(0.5, 0.6, 0.8) (0.3, 0.2 .1) ,0 b (0.3, 0.2

c(0.2, 0.5, 0.9)

b

(0.2, 0.4, 0.6) (0.2, (0.2, 0.4, 0.9) 0.9) 0.6) 0.4, 0.6) 0.4, 0.4, (0.2, (0.2, 0.9) 0.4, 0.9) 0.9) 0.4, 0.9) 0.4, 0.4, (0.2, (0.2, (0.2, (0.2,

d(0.2, 0.5, 0.9)

(0 (0 (0 (0.(20.2 .2, .2 ( .2, , 0.,30.3 0.2, 0.3 0.5, 0.

0.5 , 0.,20.2 , 0.2 , 05, 0 ,0 ) ) ) .5) .5 .

5 ) ) (0.2, 0.4, (0.2, (0.2, 0.5) 0.4, 0.5) 0.4, 0.5)

(0 (0 (0 .1 .1 .1 ,0 ,0 ,0 .3 .3 .3 ,0 ,0 ,0 .8 .8 .8 ) ) ) (0 (0 (0 .2 .2 .2 ,0 ,0 ,0 .3 .3 .3 ,0 ,0 ,0 .6 .6 .6 ) ) )

(0.2, 0.3, (0.2, (0.2, 0.3, 0.6) 0.6) 0.3, 0.6)

(0 (0 (0 .1, .1 .1, 0.1, 0.

0.1 , 01, 0 ,0 .7) .7 .7) )

(0.1, 0.1, 0.7)

b(0.3, 0.8, 0.3) (0.2, 0.3, 0.2), 0.4, 0) b(0.3, 0.8, 0.3) .1 0 ( 0.2) (0.2, 0.3, b

.5) .5) 5) ,0 , 0 , 0.

0.5 0.5 .5 2, 2, 2, 0 (0.

(0. (0.

0.6) 0.6) 0.6) , 0.4, , 0.4, (0.2, 0.4, (0.2 (0.2

b

b b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

Figure 9: G |G = (RV |RV , SE |SE ) b

of 19

.2) .2) .2) ,0 ,0 ,0 0.3 0.30.3 .2, 2, (0 (0.

b b

) ) ) .5 .5 .5 ,0 ,0 ,0 .4 .4 .4 ,0 ,0 ,0 .2 .2 .2 (0 (0 (0 ) ) .6 .6 .6) ,0 ,0 0 .4 .4 .4, ,0 ,0 0 .2) ) .2 .2, ) (0 (0 , 0.6 , 0(0.,60.6 , 0.4 0.2, .02.,40.4 (0.2 ( (0

b

.3) .3) .3) ,0 ,0 ,0 .7) .7) 7) 0.2 0.20.2 ,0 , 0 , 0. .1, 1, 0.1 0.1 .1 (0 (0.

1, 1, 1, 0 (0.

(0. (0.

0.8) 0.8) 0.8) , 0.3, , 0.3, (0.1, 0.3, (0.1 (0.1

b b

(0.1, 0.2, 0.3) (0.1, 0.2, 0.3)

b b

a(0.3, 0.8, 0.3) a(0.3, 0.8, 0.3)

b(0.2, 0.4, 0.6) b(0.2, 0.4, 0.6)

11 b(0.3, 0.8, 0.3)

(0.2, 0.3, 0.2)

.2, (0

.1, (0

b b

a(0.2, 0.4, 0.6) a(0.2, 0.4, 0.6)

a(0.3, 0.8, 0.3)

b(0.2, 0.4, 0.6)

(0.1, 0.2, 0.3)

) ) ) .6 .6 .6 ,0 ,0 ,0 .3 .3 .3 ,0 ,0 ,0 .2 .2 2 .

(0 (0 (0 ) ) ) .8 .8 .8 ,0 ,0 ,0 .3 .3 .3 ,0 ,0 ,0 .1) ) .1 .1 ) (0 (0 , 0.8 , 0(0.,80.8 , 0.3 0.1, .01.,30.3 (0.1 ( (0

(0.(10.1 ( , 0.,20.2 0.1, 0.2 , 0.,30.3 , 0.3 ) ) )

0.6) Mathematics 2018, 18 0.4,6, a(0.2,

b

b

b

Figure 9: G1 |G2 = (RV1 |RV2 , SE1 |SE2 ) Figure 9:=G(µ |RV 2 , SE1 |SE2 ) (x)(RV ∧ µ1RV 1 |G 21= RV 2 (x2 )) ∧ (µRV1 (x) ∧ µRV2 (y2 )) = µRV |RV2 (x, x2 ) ∧ µRV1 |RV2 (x, y2 ) = (µRV1 1 (x) ∧ µRV2 (x2 )) ∧ (µRV1 (x) ∧ µRV2 (y2 )) = (µRV1 (x) ∧ µRV 2 (x2 )) ∧ (µRV1 (x) ∧ µRV2 (y2 )) = µRV1 |RV2 (x, x2 ) ∧ µRV1 |RV2 (x, y2 ) = µRV1 |RV2 (x, x2 ) ∧ µRV1 |RV2 (x, y2 ) 12 12 12

Mathematics 2018, 6, 18

12 of 19

((a, b), 0.3, 0.6, 0.3) ((a, c), 0.3, 0.5, 0.6) (0.3, 0.6, 0.3) (0.3, 0.5, 0.6) (0.3, 0.5, 0.6)

((c, a), 0.5, 0.6, 0.8) b

b

((a, d), 0.3, 0.5, 0.6)

b

((b, d), 0.3, 0.5, 0.6)

(0.3, 0.5, 0.6)

(0.3, 0.5, 0.6) b

(0.3, 0.5, 0.6)

(0 .3 ,0 .5 ,0 .8 ) ((c, b), 0.5, 0.6, 0.8) ((c, c), 0.5, 0.5, 0.8) (0.5, 0.5, 0.8) (0.5, 0.5, 0.8) (0 .3 ,0 .5 ,0 .8 )

b

b

(0.5, 0.6, 0.8)

(0.5, 0.6, 0.8) (0.5, 0.6, 0.8)

(d, a), 0.5, 0.6, 0.8)

(0 .3 ,0 .6 , .0 .8 )

b

(0.3, 0.5, 0.6)

(0.3, 0.6, 0.3)

(0.3, 0.6, 0.3)

(0.5, 0.5, 0.8)

b

(0 (0 .3 .3 ,0 ,0 .5 .5 ,0 ,0 .6 .6 ) ) ((b, b), 0.3, 0.6, 0.3) ((b, c), 0.3, 0.5, 0.6) (0.3, 0.5, 0.6) (0.3, 0.5, 0.6)

(0.3, 0.6, 0.3)

((b, a), 0.3, 0.6, 0.3)

(0 .3 ,0 .6 ,0 .3 )

b

b

(0.3, 0.6, 0.3)

b

(0.3, 0.6, 0.3)

b

(0 (0 (0 .5 .5 .5 ,0 ,0 ,0 .5 .6 .5 ,0 ,0 ,0 .8 .8 .8 ) ) ) (0.5, 0.5, 0.8) (0.5, 0.6, 0.8) (0.5, 0.5, 0.8) (d, b), 0.5, 0.6, 0.8) ((d, c), 0.5, 0.5, .8)

b

b

b

((c, d), 0.5, 0.5, 0.8) b

(0.5, 0.5, 0.8)

((a, a), 0.3, 0.6, 0.3)

((d, d), 0.5, 0.5, 0.8)

Figure 10. Neutrosophic digraph G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ).

Figure 10: G1 |G2 = (RV1 |RV2 , SE1 |SE2 )

Theorem 3. The rejection of two rough neutrosophic digraphs is a rough neutrosophic digraph.

µSE1 |SE2 ((x, x2 ), (x, y2 )) = min{µRV1 |RV2 (x, x2 ), µRV1 |RV2 (x, y2 )},

Proof. Let G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) be two rough neutrosophic digraphs. Let σ =1 |SE ((x, x ), (x, y2 )) rejection = σRV1 (x) ∧ σRVG2 (x 2 ) ∧ σRV2 (y2 ) G = G1 | GSE ( G12| G2 , G12| G2 ) be the of G1 and 2 2 , where G1 | G2 = ( RV1 | RV2 , SE1 | SE2 ) and G1 | G2 = ( RV1 | RV2 , SE1 |SE2 ). To prove G1= G1∧ is 2a(x rough neutrosophic is enough | Gσ2RV (x) (σRV1 (x) ∧digraph, σRV2 (yit2 )) = that (σRV 2 )) ∧ to show that SE1 |SE2 and SE1 |SE2 are neutrosophic relations on RV1 | RV2 and RV1 | RV2 , respectively. = σRV |RV2 (x, x2 ) ∧ σRV1 |RV2 (x, y2 ) First, we show that SE1 |SE2 is a neutrosophic1 relation on RV1 | RV2 .

σSE1 |SE2 ((x, x2 ), (x, y2 )) = min{σRV1 |RV2 (x, x2 ), σRV1 |RV2 (x, y2 )},

If xλ∈ RV1 , ( x((x, SE(x, 2 , y2 )x6 ∈ ), 2 , then y )) SE1 |SE2

2

2

= λRV1 (x) ∨ λRV2 (x2 ) ∨ λRV2 (y2 )

λRV ))RV∨2 ((λ µSE1 |SE2 (( x, x2 ), ( x, y2 ))==(λ µ RV x ) ∧∨ µ RV ( x (x ) ∧2µ y2 )RV1 (x) ∨ λRV2 (y2 )) RV11((x) 2 22 ==λ(RV x22)( x∨2 ))λRV ( x2) (x, ∧ µ RV x ) ∧yµ µ RV ∧ (1µ|RV 2 )RV2 (y2 )) RV12((x, 1 |RV 1

µ RV1 | RVRV ( x, x2 ) ∧ µ RV ( x, y |RV ) (x, y2 )}, λSE1 |SE2 ((x, x2 ), (x, y2 )) ==max{λ (x, x12|), RVλ 2 1 |RV 2 RV1 2 2 2 µSE1 |SE2 (( x, x2 ), ( x, y2 )) = min{µ RV1 | RV2 ( x, x2 ), µ RV1 | RV2 ( x, y2 )} σSE1 |SE2 (( x, x2 ), ( x, y2 )) = σRV1 ( x ) ∧ σRV2 ( x2 ) ∧ σRV2 (y2 )

If x1 y1 6∈ SE1 , z ∈ RV2 , then

= (σRV1 ( x ) ∧ σRV2 ( x2 )) ∧ (σRV1 ( x ) ∧ σRV2 (y2 ))

σRV1 | RV2 ( x, x2 ) ∧ σRV1 | RV2 ( x, y2 ) µSE1 |SE2 (x1 , z)(y1 , z) = = µRV 1 (x1 ) ∧ µRV1 (y1 ) ∧ µRV2 (z)

σSE1 |SE2 (( x, x2 ), ( x, y2 )) = min{σRV1 | RV2 ( x, x2 ), σRV1 | RV2 ( x, y2 )}

= (µRV1 (x1 ) ∧ µRV2 (z)) ∧ (µRV1 (y1 ) ∧ µRV2 (z)

λSE1 |SE2 (( x, x2 ), ( x, y2 )) = λ RV1 ( x ) ∨ λ RV2 ( x2 ) ∨ λ RV2 (y2 )

= µRV1 |RV2 (x1 , z) ∧ µRV1 |RV2 (y1 , z)

= (λ RV ( x ) ∨ λ RV ( x2 )) ∨ (λ RV ( x ) ∨ λ RV (y2 ))

2 1 (y1 , z)}, µSE1 |SE2 (x1 , z)(y1 , z) = min{µ1RV1 |RV2 (x2 1 , z), µRV1 |RV 2

= λ RV1 | RV2 ( x, x2 ) ∨ λ RV1 | RV2 ( x, y2 )

σSE1 |SE2 (x1 , z)(y1 , z) = σRV1 (x1 ) ∧ σRV1 (y1 ) ∧ σRV2 (z)

λSE1 |SE2 (( x, x2 ), ( x, y2 )) = max{λ RV1 | RV2 ( x, x2 ), λ RV1 | RV2 ( x, y2 )}

If x1 y1 6∈ SE1 , z ∈ RV2 , then

= (σRV1 (x1 ) ∧ σRV2 (z)) ∧ (σRV1 (y1 ) ∧ σRV2 (z) = σRV1 |RV2 (x1 , z) ∧ σRV1 |RV2 (y1 , z)

σSE1 |SE2 (x1 , z)(y1 , z) = min{σRV1 |RV2 (x1 , z), σRV1 |RV2 (y1 , z)}, µSE1 |SE2 ( x1 , z)(y1 , z) = µ RV1 ( x1 ) ∧ µ RV1 (y1 ) ∧ µ RV2 (z)

λSE1 |SE2 (x1 , z)(y1 , z) ==λ(RV )∨ µ 1 (x ( x1 )) ∨ ∧λ µ RV1((y z))1∧ (µλRV(y2 (z) )∧µ RV1

RV2

1

RV1

1

RV2 ( z )

(x ) ∨ λ∧ µ2RV (z)) ∨(y(λ ==(λµRV z) 1 (y1 ) ∨ λRV2 (z) 1 , RV RV11| RV21( x1 , z ) RV 1 | RV2

(x1 2, (z) z) ) =λmin µ RV x1 ,∨z)λ , µRV y11, z, )} µSE1 |SE2 ( x1 , z)(y1 , z= RV1{ |RV 2 1 | RV RV11|RV | RV22((y

λSE1 |SE2 (x1 , z)(y1 , z) = max{λRV1 |RV2 (x1 , z), λRV1 |RV2 (y1 , z)},

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σSE1 |SE2 ( x1 , z)(y1 , z) = σRV1 ( x1 ) ∧ σRV1 (y1 ) ∧ σRV2 (z)

= (σRV1 ( x1 ) ∧ σRV2 (z)) ∧ (σRV1 (y1 ) ∧ σRV2 (z)

= σRV1 | RV2 ( x1 , z) ∧ σRV1 | RV2 (y1 , z)

σSE1 |SE2 ( x1 , z)(y1 , z) = min{σRV1 | RV2 ( x1 , z), σRV1 | RV2 (y1 , z)}

λSE1 |SE2 ( x1 , z)(y1 , z) = λ RV1 ( x1 ) ∨ λ RV1 (y1 ) ∨ λ RV2 (z)

= (λ RV1 ( x1 ) ∨ λ RV2 (z)) ∨ (λ RV1 (y1 ) ∨ λ RV2 (z) = λ RV1 | RV2 ( x1 , z) ∨ λ RV1 | RV2 (y1 , z)

λSE1 |SE2 ( x1 , z)(y1 , z) = max{λ RV1 | RV2 ( x1 , z), λ RV1 | RV2 (y1 , z)} If ( x1 , y1 ) 6∈ SE1 , ( x2 , y2 ) 6∈ SE2 , then σSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = σRV1 ( x1 ) ∧ σRV1 (y1 ) ∧ σRV2 ( x2 )σRV2 (y2 )

= (σRV1 ( x1 ) ∧ σRV2 ( x2 )) ∧ (σRV1 (y1 ) ∧ σRV2 (y2 ))

= σRV1 | RV2 ( x1 , x2 ) ∧ σRV1 | RV2 (y1 , y2 )

σSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = min{σRV1 | RV2 ( x1 , x2 ), σRV1 | RV2 (y1 , y2 )}

λSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = λ RV1 ( x1 ) ∨ λ RV1 (y1 ) ∨ λ RV2 ( x2 )λ RV2 (y2 )

= (λ RV1 ( x1 ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 (y1 ) ∨ λ RV2 (y2 )) = λ RV1 | RV2 ( x1 , x2 ) ∨ λ RV1 | RV2 (y1 , y2 )

λSE1 |SE2 (( x1 , x2 ), (y1 , y2 )) = max{λ RV1 | RV2 ( x1 , x2 ), λ RV1 | RV2 (y1 , y2 )} Thus, from the above, it is clear that SE1 |SE2 is a neutrosophic relation on RV1 | RV2 .

Similarly, we can show that SE1 |SE2 is a neutrosophic relation on RV1 | RV2 . Hence, G = ( G1 | G2 , G1 | G2 ) is a rough neutrosophic digraph.

Definition 8. The tensor product of two rough neutrosophic digraphs G1 and G2 is a rough neutrosophic digraph G = ( G1 ? G2 , G1 ? G2 ), where G1 ? G2 = ( RV1 ? RV2 , SE1 ? SE2 ) and G1 ? G2 = ( RV1 ? RV2 , SE1 ? SE2 ) are neutrosophic digraphs, respectively, such that

(1)

µ RV1 ? RV2 ( x, y) = min{µ RV1 ( x ), µ RV2 (y)} σRV1 ? RV2 ( x, y) = min{σRV1 ( x ), σRV2 (y)}

λ RV1 ? RV2 ( x, y) = max{λ RV1 ( x) , λ RV2 (y)} ∀ ( x, y) ∈ RV1 × RV2 µ RV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = min{µSE1 ( x1 , y1 ), µSE2 ( x2 , y2 )} σRV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = min{σSE1 ( x1 , y1 ), σSE2 ( x2 , y2 )}

λ RV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = max{λSE1 ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2

(2)

µ RV1 ? RV2 ( x, y) = min{µ RV1 ( x ), µ RV2 (y)} σRV1 ? RV2 ( x, y) = min{σRV1 ( x ), σRV2 (y)} λ RV1 ? RV2 ( x, y) = max{λ RV1 ( x) , λ RV2 (y)} ∀ ( x, y) ∈ RV1 × RV2

µ RV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = min{µSE ( x1 , y1 ), µSE2 ( x2 , y2 )} 1

σRV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = min{σSE ( x1 , y1 ), σSE2 ( x2 , y2 )} 1

λ RV1 ? RV2 (( x1 , x2 ), (y1 , y2 )) = max{λSE ( x1 , y1 ), λSE2 ( x2 , y2 )} ∀ ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2 1

Mathematics 2018, 6, 18

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Example 6. Let V1∗ = { a, b, c} and V2∗ = {w, x, y, z} be two crisp sets. Let G1 = ( G1 , G1 ) and G2 = ( G2 , G2 ) be two rough neutrosophic digraphs on V1∗ and V2∗ , respectively, where G1 = ( RV1 , SE1 ) and G1 = ( RV1 , SE1 ) are neutrosophic digraphs as shown in Figure 11. G2 = ( RV2 , SE2 ) and G2 = ( RV2 , SE2 ) are also neutrosophic digraphs, as shown in Figure 12. a(0.2, a(0.2,0.1, 0.1,0.6) 0.6)

a(0.2, a(0.2,0.1, 0.1,0.3) 0.3) b

a(0.2, 0.1, 0.6)

(0.2, (0.2,0.1, 0.1,0.5) 0.5) c(0.2, b(0.8, c(0.2,0.1, 0.1,0.6) 0.6) b(0.8,0.6, 0.6,0.5) 0.5) G G11 b

(0.

2, 0.1 ,0 .5)

b

b

b

a(0.2, 0.1, 0.3) b

(0.2, (0.2,0.1, 0.1,0.5) 0.5) c(0.9, b(0.8, 0.6, 0.5) c(0.9,0.3, 0.3,0.4) 0.4) b(0.8, 0.6, 0.5) G G11

(0.

2, 0.1 ,0 .5)

b

b

) 3)) 0.3 00..3 .11,, .1, 0 ,00. .3)) .3) ,00.3 .2, ..22, ,0 0 ( ((00 0..11, 0.1 2, .22,,0 (0. ((00.

((00..2 2,, 0 0..11 ,,00.5 .5))

b

) ) ..55) 0.5 , ,,00 0..11 0.1 )) .5) .11,,0 0..55 1, ,0 1,,0 (0. ((00. .

1 .

0 0.1 1, .11,,0 (0. ((00.

((00..2 2,, 0 0..11 ,,00.5 .5))

b

b

b

b

b

(0.2, 0.5) Figure 11.

Roughneutrosophic neutrosophic digraph G(0.2, = ( G= , (G G 1G 1 )1.,, G Figure 11: Rough digraph ) Figure 11:0.1, Rough neutrosophic digraph G110.1, =10.5) (G 1 G11 ) b

b

c(0.2, 0.1, 0.6)

b(0.8, 0.6, 0.5)

G10.6) w(0.4, w(0.4,0.2, 0.2, 0.6) b

b

b

b

4)) 00..4 .11,, ,00. ..44, ((00

4)) 00..4 .11,, ,00. ..44, ((00

((00. .22,, 00..11 ,,00. .11))

b

b

c(0.9, 0.3, 0.4) b(0.8, 0.6, 0.5) w(0.8, G10.5, w(0.8, 0.5,0.1) 0.1)

((00. .33,, 00..22 ,,00. .11))

Figure 11: Rough neutrosophic digraph G1 = (G1 , G1 ) w(0.4, 0.2, 0.6)

w(0.8, 0.5, 0.1)

(0 .2, 0.1 ,0 .1) b

b

b

G G22

(0.2, 0.1, 0.1)(0.4, 0.2, 0.3)

b

b

b

(0.3, (0.3,0.2, 0.2,0.1) 0.1) (0.4, (0.4,0.2, 0.2,0.3) 0.3) x(0.8, x(0.8,0.5, 0.5,0.1) 0.1) y(0.8, y(0.8,0.5, 0.5,0.1) 0.1) z(0.9, z(0.9,0.8, 0.8,0.4) 0.4) b

b

b

.4) ,0 0.1 .4, (0

b

.4) ,0 0.1 .4, (0

(0.2, (0.2,0.1, 0.1,0.1)(0.4, 0.1)(0.4,0.2, 0.2,0.3) 0.3) x(0.4, z(0.9,0.8, 0.8,0.4) 0.4) x(0.4,0.2, 0.2,0.6) 0.6) y(0.4, y(0.4,0.2, 0.2,0.6) 0.6) z(0.9,

(0 .3, 0.2 ,0 .1)

b

b

G G22

(0.3, 0.2, 0.1) (0.4, 0.2, 0.3)

b

b

Figure Roughneutrosophic neutrosophic digraph G2 G = G , (G Figure 12: Rough digraph 2 )2.,, G Figure 12:12.

Rough neutrosophic digraph G22(G= =20.5, (G 22)) 0.8, 0.4) 2 G x(0.8, 0.5, 0.1) y(0.8, 0.1) z(0.9, x(0.4, 0.2, 0.6) y(0.4, 0.2, 0.6) z(0.9, 0.8, 0.4) The tensor product of G1 and is G = G1 ? G2 = ( G1 ? G2 , G1 ? G2 )((a, , where G1 ?0.6) G = G2 G2 ((a, ((a, ((a, ((a,w), w),0.2, 0.2,0.1, 0.1,0.6) 0.6) z),0.2, 0.2,0.1, 0.1, 0.6)2 ((a,y), y),0.2, 0.2,0.1, 0.1,0.6) 0.6)G2 ((a,z), ((a,x), x),0.2, 0.2,0.1, 0.1,0.6) 0.6) ( RV1 ? RV2 , SE1 ? SE2 ) and G1 ? G2 = ( RV1 ? RV2 , SE1 ? SE2 ) are neutrosophic digraphs, as shown in ) , 0..55) b

b

b

b

b

b

b

b

b

b

b

b

b

)) 00..55 .0.11,, 0 1,, ((00..1

b

b

b

b

((c, ((c,x), x),0.2, 0.2,0.1, 0.1,0.6) 0.6)

((c, ((c,w), w),0.2, 0.2,0.1, 0.1,0.6) 0.6)

) 0.5 .1, 0 1, Figure 13: Figure 13: (0.

b

((00..1 1,,00 ..11,, 0 0..55) )

b

b

b b

b

b

b

((00..2 2,,00..1 1,,00..5 5))

b

b

((c, ((c,z), z),0.2, 0.2,0.1, 0.1,0.6) 0.6) ((c,y), y),0.2, 0.2,0.1, 0.1,0.6) 0.6) ((c,

(0.2 , 01.1⋆ G G11 ⋆⋆ G RV22,,SE SE SE22)) G22 = = (RV (RV11 ⋆⋆ RV 1 ,⋆0.SE 5 b

b

((c, x), 0.2, 0.1, 0.6)

((c, w), 0.2, 0.1, 0.6)

b

(0. ((00.

1, 0 .22, .1, , 00.

0.5 .11, , 00. ) .55) )

b

b

(0 .2 ,0 .1 ,0 .5 )

b

b

(0.1 ((00..11 , 0. ,,00..1 1, 0 1,,00. .5) .55)) (0 ((00. .2 .22, ,0 , 00. .1 .11, ,0 , 00. .5 .55) ) )

b

b

) 6)) .6 0..6 ,0 2,,0 .2 0..2 ,0 4,,0 .4 0..4 ,0 ),,0 y) yy) b, bb,, (( ((((

((b, w),0.4, 0.4,0.2, 0.2,0.6) 0.6) ((b, w), 0.4, 0.2, 0.6) ((b, w), (0. ((00..1 1, 0 1 , 0 , 0..1 .1, ( 1,,00.

(0 0.(500. .55)) .2 ).22, ,0 , 0 0..1 .1 1,, 0 ,0 0..5 .5 5)) )

b

0.5)z), 0.8, 0.6, 0.5) 0.6,((b, 0.8,0.6, z),0.8, ((b,z), 0.5) ((b, ) 5)) ) 5)) 0.5 ,00..5 .5 0..5 .1, 0..11, ,0 1,,0 1, 0 .1 0..11,,0 0..1 (0. , 0 ((0 ,,0 .2 ..22 (0 ((00

b

) .55)) , 0.5 .11,,00. , 0.1 .1,,00.

(0.1 ((00.1

((00. , 0..11, 0 .11,,0 ((00..22, 0 14, respectively.

Figure 12: Rough neutrosophic digraph G 0 .

2 = (G2 , G2 ) 1 . ((00 1,,0 0..55 ..22 )) ,,00 1,,0 ((a, w), 0.2, 0.1,..10.6) ((a, z), 0.2, 0.1, 0.6) ((a, y), 0.2, 0.1, 0.6) ((a, x), 0.2, 0.1, 0.6) 0..5 5)) 5) .

0 , (0. , 0.1 1, (0.2 0.1 (0 ,0 .2 .5) ,0 .1 ,0 .5 )

)) 5) ..55 0. ,,00 1, ) . ..11 ) ) 6 .6 ,,00 ) ,0 0.).6 ..22 ,0 0..55) .2 2,.,05 0 0 , 0 .

2 , .2 ( ((00 0..11, 0.,100. ,0 .11,,0 .1, .44, .4 0 ( ((00. ,00. ,0 , ) x) xx) b, bb,, (( ((((

Figures 13 and

b

b

b

)

b

((c, y), 0.2, 0.1, 0.6) ((c, z), 0.2, 0.1, 0.6)

Theorem Theorem 2.4.

2.4. The The tensor tensor product product of of two two rough rough neutrosophic neutrosophic digraphs digraphs is is aa rough rough neutrosophic neutrosophic FigureFigure 13. Neutrosophic G1 ?1G⋆2 RV = (2RV , SE12 )? SE2 ).

1 ? RV 13: G1 ⋆digraph , SE G2 = (RV digraph.

1 ⋆2SE digraph.

Proof.

Proof. Let Let G G11 = = (G (G11,,G G11)) and and G G22 = = (G (G22,,G G22)) be be two two rough rough neutrosophic neutrosophic digraphs. digraphs. Let Let G G= = Theorem 2.4.

The tensor product of two rough neutrosophic digraphs is a rough neutrosophic ⋆G , = (RV G G ⋆G ) be the tensor product of G and G , where G ⋆RV , SE ⋆G ⋆G , = (RV G11⋆G ⋆G22 = = (G (G G ⋆G ) be the tensor product of G and G , where G ⋆RV , SE ⋆G 11 22 11 22 11 22 11⋆⋆ 11 22 11 22 digraph. ) and G ⋆ G = (RV ⋆ RV , SE ⋆ SE ).

To prove that G = G G is a rough neutrosophic ⋆ SE SE22 ) and G11 ⋆ G22 = (RV11 ⋆ RV22 , SE11 ⋆ SE22 ). To prove that G = G11 ⋆ G22 is a rough neutrosophic digraph it digraphLet it is is enough enough to to show show that that SE SE11 ⋆⋆ SE SE22 and and SE SE11 ⋆⋆ SE SE22 are are neutrosophic neutrosophic relations relations on on Proof.

G 1 = (G1 , G1 ) and G2 = (G2 , G2 ) be two rough neutrosophic digraphs. Let G = ⋆ RV and RV ⋆ RV , respectively.

First, we show that SE ⋆ SE is a neutrosophic relation RV RV⋆G 11 ⋆ RV22 and RV11 ⋆ RV22 , respectively. First, we show that SE11 ⋆ SE22 is a neutrosophic relation G 1 2 = (G1 ⋆G2 , G1 ⋆G2 ) be the tensor product of G1 and G2 , where G1 ⋆G2 = (RV1 ⋆RV2 , SE1 ⋆ . on RV11 ⋆⋆ RV on ) and RV G 22⋆. G = (RV ⋆ RV , SE ⋆ SE ). To prove that G = G ⋆ G is a rough neutrosophic SE RV 2

1

2

1

2

1

2

1

2

digraph it is enough to show that SE1 ⋆ SE2 and SE1 ⋆ SE2 are neutrosophic relations on RV1 ⋆ RV2 and RV1 ⋆ RV2 , respectively. First, we show that SE1 ⋆ SE2 is a neutrosophic relation 15 on RV1 ⋆ RV2 .

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Mathematics 2018, 6, 18

15 of 19

((a, x), 0.2, 0.1, 0.3)

((a, z), 0.2, 0.1, 0.4) ) 5 .

0 , , 0.1 (0.2

((a, y), 0.2, 0.1, 0.3)

b

b

((b, w), 0.8, 0.5, 0.5) (0.

2, 0 .1, (0 0.3 .2 ) ,0 .1 ,0 .5 )

b

(0.

2, 0 .1, 0.3 )

b

.4) ,0 0.1 , 2 (0.

b

b

(0 .2 ,0 .1 ,0 .5 )

(0.2 , 0.

1, 0 .3)

b

((c, x), 0.8, 0.3, 0.4)

((b, z), 0.8, 0.6, 0.5) ) ) 0.3 .5 .1, ,0 2, 0 .1 (0. ,0 .2 (0

5) 0.

5, 0.

8, 0. ),

5) 0.

5, 0.

8, 0. ),

x b, ((

y b, ((

) , 0.3 , 0.1 (0.2

(0.

2, 0.1 (0 ,0 .4) .2 ,0 .1 ,0 .5 )

((c, w), 0.8, 0.3, 0.4)

b

) .5 ,0 .1 ,0 .3) .2 1, 0 (0 , 0.

(0.2

b

b

(0 .2 ,0 .1 ,0 .5 )

((a, w), 0.2, 0.1, 0.3)

(0.2 , 0.1 , 0.5 ) b

b

((c, y), 0.8, 0.3, 0.4) ((c, z), 0.9, 0.3, 0.4)

Figure 14.Figure Neutrosophic G1 ?1 G⋆2RV = 2(,RV RV 14: G1 ⋆digraph G2 = (RV SE SE 2 , SE 1?

1⋆ 2 ) 1 ? SE2 ).

Theorem tensor two rough neutrosophic digraphs is a rough neutrosophic digraph.

SE If (x1 ,4.y1 )The ∈ SE 1 , (xproduct 2 , y2 ) ∈ of 2 , then , x2 ), ))2 ,= ⋆SE 1 , y(2G 1 , yrough 1 ) ∧ µSE Proof. Let G1 µ=SE(1G G(y G2µ)SE be1 (x two neutrosophic digraphs. Let G = G1 ? G2 = 2 ((x 2 (x2 , y2 ) 2 = 1 ) 1and 1, G (x1 )G∧2 ,µwhere (x(2RV ) ∧1µ?RV )) 1 ? SE2 ) and ( G1 ? G2 , G1 ? G2 ) be the tensor product≤of(µGRV G∧ GRV RV RV1 (y1 )) 2 ,2SE 1 1and 2 (y 1 ?(µ 2 2= G1 ? G2 = ( RV1 ? RV2 , SE1 ? SE2 ). To prove that G = G G is a rough neutrosophic ? (x ) ∧ µ (x )) ∧ (µ (y ) ∧ µ (y = (µRV 2 1 2 1 RV RV1 1 RV2 2 )) digraph, it is 1 enough to show that SE1 ? SE2 and SE1 ?=SE are neutrosophic relations on RV 2 1 ⋆RV2 (x1 , x2 ) ∧ µRV1 ⋆RV2 (y1 , y2 ) 1 ? RV2 and RV1 ? RV2 , µRV respectively. First, we show is a neutrosophic ((x , xthat ), (ySE, 1y ?))SE ≤2 min{µ (x , xrelation ), µ on RV (y 1,?y RV )},2 . µ SE1 ⋆SE2

1

2

1

2

RV1 ⋆RV2

1

2

RV1 ⋆RV2

1

2

σSE1 ⋆SE2 ((x1 , x2 ), (y1 , y2 )) = σSE1 (x1 , y1 ) ∧ σSE2 (x2 , y2 ) If ( x1 , y1 ) ∈ SE1 , ( x2 , y2 ) ∈ SE2 , then ≤ (σRV1 (x1 ) ∧ σRV1 (y1 )) ∧ (σRV2 (x2 ) ∧ σRV2 (y2 ))

σRV ∧ (σ (y ) ∧ σRV2 (y2 )) 1 ) )∧∧ 2 (x µSE1 ?SE2 (( x1 , x2 ), (y1 , y2 )) ==µ(σ ( x1 (x µSE ( 2x)) 2 , y2 ) RV1 1 1, y 1 SERV 2 1 1 ⋆RV2 (x1 , x2 ) ∧ σRV1 ⋆RV2 (y1 , y2 ) ≤=(σ µRV RV1 ( x1 ) ∧ µ RV1 ( y1 )) ∧ ( µ RV2 ( x2 ) ∧ µ RV2 ( y2 )) σSE1 ⋆SE2 ((x1 , x2 ), (y1 , y2 )) ≤ min{σRV1 ⋆RV2 (x1 , x2 ), σRV1 ⋆RV2 (y1 , y2 )}, = (µ RV ( x1 ) ∧ µ RV2 ( x2 )) ∧ (µ RV (y1 ) ∧ µ RV2 (y2 )) λSE1 ⋆SE2 ((x1 , x2 ), (y1 , y2 )) = λSE11 (x1 , y1 ) ∨ λSE2 (x2 , y2 ) 1 = µ RV1 ? RV2 ( x1 , x2 ) ∧ µ RV1 ? RV2 (y1 , y2 ) ≤ (λRV1 (x1 ) ∨ λRV1 (y1 )) ∨ (λRV2 (x2 ) ∨ λRV2 (y2 )) µSE1 ?SE2 (( x1 , x2 ), (y1 , y2 )) ≤ min{µ RV1 ? RV2 ( x1 , x2 ), µ RV1 ? RV2 (y1 , y2 )} = (λRV1 (x1 ) ∨ λRV2 (x2 )) ∨ (λRV1 (y1 ) ∨ λRV2 (y2 )) σSE1 ?SE2 (( x1 , x2 ), (y1 , y2 )) ==σλSE1 ( x1 , y1(x ) ∧ σ ) 2∨( xλ2RV , y2 ) (y , y ) 1 2 RV1 ⋆RV2 1 , x2SE 1 ⋆RV2 σRV ( y )) ∧ ( σ σRV1 ( xRV ( x22(y ) 1∧, yσ2RV RV 1 ) 1∧⋆RV 1 λSE1 ⋆SE2 ((x1 , x2 ), (y1 , y2 ))≤≤(max{λ (x , x ), λ )}.2 (y2 )) 2 1 1 2 RV ⋆RV 2 1

= (σRV1 ( x1 ) ∧ σRV2 ( x2 )) ∧ (σRV1 (y1 ) ∧ σRV2 (y2 ))

= σ 1 ? RV2 ( x1 , x2 ) ∧ σRV1 ? RV2 (y1 , y2 ) SE2 is a neutrosophic relation on RV1 ⋆ RV2 .

Thus, from above it is clear that SE1 ⋆RV x1 , x2that ), (y1SE , y21))⋆ SE ≤ min ( x1 , x2 )relation , σRV1 ? RV (y1 , y12⋆)}RV2 . Hence, G = neutrosophic on Similarly,σSE we1 ?can show SE2 (( 2 is {aσRV 2 RV 1 ? RV2 (G1 ⋆ G2 ,λG ⋆ G ) is a rough neutrosophic digraph.

1 ?SE 2(( x1 , x2 ), ( y1 , y2 )) = λ SE ( x1 , y1 ) ∨ λ SE ( x2 , y2 ) SE 1

2

1

2

(λ RV1 ( x1 ) ∨digraphs λ RV1 (y1 )) (λ RV2 ( x2 ) ∨ λ RV (y2 )) digraphs, Remark 2.1. Hybrid model rough ≤ neutrosophic are∨generalization of 2fuzzy which can be used to represent the = relations and flows between data.

Rough neutrosophic di(λ RV1 ( x1 ) ∨ λ RV2 ( x2 )) ∨ (λ RV1 (y1 ) ∨ λ RV2 (y2 )) graphs can be incrementally modified by deleting or adding elements, or they can be built by = λdigraphs ∨ λ RVneutrosophic (y1 , y2 ) operations. Our proRV1 ? RV2 ( xusing 1 , x2 )rough 1 ? RV2 combining multiple rough neutrosophic λSE1 ?SE2are (( x1methods , x2 ), (y1of , y2construction )) ≤ max{λof ( x1 , x2neutrosophic ), λ RV1 ? RV2 (ydigraphs posed operations RV1new ? RV2rough 1 , y2 )} from old ones.

Thus, from the above, it is clear that SE1 ? SE16 2 is a neutrosophic relation on RV1 ? RV2 .

Similarly, we can show that SE1 ? SE2 is a neutrosophic relation on RV1 ? RV2 . Hence, G = ( G1 ? G2 , G1 ? G2 ) is a rough neutrosophic digraph.

Remark 1. Hybrid-model rough neutrosophic digraphs are generalization of fuzzy digraphs and can be used to represent the relations and flows between data. Rough neutrosophic digraphs can be incrementally modified

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by deleting or adding elements, or they can be built by combining multiple rough neutrosophic digraphs using rough neutrosophic operations. Our proposed operations are methods of construction of new rough neutrosophic digraphs from old digraphs. By introducing the rough neutrosophic digraph theory, we have proposed a novel decision-making method based on rough neutrosophic information. It provides a new viewpoint for rough neutrosophic information. The given decision-making method can be used to evaluate upper and lower approximations to develop deep considerations of the problem.

3. Application In this application, we use the concept of a rough neutrosophic digraph for decision-making in real-life problems. To obtain the optimal decision, we use the following formula: Sij = ( TSij , ISij , FSij ) where  TRV (vi )∗ TRV (v j ) 

TSij = TSE ⊕ TSE (vi , v j ) =   3 − T ( v ,v )+ T  SE i j SE ( vi ,v j )− TSE ( vi ,v j )∗ TSE ( vi ,v j )   IRV (vi )∗ IRV (v j )

ISij = ISE ⊕ ISE (vi , v j ) = 3− ISE (vi ,v j )+ ISE (vi ,v j )− ISE (vi ,v j )∗ ISE (vi ,v j )    FRV (vi )∗ FRV (v j )  

 FSij = FSE ⊕ FSE (vi , v j ) =

(1)

3− FSE (vi ,v j )+ FSE (vi ,v j )− FSE (vi ,v j )∗ FSE (vi ,v j )

Flight planning is the process of producing a flight plan to describe a proposed aeroplane fight.

Flight plans generally include basic information such as departure and arrival points, estimated time en route, and alternate airports in case of bad weather. The presented application provides alternate airports for a plane in the case of bad weather.

We suppose V ∗ = {Chicago(CHI),Beijing(BJ),Lahore(LHR),Paris(PAR),Istanbul(IST)} is the set of cities under consideration and R is an equivalence relation on V ∗ , where equivalence classes represent cities having the same characteristics.   1 0 0 1 0  0 1 1 0 1      R= 0 1 1 0 1     1 0 0 1 0  0 1 1 0 1

We assume that a flight Boeing 747 of Pakistan International Airways (PIA) travels to these cities.

In the case of bad weather, the flight will be directed to the city with the best weather condition among the cities under consideration.

Let V = {(CHI, 0.1, 0.2, 0.8), (BJ, 0.9, 0.7, 0.5), ( LHR, 0.8, 0.4, 0.3), (PAR, 0.6, 0.5, 0.4), ( IST, 0.2, 0.4, 0.6)} be a NS on V ∗ that describes the characteristic of each city, and RV = ( RV, RV ) be a rough NS, where RV and RV are lower and upper approximations of V, respectively, as follows: RV = RV =

{(CH I, 0.1, 0.2, 0.8), ( BJ, 0.2, 0.4, 0.6), ( LHR, 0.2, 0.4, 0.6), ( PAR, 0.1, 0.2, 0.8), ( IST, 0.2, 0.4, 0.6)}} {(CH I, 0.6, 0.5, 0.4), ( BJ, 0.9, 0.7, 0.3).( LHR, 0.9, 0.7, 0.3), ( PAR, 0.6, 0.5, 0.4), ( IST, 0.9, 0.7, 0.3)}

Let E∗ = {( BJ, CH I ), ( LHR, CH I ), ( BJ, LHR), ( IST, BJ ), ( PAR, BJ ), ( PAR, LHR)} be a subset of V ∗ × V ∗ and S be an equivalence relation on E∗ defined as follows:   1 0 0 1 1 0  0 1 0 0 0 0     0 0 1 0 0 0    S=   1 0 0 1 1 0     1 0 0 1 1 0  0 0 0 0 0 1

   1 0 0 1 1 0  0 0 0 0 0 1

where S represents the equivalence classes of ”weather between different cities”. For example the relationships (BJ,CHI),(IST,BJ) and (PAR,BJ) belong to the same equivalence class. This Mathematics 2018, 6, 18 17 of 19 means that weather between Beijing,Chicago is the same as the weather between Paris, Beijing and the weather between Paris, Beijing. whereLet S represents equivalence classes of “weather cities”.

For0.2), example, the E = {((BJ,the CHI), 0.1, 0.1, 0.3), ((LHR, CHI), 0.1,between 0.2, 0.3), different ((BJ, LHR), 0.1, 0.3, relationships (BJ,CHI),(IST,BJ) and (PAR,BJ) belong to the same equivalence class. This means that the ((IST, BJ), 0.2, 0.1, 0.1), ((P AR, BJ), 0.1, 0.1, 0.4), ((P AR, LHR), 0.2, 0.2, 0.3))} weather between Beijing and Chicago is the same as the weather between Paris and Beijing. ∗ which E = {(( BJ, CHset I ), on 0.1,E0.1, 0.3), ((describes LHR, CH the I ), 0.1, 0.2, 0.3), ((of BJ,weathers LHR), 0.1, ), (( IST, BJ ), beLet a neutrosophic comparison of 0.3, the0.2 cities under ∗ that describes the 0.2,consideration.Let 0.1, 0.1), (( PAR, BJ ) , 0.1, 0.1, 0.4 ) , (( PAR, LHR ) , 0.2, 0.2, 0.3 ))} be a NS on E SE = (SE, SE) be a rough neutrosophic set, where SE and SE are lower and upper approximations E,cities respectively, as follows: Let SE = (SE, SE) be a rough NS, where SE comparison of weathers ofofthe under consideration. and SE are lower and upper approximations of E, respectively, as follows: SE = {((BJ, CHI), 0.1, 0.1, 0.4), ((LHR, CHI), 0.1, 0.2, 0.3), ((BJ, LHR)0.1, 0.3, 0.2),

BJ), 0.1, ((P),AR, BJ),CH 0.1,I )0.1, LHR), = {(( BJ, CH I ),0.1, 0.1,0.4), 0.1, 0.4 (( LHR, , 0.1,0.4), 0.2, ((P 0.3)AR, , (( BJ, LHR)0.2, 0.1,0.2, 0.3,0.3))}, 0.2) SE((IST, SE = {((BJ, 0.2, 0.1, ((LHR, 0.1, ((BJ, 0.3,))} 0.2), (( IST,CHI), BJ ), 0.1, 0.1, 0.40.1), ), (( PAR, BJ )CHI), , 0.1, 0.1, 0.40.2, ), ((0.3), PAR, LHRLHR)0.1, ), 0.2, 0.2, 0.3 0.1, 0.1), BJ), 0.2, ((P),AR, LHR), SE =((IST, {(( BJ,BJ), CH I )0.2, , 0.2, 0.1, 0.1)((P , (( J, LHR, CH I ),0.1, 0.1,0.1), 0.2, 0.3 (( BJ, LHR)0.2, 0.1,0.2, 0.3, 0.3))}.

0.2)

IST,SE) BJ ), and 0.2, 0.1, 0.1(RV, ), (( PJ, BJ )are , 0.2,neutrosophic 0.1, 0.1), (( PAR, LHR)as , 0.2, 0.2, 0.3in))} Thus, G = (( (RV, G = SE) digraphs shown Fig.15 and Fig.16 Thus, = city ( RV,with SE) good and Gweather = ( RV,condition, SE) are neutrosophic digraphs, as shown in Figuresin15equaand 16.

To find Gthe we use the formula which we mentioned (Be, 0.2, 0.4, 0.6)

(Is, 0.2, 0.4, 0.6)

(0.1, 0.3, 0.2)

4) 0.

1, 0.

(Ch, 0.1, 0.2, 0.8) (0 .1, 0.2 ,0 .3)

(Lh, 0.2, 0.4, 0.6)

) .4 ,0 .1 ,0 .1 (0

, .1 (0

(0.1, 0.1, 0.4)

(0.2, 0.2, 0.3)

(P a, 0.1, 0.2, 0.8)

Figure 15. Neutrosophic digraph G = ( RV, SE).

Figure 15: G = (RV, SE)

(0.2, 0.1, 0.1) tion (i).

(Is, 0.9, 0.7, 0.3) (Be, 0.9, 0.7, 0.3) ⊕ T )(e ), where e = (v , v ).

By direct calculations, we Our decision is ek if ek = max(TSE ) i i i j SE 1 . i

,0

(0.1, 0.3, 0.2)

0.2 ,0 .3)

(Lh, 0.9, 0.7,18 0.3)

) .1 ,0 .1 ,0 .2 (0

.1 have ,0 .2 0 ( TSE ⊕ TSE (BJ, CHI) = 0.044, ISE ⊕ ISE (BJ, CHI) = 0.071, CHI) = 0.094.

FSE ⊕ FSE (BJ,(Ch, 0.6, 0.5, 0.4) TSE ⊕ TSE (LHR, CHI) = 0.043, ISE ⊕ ISE (LHR, CHI) = 0.076, (0 .1, FSE ⊕ FSE (LHR, CHI) = 0.096.

(0.2, 0.2, 0.3)

(P a, 0.6, 0.5, 0.4)

Figure 16. Neutrosophic digraph G = ( RV, SE).

Figure 16: G = (RV, SE)

To find the city with the best weather condition, we use the formula that we mentioned in TSE ⊕ TSE (BJ, LHR) = 0.064, ISE ⊕ ISE (BJ, LHR) = 0.112, Equation (1).

FSE ⊕ FSE (BJ, LHR) = 0.068. is BJ) ek if = ek 0.066, = maxI( TSE⊕⊕I TSE(IST, )(ei ), BJ) where ei = (vi , v j ). By direct calculations, we have ⊕ decision T (IST, = 0.100, T Our SE

SE

i

SE

SE

FSE ⊕ FSE (IST, BJ) = 0.070.

TSE⊕⊕I TSE(P ( BJ, CH I) = AR, BJ) = 0.044 0.050, TSE ⊕ TSE (P AR, BJ) = 0.033, ISE SE FSE ⊕ FSE (P AR, BJ) = 0.094. I ⊕ I ( BJ, CH I ) = 0.071 SE TSE ⊕ TSE (P AR, LHR) = 0.034, ISE ⊕ ISE SE (P AR, LHR) = 0.155, FSE ⊕ FSE (P AR, LHR) = 0.096. FSE ⊕ FSE ( BJ, CH I ) = 0.094 Hence the weather condition between Istanbul and Beijing is good, Boeing 747 can use this TSE ⊕ TSE ( LHR, CH I ) = 0.043 path in case of weather emergency.

ISE ⊕ ISE ( LHR, CH I ) = 0.076 We present an algorithm for above mention application. The presented algorithm can be applied to avoid lengthy calculations when dealing with a large number of objects.

Algorithm 3.1.

1. Input the vertex set V ∗ .

2. Construct an equivalence relation T on the set V ∗ .

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FSE ⊕ FSE ( LHR, CH I ) = 0.096 TSE ⊕ TSE ( BJ, LHR) = 0.064 ISE ⊕ ISE ( BJ, LHR) = 0.112 FSE ⊕ FSE ( BJ, LHR) = 0.068 TSE ⊕ TSE ( IST, BJ ) = 0.066 ISE ⊕ ISE ( IST, BJ ) = 0.100 FSE ⊕ FSE ( IST, BJ ) = 0.070 TSE ⊕ TSE ( PAR, BJ ) = 0.033 ISE ⊕ ISE ( PAR, BJ ) = 0.050 FSE ⊕ FSE ( PAR, BJ ) = 0.094 TSE ⊕ TSE ( PAR, LHR) = 0.034 ISE ⊕ ISE ( PAR, LHR) = 0.155 FSE ⊕ FSE ( PAR, LHR) = 0.096 Hence the weather conditions between Istanbul and Beijing are good; Boeing 747 can use this path in the case of a weather emergency.

We present an Algorithm 1 for the above-mentioned application. The presented algorithm can be applied to avoid lengthy calculations when dealing with a large number of objects.

Algorithm 1: 1.

2.

3.

4.

5.

6.

7.

Input, the vertex set V ∗ .

Construct an equivalence relation T on the set V ∗ .

Calculate the approximation sets TV and TV. Input, the edge set E∗ ⊆ V ∗ × V ∗ .

Construct an equivalence relation S on E∗ .

Calculate the approximation sets SE and SE. Calculate the score value, by using the following formula: TSE ⊕ TSE (vi , v j )

=

ISE ⊕ ISE (vi , v j )

=

FSE ⊕ FSE (vi , v j )

=

TRV (vi ) ∗ TRV (v j )

3 − TSE (vi , v j ) + TSE (vi , v j ) − TSE (vi , v j ) ∗ TSE (vi , v j ) IRV (vi ) ∗ IRV (v j )

3 − ISE (vi , v j ) + ISE (vi , v j ) − ISE (vi , v j ) ∗ ISE (vi , v j ) FRV (vi ) ∗ FRV (v j )

3 − FSE (vi , v j ) + FSE (vi , v j ) − FSE (vi , v j ) ∗ FSE (vi , v j )

8.

Decision is ek if ek = max( TSE ⊕ TSE )(ei ), where ei = (vi , v j ).

9.

If ek has more than one value, then any one of S(vk ) may be chosen.

i

4. Conclusions The NS model is suitable for modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary and human evaluation is needed. Various sources of uncertainty can make it a challenge to make a reliable decision. The NS model and rough set model are used to handle uncertainty, combining these two models with another remarkable model of soft sets, giving more precise results for decision-making problems. In this paper, we

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have presented certain operations, including Lexicographic products and tensor products on rough neutrosophic digraphs. This research work can be extended to (1) rough bipolar neutrosophic soft graphs, (2) bipolar neutrosophic soft rough graphs, (3) interval-valued bipolar neutrosophic rough graphs, and (4) neutrosophic soft rough graphs.

Acknowledgments: The authors are very thankful to the editor and referees for their valuable comments and suggestions for improving the paper.

Author Contributions: Nabeela Ishfaq and Sidra Sayed conceived and designed the experiments; Muhammad Akram performed the experiments; Florentin Smarandache contributed reagents/materials/analysis tools.

Conflicts of Interest: The authors declare that they have no conflict of interest regarding the publication of the research article.

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article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).