An Introduction to Bipolar Single Valued Neutrosophic Graph Theory
Said Broumi
1,a
, Florentin Smarandache
2,b
, Mohamed Talea
3,c
, Assia Bakali
4,d
1,3
Laboratory of Information Processing, University Hassan II, B.P 7955, Sidi Othman,
Casablanca, Morocco
2
Department of Mathematics, University of New Mexico,705 Gurley Avenue, Gallup, NM 87301,
USA
4
Ecole Royale Navale, Boulevard Sour Jdid, B.P 16303 Casablanca, Morocco
a
broumisaid78@gmail.com,
b
fsmarandache@gmail.com,
c
taleamohamed@yahoo.fr,
d
assiabakali@yahoo.fr
Keywords: Single valued neutrosophic set, bipolar neutrosophic set, single valued neutrosophic
graph, bipolar single valued neutrosophic graphs.
Abstract. In this paper, we first define the concept of bipolar single neutrosophic graphs as the
generalization of bipolar fuzzy graphs, N-graphs, intuitionistic fuzzy graph, single valued
neutrosophic graphs and bipolar intuitionistic fuzzy graphs.
1. Introduction
Zadeh [9] coined the term ‘degree of membership’ and defined the concept of fuzzy set in order to
deal with uncertainty. Atanassov [8] incorporated the degree of non-membership in the concept of
fuzzy set as an independent component and defined the concept of intuitionistic fuzzy set.
Smarandache [2] grounded the term ‘degree of indeterminacy’ as an independent component and
defined the concept of neutrosophic set from the philosophical point of view to deal with
incomplete, indeterminate and inconsistent information in real world. The concept of neutrosophic
set is a generalization of the theory of fuzzy set, intuitionistic fuzzy set. Each element of a
neutrosophic set has three membership degrees including a truth membership degree, an
indeterminacy membership degree, and a falsity membership degree which are within the real
standard or nonstandard unit interval ]
−
0, 1
+
[. Therefore, if their range is restrained within the real
standard unit interval [0, 1], the neutrosophic set is easily applied to engineering problems. For this
purpose, Wang et al. [6] introduced the concept of the single-valued neutrosophic set (SVNS) as a
subclass of the neutrosophic set. Recently, Deli et al. [7] defined the concept of bipolar
neutrosophic, as a generalization of single valued neutrosophic set, and bipolar fuzzy graph, also
studying some of their related properties. The neutrosophic set theory of and their extensions have
been applied in various domains [22] (refer to the site http://fs.gallup.unm.edu/NSS/).
When the relations between nodes (or vertices) in problems are indeterminate, the concept of
fuzzy graphs [15] and its extensions, such as intuitionistic fuzzy graphs [11, 16], N-graphs [13],
bipolar fuzzy graphs [11, 12, 14], bipolar intuitionistic fuzzy graphs [1] are not suitable. For this
purpose, Smarandache [3] defined four main categories of neutrosophic graphs, two based on literal
indeterminacy (I), calling them I-edge neutrosophic graph and I-vertex neutrosophic graph; these
concepts are deeply studied and gained popularity among some researchers [4, 5, 19, 20, 21] due to
their applications in the real world problems. The two others graphs are based on (t, i, f)
components, and are called: (t, i, f)-edge neutrosophic graph and (t, i, f)-vertex neutrosophic graph;
but these new concepts are not developed at all yet. Later on, Broumi et al. [18] introduced a third
neutrosophic graph model. The single valued neutrosophic graph is the generalization of fuzzy
graph and intuitionstic fuzzy graph. Also, the same authors [17] introduced neighborhood degree of
a vertex and closed neighborhood degree of a vertex in the single valued neutrosophic graph, as a
generalization of neighborhood degree of a vertex and closed neighborhood degree of vertex in
fuzzy graph and intuitionistic fuzzy graph.
Applied Mechanics and Materials Submitted: 2016-02-26
ISSN: 1662-7482, Vol. 841, pp 184-191 Revised: 2016-03-01
doi:10.4028/www.scientific.net/AMM.841.184 Accepted: 2016-03-07
© 2016 Trans Tech Publications, Switzerland Online: 2016-06-22
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, www.ttp.net. (#71057451-04/11/16,18:52:36)
In this paper, motivated by the works of Deli et al. [7] and Broumi et al. [18], we introduced the
concept of bipolar single valued neutrosophic graph and proved some propositions.
2. Preliminaries
In this section, we mainly recall some notions, which we are also going to use in the rest of the
paper. The readers are referred to [6, 7, 10, 11, 13, 15, 18] for further details and background.
Definition 2.1 [6]
Let U be an universe of a discourse; then, the neutrosophic set A is an object having the form A
= {< x: , , >, x ∈ U}, where the functions T, I, F: U→]
−
0,1
+
[ define respectively
the degree of membership, the degree of indeterminacy, and the degree of non-membership of the
element x ∈ U to the set A with the condition:
−
0 ≤ + + ≤ 3
+
.
Definition 2.2 [7]
A bipolar neutrosophic set A in X is defined as an object of the form A={<x, (x), (x),
(x), (x), (x), (x)>: x X}, where , , :X [1, 0] and , , : X [-1, 0].
The positive membership degree (x), (x), (x) denotes the truth membership, indeterminate
membership and false membership of an element X corresponding to a bipolar neutrosophic set
A, and the negative membership degree (x), (x), (x) denotes the truth membership,
indeterminate membership and false membership of an element X to some implicit counter-
property corresponding to a bipolar neutrosophic set A.
Example 2.1
Let X = { , , };
A = is a bipolar neutrosophic subset of X.
Definition 2.3 [7]
Let = {<x, (x), (x), (x), (x), (x), (x)>} and = {<x, (x), (x), x),
(x), (x), (x) >} be two bipolar neutrosophic sets. Then, if and only if (x)
(x) , (x) (x), (x) (x) and (x) (x) , (x) (x) , (x) (x) for all
x X.
Definition 2.4 [15]
A fuzzy graph with V as the underlying set is a pair G = (σ, μ), where σ: V → [0, 1] is a fuzzy
subset and μ: V × V → [0, 1] is a fuzzy relation on σ such that μ(x, y) σ(x) σ(y) for all x, y ∈ V
where stands for minimum.
Definition 2.5 [13]
By a N-graph G of a graph , we mean a pair G= ( , ) where is an N-function in V and
is an N-relation on E such that (u,v) max ( (u), (v)) all u, v V.
Definition 2.6 [10]
An intuitionistic fuzzy graph is of the form G = (V, E), where
i. V = { , ,…., } such that : V [0,1] and : V [0,1] denoting the degree
of membership
and non-membership of the element V, respectively, and 0≤ ( ) +
( )) ≤ 1 for every V, (i = 1, 2, ……. n), (1)
ii. E V x V where : VxV [0,1] and : VxV [0,1] are such that ( , ) ≤
min [ ( ), ( )] and ( , ) max [ ( ), ( )] and 0 ≤ ( , ) + ( , ) ≤ 1 for
every ( , ) E, ( i, j = 1,2, ……. n). (2)
Definition 2.7 [11]
Let X be a non-empty set. A bipolar fuzzy set A in X is an object having the form A = {(x, (x),
(x)) | x X}, where (x): X → [0, 1] and (x): X → [−1, 0] are mappings.
Applied Mechanics and Materials Vol. 841 185
Definition 2.8 [11]
A bipolar fuzzy graph of a graph = (V, E) is a pair G = (A,B), where A = ( , ) is a
bipolar fuzzy set in V and B = ( , ) is a bipolar fuzzy set on E V x V such that (xy)
min{ (x), (y)} for all xy , (xy) min{ (x), (y)} for all xy and (xy) = (xy)
= 0 for all xy –E. Here A is called bipolar fuzzy vertex set of V, and B - the bipolar fuzzy edge
set of E.
Definition 2.9 [18]
A single valued neutrosophic graph (SVNG) of a graph = (V, E) is a pair G = (A, B), where:
i. V = { , ,…, } such that :V [0, 1], :V [0, 1] and :V [0, 1] denote the degree
of truth-membership, degree of indeterminacy-membership and falsity-membership of the
element V, respectively, and 0 + ( ) + 3 for every V (i=1, 2,
…, n). (3)
ii. E V x V, where :V x V [0, 1], :V x V [0, 1] and :V x V [0, 1] are such that
min [ , ], max [ , ] and max
[ , ] and 0 + + 3, for every E (i, j =
1, 2,…, n). (4)
3. Bipolar Single Valued Neutrosophic Graphs
In this section, we firstly define the concept of a bipolar single valued neutrosophic relation.
Definition 3.1
Let X be a non-empty set. Then we call a mapping A = (x, (x), (x), (x), (x), (x),
(x)):X × X → [−1, 0] × [0, 1] a bipolar single valued neutrosophic relation on X such that (x,
y) [0, 1], (x, y) [0, 1], (x, y) [0, 1], and (x, y) [−1, 0], (x, y) [−1, 0], (x, y)
[−1, 0].
Definition 3.2
Let A = ( , , , , ) and B = ( , , , , ) be a bipolar single valued
neutrosophic graph on a set X. If B = ( , , , , ) is a bipolar single valued
neutrosophic relation on A = ( , , , , ) then:
(x, y) min( (x), (y)), (x, y) max( (x), (y)) (5)
(x, y) max( (x), (y)), (x, y) min( (x), (y)) (6)
(x, y) max( (x), (y)), (x, y) min( (x), (y)), for all x, y X. (7)
A bipolar single valued neutrosophic relation B on X is called symmetric if (x, y) = (y,
x), (x, y) = (y, x), (x, y) = (y, x) and (x, y) = (y, x), (x, y) = (y, x), (x, y) =
(y, x), for all x, y X.
Definition 3.3
A bipolar single valued neutrosophic graph of a graph = (V, E) is a pair G = (A, B), where A
= ( , , , , ) is a bipolar single valued neutrosophic set in V, and B = ( , ,
, , ) is a bipolar single valued neutrosophic set in , such that
(x, y) min( ( ), ( )), (x, y) max( ( ), ( )) (8)
(x, y) max( ( ), ( )), (x, y) min( ( ), ( )), and (9)
(x, y) max( ( ), ( )), (x, y) min( ( ), ( )), for all xy . (10)
Notation
An edge of BSVNG is denoted by E or E.
Here, the sextuple ( , , , , , ) denotes the positive degree of truth-membership,
the positive degree of indeterminacy-membership, the positive degree of falsity-membership, the
negative degree of truth-membership, the negative degree of indeterminacy-membership, the
negative degree of falsity- membership of the vertex .
The sextuple ( , , , , , ) denotes the positive degree of truth-membership, the
positive degree of indeterminacy-membership, the positive degree of falsity-membership, the
186 Recent Tendency in Aerospace, Robotics, Manufacturing Systems, Energy
and Mechanical Engineering
negative degree of truth-membership, the negative degree of indeterminacy-membership, the
negative degree of falsity- membership of the edge relation = ( , ) on V V.
Notes
i. When = = = 0 and = = = 0 for some i and j, then there is no edge
between and . Otherwise there exists an edge between and .
ii. If one of the inequalities is not satisfied, then G is not a BSVNG.
Fig.1: Bipolar single valued neutrosophic graph.
Proposition 3.1
A bipolar single valued neutrosophic graph is the generalization of the fuzzy graph.
Proof
Suppose G = (A, B) is a bipolar single valued neutrosophic graph. Then, by setting the positive
indeterminacy-membership, positive falsity-membership and negative truth-membership, negative
indeterminacy-membership, negative falsity-membership values of vertex set and edge set equals to
zero, it reduces the bipolar single valued neutrosophic graph to a fuzzy graph.
Example 3.1
Fig. 2: Fuzzy graph
Proposition 3.2
A bipolar single valued neutrosophic graph is the generalization of the bipolar intuitionstic fuzzy
graph.
Applied Mechanics and Materials Vol. 841 187
Proof
Suppose G = (A, B) is a bipolar single valued neutrosophic graph. Then, by setting the positive
indeterminacy-membership, negative indeterminacy-membership values of vertex set and edge set
equals to zero, it reduces the bipolar single valued neutrosophic graph to a bipolar intuitionistic
fuzzy graph.
Example 3.2
Fig.3: Intuitionistic fuzzy graph.
Proposition 3.3
A bipolar single valued neutrosophic graph is the generalization of the single valued
neutrosophic graph.
Proof
Suppose G = (A, B) is a bipolar single valued neutrosophic graph. Then, by setting the negative
truth-membership, negative indeterminacy-membership, negative falsity-membership values of
vertex set and edge set equals to zero, it reduces the bipolar single valued neutrosophic graph to a
single valued neutrosophic graph.
Example 3.3
Fig. 4: Single valued neutrosophic graph.
Proposition 3.4
A bipolar single valued neutrosophic graph is the generalization of the bipolar intuitionstic fuzzy
graph.
188 Recent Tendency in Aerospace, Robotics, Manufacturing Systems, Energy
and Mechanical Engineering