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Regular bipolar single valued neutrosophic hypergraphs.

1 Introduction

The notion of neutrosophic sets (NSs) was proposed by Smarandache [8] as a generalization of the fuzzy sets [14], intuitionistic fuzzy sets [12], interval valued fuzzy set [11] and interval-valued intuitionistic fuzzy sets [13] theories. The neutrosophic set is a powerful mathematical tool for dealing with incomplete, indeterminate and inconsistent information in real world. The neutrosophic sets are characterized by a truth-membership function (t), an indeterminacy-membership function (i) and a falsity membership function (f independently, which are within the real standard or nonstandard unit interval ][sup.-]0, [1.sup.+][. In order to conveniently use NS in real life applications, Wang et al. [9] introduced the concept of the single-valued neutrosophic set (SVNS), a subclass of the neutrosophic sets. The same authors [10] introduced the concept of the interval valued neutrosophic set (IVNS), which is more precise and flexible than the single valued neutrosophic set. The IVNS is a generalization of the single valued neutrosophic set, in which the three membership functions are independent and their value belong to the unit interval [0, 1]. More works on single valued neutrosophic sets, interval valued neutrosophic sets and their applications can be found on http://fs.gallup.unm.edu/NSS/.

Hypergraph is a graph in which an edge can connect more than two vertices, hypergraphs can be applied to analyse architecture structures and to represent system partitions, Mordesen J.N and P.S Nasir gave the definitions for fuzzy hypergraphs. Parvathy. R and M. G. Karunambigai's paper introduced the concepts of Intuitionistic fuzzy hypergraphs and analyse its components, Nagoor Gani. A and Sajith Begum. S defined degree, order and size in intuitionistic fuzzy graphs and extend the properties. Nagoor Gani. A and Latha. R introduced irregular fuzzy graphs and discussed some of its properties.

Regular intuitionistic fuzzy hypergraphs and totally regular intuitionistic fuzzy hypergraphs are introduced by Pradeepa. I and Vimala. S in [0]. In this paper we extend regularity and totally regularity on bipolar single valued neutrosophic hypergraphs.

2 Preliminaries

In this section we discuss the basic concept on neutrosophic set and neutrosophic hyper graphs.

Definition 2.1 Let X be the space of points (objects) with generic elements in X denoted by x. A single valued neutrosophic set A (SVNS A) is characterized by truth membership function [T.sub.A](x), indeterminacy membership function [I.sub.A](x) and a falsity membership function [F.sub.A](x). For each point x [member of] X; [T.sub.A](x), [I.sub.A](x), [F.sub.A](x) [member of] [0, 1].

Definition 2.2 Let X be a space of points (objects) with generic elements in X denoted by x. A bipolar single valued neutrosophic set A (BSVNS A) is characterized by positive truth membership function P[T.sub.A](x), positive indeterminacy membership function P[I.sub.A](x) and a positive falsity membership function P[F.sub.A](x) and negative truth membership function N[T.sub.A](x), negative indeterminacy membership function N[I.sub.A](x) and a negative falsity membership function N[F.sub.A](x).

For each point x [member of] X; P[T.sub.A](x), P[I.sub.A](x), P[F.sub.A](x) [member of] [0, 1] and N[T.sub.A](x), N[I.sub.A](x), N[F.sub.A](x) [member of] [-1, 0].

Definition 2.3 Let A be a BSVNS on X then support of A is denoted and defined by

Supp(A) = {x : x [member of] X, P[T.sub.A](x) > 0, P[I.sub.A](x) > 0, PFJx) > 0, N[T.sub.A](x) < 0, N[I.sub.A](x) < 0, N[F.sub.A](x) < 0}.

Definition 2.4 A hyper graph is an ordered pair H = (X, E), where

(1) X = {[x.sub.1], [x.sub.2], ..., [x.sub.n]} be a finite set of vertices.

(2) E = {[E.sub.1], [E.sub.2], ..., [E.sub.m]} be a family of subsets of X.

(3) [E.sub.j] for j = 1,2,3, ..., m and [U.sub.j]([E.sub.j]}) = X.

The set X is called set of vertices and E is the set of edges (or hyper edges).

Definition 2.5 A bipolar single valued neutrosophic hypergraph is an ordered pair H = (X, E), where

(1) X = {[x.sub.1], [x.sub.2], ..., [x.sub.m]} be a finite set of vertices.

(2) E = {[E.sub.1], [E.sub.2], ..., [E.sub.m]} be a family of BSVNSs of X.

(3) [E.sub.j] [not equal to] O = (0, 0, 0) for j = 1,2,3, ..., m and [U.sub.j] Supp([E.sub.j]) = X.

The set X is called set of vertices and E is the set of BSVN-edges (or BSVN-hyper edges).

Proposition 2.6 The bipolar single valued neutrosophic hyper graph is the generalization of fuzzy hyper graphs, intuitionistic fuzzy hyper graphs, bipolar fuzzy hyper graphs and single valued neutrosophic hypergraphs.

3 Regular and totally regular BSVNHGs

Definition 3.1 The open neighbourhood of a vertex x in bipolar single valued neutrosophic hypergraphs (BSVNHGs) is the set of adjacent vertices of x, excluding that vertex and is denoted by N(x).

Definition 3.2 The closed neighbourhood of a vertex x in bipolar single valued neutrosophic hypergraphs (BSVNHGs) is the set of adjacent vertices of x, including that vertex and is denoted by N[x].

Example 3.3 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E = {P, Q, R, S}, which is defined by

P = {<a, 0.1, 0.2, 0.3, -0.4, -0.6 -0.8), (b, 0.4, 0.5, 0.6, -0.4, -0.6 -0.8>}

Q = {<c, 0.1, 0.2, 0.3, -0.4, -0.4 -0.9), (d, 0.4,.5, 0.6, -0.3, -0.5 -0.6), (e, 0.7, 0.8, 0.9, -0.7, -0.9, -0.2>}

R = {<b, 0.1, 0.2, 0.3, -0.2, -0.5, -0.8), (c, 0.4, 0.5, 0.6, -0.9, -0.7 -0.4>}

S = {<a, 0.1, 0.2, 0.3, -0.7, -0.6, -0.9), (d, 0.9, 0.7, 0.6, -0.4, -0.7, -0.9>}

Then the open neighbourhood of a vertex a is the b and d, and closed neighbourhood of a vertex b is b, a and c.

Definition 3.4 Let H = (X, E) be a BSVNHG, the open neighbourhood degree of a vertex x, which is denoted and defined by

deg(x) = ([deg.sub.P]T(x), [deg.sub.P]I(x), [deg.sub.P]F(x), [deg.sub.N]T(x), [deg.sub.NI](x), [deg.sub.NF](x))

where

[deg.sub.PT](x) = [[summation].sub.x[member of]N(x)] P[T.sub.E](x)

[deg.sub.PI](x) = [[summation].sub.x[member of]N(x)] P[I.sub.E](x)

[deg.sub.PF](x) = [[summation].sub.x[member of]N(x)] P[F.sub.E](x)

[deg.sub.NT](x) = [[summation].sub.x[member of]N(x)] N[T.sub.E](x)

[deg.sub.NI](x) = [[summation].sub.x[member of]N(x)] N[I.sub.E](x)

[deg.sub.NF](x) = [[summation].sub.x[member of]N(x)] N[F.sub.E](x)

Example 3.5 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E = {P, Q, R, S}, which are defined by

P = {<a,.1,.2,.3, -0.1, -0.2, -0.3), (b,.4,.5,.6, -0.1, -0.2, -0.3>}

Q = {<c,.1,.2,.3, -0.1, -0.2, -0.3), (d,.4,.5,.6, -0.1, -0.2, -0.3), (e,.7,.8,.9, -0.1, -0.2, -0.3>}

R = {<b,.1,.2,.3, -0.1, -0.2, -0.3), (c,.4,.5,.6, -0.1, -0.2, -0.3>}

S = {<a,.1,.2,.3, -0.1, -0.2, -0.3), (d,.4,.5,.6, -0.1, -0.2, -0.3>}

Then the open neighbourhood of a vertex a contain b and d and therefore open neighbourhood degree of a vertex a is (.8, 1, 1.2, -0.2, -0.4, -0.6).

Definition 3.6 Let H = (X, E) be a BSVNHG, the closed neighbourhood degree of a vertex x is denoted and defined by,

[deg.sub.][x] = ([deg.sub.PT][x], [deg.sub.PI][x], [deg.sub.PF][x], [deg.sub.NT][x], [deg.sub.NI][x], [deg.sub.NF][x])

which are defined by

[deg.sub.PT](x) = [deg.sub.PT](x) + P[T.sub.E](x)

[deg.sub.PI](x) = [deg.sub.PI](x) + P[I.sub.E](x)

[deg.sub.PF](x) = [deg.sub.PF](x) + P[F.sub.E](x)

[deg.sub.NT](x) = [deg.sub.NT](x) + N[T.sub.E](x)

[deg.sub.NI](x) = [deg.sub.NI](x) + N[I.sub.E](x)

[deg.sub.NF](x) = [deg.sub.NF](x) + N[F.sub.E](x)

Example 3.7 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d, e} and E = {P, Q, R, S}, which is defined by

P = {<a, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3>}

Q = {<c, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3), (e, 0.7, 0.8, 0.9, -0.1, -0.2, -0.3>}

R = {<b, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (c, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3>}

S = {<a, 0.1, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.4, 0.5, 0.6, -0.1, -0.2, -0.3>}

The closed neighbourhood of a vertex a contain a, b and d, hence the closed neighbourhood degree of a vertex a is (0.9,.1.2, 1.5, -0.3, -0.6, -0.9).

Definition 3.8 Let H = (X, E) be a BSVNHG, then H is said to be an n-regular BSVNHG if all the vertices have the same open neighbourhood degree n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4], [n.sub.5], [n.sub.6])

Definition 3.9 Let H = (X, E) be a BSVNHG, then H is said to be m-totally regular BSVNHG if all the vertices have the same closed neighbourhood degree m = ([m.sub.1], [m.sub.2], [m.sub.3], [m.sub.4], [m.sub.5], [m.sub.6]).

Proposition 3.10 A regular BSVNHG is the generalization of regular fuzzy hypergraphs, regular intuitionistic fuzzy hypergraphs, regular bipolar fuzzy hypergraphs and regular single valued neutrosophic hypergraphs.

Proposition 3.11 A totally regular BSVNHG is the generalization of totally regular fuzzy hypergraphs, totally regular intuitionistic fuzzy hypergraphs, totally regular bipolar fuzzy hypergraphs and totally regular single valued neutrosophic hypergraphs.

Example 3.12 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and E = {P, Q, R, S} which is defined by

P = {<a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

Q = {<b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

R = {<c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

S = {<d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

Here the open neighbourhood degree of every vertex is (1.6, 0.4, 0.6, -0.2, -0.4, -0.6) hence H is regular BSVNHG and closed neighbourhood degree of every vertex is (2.4, 0.6, 0.9, -0.3, -0.6, -0.9), Hence H is both regular and totally regular BSVNHG.

Theorem 3.13 Let H = (X, E) be a BSVNHG which is both regular and totally regular BSVNHG then E is constant.

Proof: Suppose H is an n-regular and m-totally regular BSVNHG. Then deg(x) = n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4], [n.sub.5], [n.sub.6]) and deg[x] = m = ([m.sub.1], [m.sub.2], [m.sub.3], [m.sub.4], [m.sub.5], [m.sub.6]) [for all] x [member of] [E.sub.i]. Consider deg[x] = m. Hence by definition, deg(x) + [E.sub.i](x) = m this implies [E.sub.i](x) = m - n for all x [member of] [E.sub.i]. Hence E is constant.

Remark 3.14 The converse of above theorem need not to be true in general.

Example 3.15 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and E = {P, Q, R, S}, which is defined by

P = {<a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

Q = {<b, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

R = {<c, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

S = {<d, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3), (a, 0.8, 0.2, 0.3, -0.1, -0.2, -0.3>}

Here E is constant but deg(a) = (1.6, 0.4, 0.6, -0.2, -0.4, 0.6) and deg(d) = (2.4, 0.6, 0.9, -0.3, -0.6, -0.9) i.e deg(a) and deg(d) are not equals hence H is not regular BSVNHG. Next deg[a] = (2.4, 0.6, 0.9, -0.3, -0.6, -0.9) and deg[d]= (3.2, 0.8, 1.2, -.4, -0.8, -1.2), hence deg[a] and deg[d] are not equals hence H is not totally regular BSVNHG, Thus that H is neither regular and nor totally regular BSVNHG.

Theorem 3.16 Let H = (X, E) be a BSVNHG then E is constant on X if and only if following are equivalent,

(1) H is regular BSVNHG.

(2) H is totally regular BSVNHG.

Proof: Suppose H = (X, E) be a BSVNHG and E is constant in H, that is [E.sub.i](x) = c = ([c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4], [c.sub.5], [c.sub.6]) [for all] x [member of] [E.sub.i]. Suppose H is n-regular BSVNHG, then deg(x) = n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4], [n.sub.5], [n.sub.6]) [for all] x [member of] [E.sub.i], consider deg[x] = deg(x) + [E.sub.i](x) = n + c

[for all] x [member of] [E.sub.i], hence H is totally regular BSVNHG.

Next suppose that H is m-totally regular BSVNHG, then deg[x] = m = ([m.sub.1], [m.sub.2], [m.sub.3], [m.sub.4], [m.sub.5], [m.sub.6]) for all x [member of] [E.sub.i], that is deg(x) + [E.sub.i](x) = m [for all] x [member of] [E.sub.i], this implies that deg(x) = m - c [for all] x [member of] [E.sub.i]. Thus H is regular BSVNHG, thus (1) and (2) are equivalent.

Conversely: Assume that (1) and (2) are equivalent. That is H is regular BSVNHG if and only if H is totally regular BSVNHG. Suppose contrary E is not constant, that is [E.sub.i](x) and [E.sub.i](y) not equals for some x and y in X. Let H = (X, E) be n-regular BSVNHG, then deg(x) = n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4], [n.sub.5], [n.sub.6]) for all x [member of] [E.sub.i]. Consider

deg[x] = deg(x) + [E.sub.i](x) = n + [E.sub.i](x) deg[y] = deg(y) + [E.sub.y](y) = n + [E.sub.i](y)

Since [E.sub.i](x) and [E.sub.i](y) are not equals for some x and y in X. Hence deg[x] and deg[y] are not equals, thus H is not totally regular BSVNHG, which contradict to our assumption.

Next let H be totally regular BSVNHG, then deg[x] = deg[y], that is deg(x) + [E.sub.i](x) = deg(y) + [E.sub.i](y) and deg(x) deg(y) = [E.sub.i](y) - [E.sub.i](x), since RHS of last equation is nonzero, hence LHS of above equation is also nonzero, thus deg(x) and deg(y) are not equals, so H is not regular BSVNHG, which is again contradict to our assumption, thus our supposition was wrong, hence E must be constant, this completes the proof.

Definition 3.17 Let H = (X, E) be a regular BSVNHG, then the order of BSVNHG H is denoted and defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For every x [member of] X and size of regular BSVNHG is denoted and defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where S([E.sub.i]) = (a, b, c, d, e, f) which is defined by

a = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

b = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

c = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

d = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

e = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

f = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 3.18 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E) where, X = {a, b, c, d} and E = {P, Q, R, S}, which is defined by

P = {<a,.8,.2,.3, -.1, -.2, -.3), (b,.8,.2,.3, -.1, -.2, -.3>}

Q = {<b,.8,.2,.3, -.1, -.2, -.3), (c,.8,.2,.3, -.1, -.2, -.3>}

R = {<c,.8,.2,.3, -.1, -.2, -.3), (d,.8,.2,.3, -.1, -.2, -.3>}

S = {<d,.8,.2,.3, -.1, -.2, -.3), (a,.8,.2,.3, -.1, -.2, -.3>}

Here order and size of H are given (3.2,.8, 1.2, -.4, -.8, 1.2) and (6.4, 1.6, 2.4, -.8, -1.6, -2.4) respectively.

Proposition 3.19 The size of an n-regular BSVNHG H = (H, E) is nk/2, where [absolute value of X] = k.

Proposition 3.20 If H = (X, E) be m-totally regular BSVNHG then 2S(H) + O(H) = mk, where [absolute value of X] = k.

Corollary 3.21 Let H = (X, E) be a n-regular and m-totally regular BSVNHG then O(H) = k(m - n), where [absolute value of X]=k.

Proposition 3.22 The dual of n-regular and m-totally regular BSVNHG H = (X, E) is again an n-regular and m-totally regular BSVNHG.

Definition 3.23 A bipolar single valued neutrosophic hypergraph (BSVNHG) is said to be complete BSVNHG if for every x in X, N(x) = {x: x in X-{x}}, that is N(x) contains all remaining vertices of X except x.

Example 3.24 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E), where X = {a, b, c, d} and E = {P, Q, R}, which is defined by

P = {<a, 0.4, 0.6, 0.3, -0.5, -0.2, -0.3), (c, 0.8, 0.2, 0.3, -0.1, -0.8, -0.3>}

Q = {<a, 0.8, 0.8, 0.3, -0.1, -0.6, -0.3), (b, 0.8, 0.2, 0.1, -0.1, -0.2, -0.3), (d, 0.8, 0.2, 0.1, -0.1, -0.9, -0.3>}

R = {<c, 0.4, 0.9, 0.9, -0.1, -0.2, -0.3), (d, 0.7, 0.2, 0.1, -0.5, -0.9, -0.3), (b,

0.4, 0.2, 0.1, -0.8, -0.4, -0.2>}. Here N(a) = {b, c, d}, N(b) = {a, c, d}, N(c) = {a, b, d}, N(d) = {a, b, c} hence H is complete BSVNHG.

Remark 3.25 In a complete BSVNHG H = (X, E), the cardinality of N(x) is same for every vertex.

Theorem 3.26 Every complete BSVNHG H = (X, E) is both regular and totally regular if E is constant in H.

Proof: Let H = (X, E) be complete BSVNHG, suppose E is constant in H, so that [E.sub.i](x) = c = ([c.sub.1], [c.sub.2], [c.sub.3], [c.sub.4], [c.sub.5], [c.sub.6]) [for all] x [member of] [E.sub.i], since BSVNHG is complete, then by definition for every vertex x in X, N(x) = {x: x in X-{x}}, the open neighbourhood degree of every vertex is same. That is deg(x) = n = ([n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4], [n.sub.5], [n.sub.6]) [for all] x [member of] [E.sub.i]. Hence complete BSVNHG is regular BSVNHG. Also, deg[x] = deg(x) + [E.sub.i](x) = n + c [for all] x [member of] [E.sub.i]. Hence H is totally regular BSVNHG.

Remark 3.27 Every complete BSVNHG is totally regular even if E is not constant.

Definition 3.28 A BSVNHG is said to be k-uniform if all the hyper edges have same cardinality.

Example 3.29 Consider a bipolar single valued neutrosophic hypergraphs H = (X, E), where X = {a, b, c, d} and E = {P, Q, R}, which is defined by

P = {<a, 0.8, 0.4, 0.2,-0.4, -0.6, -0.2), (b, 0.7, 0.5, 0.3, -0.7, -0.1, -0.2>}

Q = {<b, 0.9, 0.4, 0.8, -0.3, -0.2, -0.9), (c, 0.8, 0.4, 0.2, -0.4, -0.3, -0.7>}

R = {<c, 0.8, 0.6, 0.4, -0.3, -0.7, -0.2), (d, 0.8, 0.9, 0.5, -0.4, -0.8, -0.9>}

4 Conclusion

Theoretical concepts of graphs and hypergraphs are utilized by computer science applications. Single valued neutrosophic hypergraphs are more flexible than fuzzy hypergraphs and intuitionistic fuzzy hypergraphs. The concepts of single valued neutrosophic hypergraphs can be applied in various areas of engineering and computer science. In this paper we defined the regular and totally regular bipolar single valued neutrosophic hyper graphs. We plan to extend our research work to irregular and totally irregular on bipolar single valued neutrosophic hyper graphs.

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Received: 10 November, 2016. Accepted: December 02, 2017.

Muhammad Aslam Malik (1), Ali Hassan (2), Said Broumi (3), Florentin Smarandache (4)

(1) Department of mathematics, University of Punjab, Lahore (Pakistan), E-mail: aslam@math.pu.edu.pk, malikpu@yahoo.com.

(2) Department of mathematics, University of Punjab, Lahore (Pakistan), E-mail: alihassan.iiui.math@gmail.com.

(3) Laboratory of Information Processing, Faculty of Science Ben M'Sik, University Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco.

(4) Department of mathematics, university of New Mexico, 705, Gurley, Avenue, Gallup, NM 87301, USA E-mail: fsmarandache@gmail.com.
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Date:Oct 1, 2016
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