J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae25.
CERTAIN BIPOLAR NEUTROSOPHIC COMPETITION
GRAPHS
Muhammad Akram1 and Maryam Nasir2 Department of Mathematics.
University of the Punjab.
New Campus.
Lahore.
Pakistan makrammath@yahoo.
Abstract.
Bipolarity plays an important role in many research domains.
A bipolar fuzzy model is a very important model in which positive information represents what is possible or preferred, while negative information represents what is forbidden or surely false.
In this research paper, we first introduce the concept of p-competition bipolar neutrosophic graphs.
We then define generalization of bipolar neutrosophic competition graphs called m-step bipolar neutrosophic competition graphs.
Moreover, we present some related concepts of bipolar neutrosophic graphs.
Finally, we describe an application of m-step bipolar neutrosophic competition graphs.
Key words and Phrases: p-competition bipolar neutrosophic graphs, m-step bipolar neutrosophic competition graphs.
Algorithm.
Abstrak.
Bipolariti memainkan peran penting dalam berbagai macam topik penelitian.
Sebuah model Fuzzy bipolar adalah sebuah model yang sangat penting dalam hal informasi postif menyatakan apa yang mungkin dipilih, sedangkan informasi negatif menyatakan apa yang dilarang atau pasti salah.
Pada paper ini, pertama kali kami memperkenalkan konsep graf neotrosopik bipolar p-kompetisi.
Kemudian kami mendefinisikan perumuman dari graf kompetisi neutrosopik bipolar yang disebut dengan graf kompetisi neutrosopik bipolar m-langkah.
Lebih jauh, kami menyajikan beberapa konsep yang terkait dengan graf neutrosopik bipolar.
Akhirnya, kami menggambarkan sebuah aplikasi dari graf kompetisi neutrosopik bipolar m-langkah.
Kata kunci: Graf neotrosopik bipolar p-kompetisi, graf kompetisi neutrosopik bipolar m-langkah, algoritma.
2000 Mathematics Subject Classification: 03E72, 68R10, 68R05.
Received: 28-02-2017, revised: 11-04-2017, accepted: 09-05-2017.
Akram and M.
Nasir Introduction The notion of competition graphs was introduced by Cohen .
in 1968, depending upon a problem in ecology.
The competition graphs have many utilizations in solving daily life problems, including channel assignment, modeling of complex economic, phytogenetic tree reconstruction, coding and energy systems.
Fuzzy set theory .
and intuitionistic fuzzy set theory .
are useful models for dealing with uncertainty and incomplete information.
But they may not be sufficient in modeling of indeterminate and inconsistent information encountered in real world.
In order to cope with this issue, neutrosophic set theory was proposed by Smarandache .
as a generalization of fuzzy sets and intuitionistic fuzzy sets.
However, since neutrosophic sets are identified by three functions called truthmembership .
, indeterminacy-membership .
and falsity-membership .
) whose values are real standard or non-standard subset of unit interval ]0Oe , 1 [.
There are some difficulties in modeling of some problems in engineering and sciences.
overcome these difficulties, in 2010, concept of single-valued neutrosophic sets and its operations defined by Wang et al.
as a generalization of intuitionistic fuzzy Ye .
, .
has presented several novel applications of neutrosophic sets.
Deli et al.
extended the ideas of bipolar fuzzy sets .
and neutrosophic sets to bipolar neutrosophic sets and studied its operations and applications in decision making problems.
Smarandache .
proposed notion of neutrosophic graph and they separated them to four main categories.
Wu .
discussed fuzzy digraphs.
The concept of fuzzy k-competition graphs and p-competition fuzzy graphs was first introduced by Samanta and Pal in .
, it was further studied in .
, 13, 16, .
Cho et al.
proposed the generalization of a digraphs known as m-step competition graphs.
Samanta et al.
introduced the generalization of fuzzy competition graphs, called m-step fuzzy competition graphs.
On the other hand, the concepts of bipolar fuzzy competition graphs and intuitionistic fuzzy competition graphs are discussed in .
, .
Samanta et al.
also introduced the concepts of fuzzy mstep neighbouthood graphs.
The notion of bipolar fuzzy graphs was first introduced by Akram .
in 2011 as a generalization of fuzzy graphs.
On the other hand.
Akram and Shahzadi .
first introduced the notion of neutrosophic soft graphs and gave its Akram .
introduced the notion of single-valued neutrosophic planar Akram and Sarwar have shown that there are some flaws in Broumi et al.
Aos definition, which cannot be applied in network models.
All the predator-prey relations cannot only be represented by bipolar neutrosophic competition graphs.
For example, in a food web, species may have a chain consisting of same number of preys by which they can reach to their common preys.
This idea motivates the necessity of m-step bipolar neutrosophic competition graphs.
In this research paper, we first introduce the concept of p-competition bipolar neutrosophic graphs.
then define generalization of bipolar neutrosophic competition graphs called m-step bipolar neutrosophic competition graphs.
Moreover, we present some related concepts of bipolar neutrosophic graphs.
Finally, we describe an application of m-step bipolar neutrosophic competition graphs.
Certain bipolar neutrosophic competition graphs Certain Bipolar Neutrosophic Competition Graphs Definition 2.
, .
A fuzzy set AA in a universe X is a mapping AA : X Ie .
, .
A fuzzy relation on X is a fuzzy set in X y X.
Definition 2.
A bipolar fuzzy set on a non-empty set X has the form A = {.
AAP A .
, a .
) : x OO X} where.
AAP A : X Ie .
, .
and a : X Ie [Oe1, .
are mappings.
The positive memP bership value a .
represents the strength of truth or satisfaction of an element x to a certain property corresponding to bipolar fuzzy set A and AAN A .
denotes the strength of satisfaction of an element x to some counter property of bipolar fuzzy set A.
If AAP A .
6= 0 and a .
= 0 it is the situation when x has only truth satisfaction degree for property A.
If AAN A .
6= 0 and a .
= 0, it is the case that x is not satisfying the property of A but satisfying the counter property to A.
It is possible for x that AAP A .
6= 0 and a .
6= 0 when x satisfies the property of A as well as its counter property in some part of X.
Definition 2.
Let X be a non-empty set.
A mapping B = (AAP B .
AAB ) : X y X Ie .
, .
y [Oe1, .
is a bipolar fuzzy relation on X such that AAB .
OO .
, .
and AAN B .
OO [Oe1, .
for x, y OO X.
Definition 2.
A bipolar fuzzy graph on X is a pair G = (A.
B) where A = (AAP A , a ) is a bipolar fuzzy set on X and B = (AAB .
AAB ) is a bipolar fuzzy relation in X such that AAP B .
O a .
O a .
and AAB .
Ou a .
O a .
for all x, y OO X.
Definition 2.
A neutrosophic set A on a non-empty set X is characterized by a truth-membership fuction tA : X Ie .
, .
, indeterminacy-membership function iA : X Ie .
, .
and a falsity-membership function fA : X Ie .
, .
There is no restriction on the sum of tA .
, iA .
and fA .
for all x OO X.
Definition 2.
A bipolar neutrosophic set A on a non-empty set X is an object of the form A = {.
, tP A .
, iA .
, fA .
, tA .
, iA .
, fA .
) : x OO X}, where tP A , iA , fA : X Ie .
, .
and tA , iA , fA : X Ie [Oe1, .
The positive values tA .
, iA .
, fA .
denote respectively the truth, indeterminacy and falseN memberships degrees of an element x OO X, whereas, tN A .
, iA .
, fA .
denote the implicit counter property of the truth,indeterminacy and false-memberships degrees of the element x OO X corresponding to the bipolar neutrosophic set A.
Akram and M.
Nasir Definition 2.
The height of bipolar neutrosophic set A = .
P A .
, iA .
, fA .
, tA .
, iA .
, fA .
) in universe of discourse X is defined as, h(A) = .
1 (A), h2 (A), h3 (A), h4 (A), h5 (A), h6 (A)) = .
up tP A .
, sup iA .
, inf fA .
, sup tA .
, sup iA .
, inf fA .
), xOOX xOOX xOOX xOOX xOOX xOOX for all x OO X.
Ie Oe Definition 2.
Let G be a bipolar neutrosophic digraph then bipolar neutrosophic out-neighbourhoods of a vertex x is a bipolar neutrosophic set
(P )
N .
= (Xx , tx
(P )
, ix
(P )
, fx
(N )
, tx
(N )
, ix
(N )
, fx OeOeOeIe OeOeOeIe OeOeOeIe OeOeOeIe OeOeOeIe Xx = .
B1P .
, .
> 0.
B2P .
, .
> 0.
B3P .
, .
> 0.
B1N .
, .
< 0.
B2N .
, .
< 0.
OeOeOeIe B3N .
, .
< .
OeOeOeIe (P ) (P ) (P ) such that tx : Xx Ie .
, .
, defined by tx .
= B1P .
, .
, ix : Xx Ie .
, .
(P )
(P )
= : Xx Ie .
, .
, defined by fx defined by ix .
= B2P .
, .
, fx OeOeOeIe (N ) OeOeOeIe (N ) (N ) : Xx Ie B3P .
, .
, tx : Xx Ie [Oe1, .
, defined by tx .
= B1N .
, .
, ix OeOeOeIe (N ) (N ) (N ) .
= : Xx Ie [Oe1, .
, defined by fx [Oe1, .
, defined by ix .
= B2N .
, .
, fx OeOeOeIe B3 .
, .
Ie Oe Definition 2.
Let G be a bipolar neutrosophic digraph then bipolar neutrosophic in-neighbourhoods of a vertex x is a bipolar neutrosophic set (P )Oe N Oe .
= (XxOe , tx (P )Oe , ix (P )Oe , fx (N )Oe , tx (N )Oe , ix (N )Oe , fx OeOeOeIe OeOeOeIe OeOeOeIe OeOeOeIe OeOeOeIe XxOe = .
B1P .
, .
> 0.
B2P .
, .
> 0.
B3P .
, .
> 0.
B1N .
, .
< 0.
B2N .
, .
< 0.
OeOeOeIe B3N .
, .
< .
OeOeOeIe (P )Oe (P )Oe (P )Oe : XxOe Ie .
, .
, such that tx : XxOe Ie .
, .
, defined by tx .
= B1P .
, .
, ix (P )Oe (P ) (P ) Oe = defined by ix .
= B2 .
, .
, fx : Xx Ie .
, .
, defined by fx OeOeOeIe (N )Oe OeOeOeIe (N )Oe (N )Oe Oe B3 .
, .
, tx : Xx Ie [Oe1, .
, defined by tx = B1 .
, .
, ix
: XxOe Ie
(N )
(N )
(N )Oe .
= B2N .
, .
, fx : XxOe Ie [Oe1, .
, defined by fx .
= [Oe1, .
, defined by ix OeOeOeIe B3 .
, .
Definition 2.
A bipolar neutrosophic competition graph of a bipolar neutroOeOeIe Ie Oe Ie Oe sophic graph G = (A.
B ) is an undirected bipolar neutrosophic graph C (G) = Ie Oe (A.
R) which has the same vertex set as in G and there is an edge between two Certain bipolar neutrosophic competition graphs vertices x and y if and only if N .
O N .
is non-empty.
The positive truthmembership, indeterminacy-membership, falsity-membership and negative truthmembership, indeterminacy-membership, falsity-membership values of the edge .
, .
are defined as, .
tP R .
, .
= .
A .
O tA .
)h1 (N .
O N .
), .
iR .
, .
= .
A .
O iA .
)h2 (N .
O N .
), .
fRP .
, .
= .
AP .
O fAP .
)h3 (N .
O N .
), .
tN R .
, .
= .
A .
O tA .
)h4 (N .
O N .
), .
iR .
, .
= .
A .
O iA .
)h5 (N .
O N .
), .
fRN .
, .
= .
AN .
O fAN .
)h6 (N .
O N .
), for all x, y OO X.
Ie Oe Example 2.
Consider G = (A.
B) is a bipolar single-valued neutrosophic digraph, such that.
X = .
, b, c, .
A = {.
, 0.
3, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, .
, 8, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
7, 0.
3, 0.
OeOeOeIe OeOeOeIe Oe0.
Oe0.
Oe0.
}, and B = {(.
, .
, 0.
2, 0.
1, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
OeOeOeIe OeOeOeIe 5, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
6, 0.
2, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
2, 0.
Oe0.
Oe0.
Oe0.
} as shown in Fig.
Figure 1.
Bipolar single-valued neutrosophic digraph By direct calculations we have Table 1 representing bipolar single-valued neutrosophic out-neighbourhoods.
Table 1.
Bipolar single-valued neutrosophic out-neighbourhoods N .
, 0.
2, 0.
1, 0.
1,-0.
4,-0.
1,-0.
, .
, 0.
3, 0.
5, 0.
6,-0.
2,-0.
2,-0.
, 0.
6, 0.
2, 0.
2,-0.
1,-0.
2,-0.
, 0.
2, 0.
2, 0.
2,-0.
2,-0.
3,-0.
Akram and M.
Nasir Then bipolar single-valued neutrosophic competition graph of Fig.
1 is shown in Fig.
Figure 2.
Bipolar single-valued neutrosophic competition graph Definition 2.
The support of a bipolar neutrosophic set A = .
, tP A , i A , fA , tA , iA , fA ) in X is the subset of X defined by supp(A) = .
OO X : tP A .
6= 0, iA .
6= 0, fA .
6= 1, tA .
6= Oe1, iA .
6= Oe1, fA .
6= .
upp(A)| is the number of elements in the set.
Example 2.
The support of a bipolar neutrosophic set A = {.
, 0.
5, 0.
7, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
1, 0.
2, 1.
Oe0.
Oe0.
Oe0.
, .
, 0.
3, 0.
5, 0.
Oe0.
Oe0.
Oe0.
, .
, 0, 0, 1.
Oe1.
Oe1, .
} in X = .
, b, c, .
is supp(A) = .
, b, .
upp(A)| = 3.
We now discuss p-competition bipolar neutrosophic graphs.
Definition 2.
Let p be a positive integer.
Then p-competition bipolar neuIe Oe Ie Oe Ie Oe trosophic graph C p ( G ) of the bipolar neutrosophic digraph G = (A.
B ) is an undirected bipolar neutrosophic graph G = (A.
B) which has same bipolar neuIe Oe trosophic set of vertices as in G and has a bipolar neutrosophic edge between two Ie Oe vertices x, y OO X in C p ( G ) if and only if .
upp(N .
O N .
)| Ou p.
The posiIe Oe .
Oe.
1 P tive truth-membership value of edge .
, .
in C p ( G ) is tP .
A .
O B .
, .
= tA .
]h1 (N .
O N .
), the positive indeterminacy-membership value of edge Ie Oe .
Oe.
1 P .
, .
in C p ( G ) is iP .
A .
O iP B .
, .
= A .
]h2 (N .
O N .
), positive Ie Oe falsity-membership value of edge .
, .
in C p ( G ) is fBP .
, .
= .
Oe.
AP .
O fA .
]h3 (N .
O N .
), the negative truth-membership value of edge .
, .
Certain bipolar neutrosophic competition graphs Ie Oe .
Oe.
1 N in C p ( G ) is tN .
A .
O tN B .
, .
= A .
]h4 (N .
O N .
), the negative Ie Oe .
Oe.
1 N indeterminacy-membership value of edge .
, .
in C p ( G ) is iN .
A .
O B .
, .
= iA .
]h5 (N .
O N .
), negative falsity-membership value of edge .
, .
in Ie Oe C p ( G ) is fBN .
, .
= .
Oe.
AN .
O fAN .
]h6 (N .
O N .
), where i = .
upp(N .
O N .
)|.
The 3Oecompetition bipolar neutrosophic graph is illustrated by the following example.
Ie Oe Ie Oe Example 2.
Consider G = (A.
B ) is a bipolar neutrosophic digraph, such that.
X = .
, y, z, a, b, .
A = {.
, 0.
7, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
7, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
6, 0.
7, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
5, 0.
6, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
5, 0.
6, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
5, 0.
6, 0.
Oe0.
Oe0.
OeOeOeIe OeOeOeIe Oe0.
}, and B = {(.
, .
, 0.
3, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
4, 0.
5, 0.
Oe0.
OeOeOeIe OeOeOeIe Oe0.
Oe0.
, (.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
OeOeOeIe OeOeOeIe Oe0.
, (.
, .
, 0.
4, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
OeOeOeIe OeOeOeIe (.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
, (.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
}, as shown in Fig.
Then N .
= {.
, 0.
3, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
N .
= {.
, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
N .
= {.
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
So.
N .
O N .
= {.
, 0.
3, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
4, 0.
5, 0.
Oe0.
Oe0.
Oe0.
Figure 3.
Bipolar neutrosophic digraph Akram and M.
Nasir Now i = .
upp(N .
O N .
)| = 3.
For p = 3, tP B .
, .
= 0.
08, iB .
, .
= 0.
1166, fB .
, .
= 0.
066, tB .
, .
= Oe0.
04, iB .
, .
= Oe0.
033, and fB .
, .
= Oe0.
As shown in Fig.
Figure 4.
3Oecompetition bipolar neutrosophic graph Ie Oe Ie Oe Theorem 2.
Let G = (A.
B ) be a bipolar neutrosophic digraph.
h1 (N .
O N .
) = 1, h2 (N .
O N .
) = 1, h3 (N .
O N .
) = 0, h4 (N .
O N .
) = 1, h5 (N .
O N .
) = 1, h6 (N .
O N .
) = 0.
Ie Oe in C [ 2 ] ( G ), then the edge .
, .
is strong, where i = .
upp(N .
O N .
)|.
(Note that for any real number x, .
=greatest integer not exceeding x.
Ie Oe Ie Oe Suppose G = (A.
B ) is a bipolar neutrosophic digraph.
Let the corIe Oe responding [ 2i ]-bipolar neutrosophic competition graph be C [ 2 ] ( G ), where i = .
upp(N .
O N .
)|.
Also, assume that h1 (N .
O N .
) = 1, h2 (N .
O N .
) = 1, h3 (N .
O N .
) = 0, h4 (N .
O N .
) = 1, h5 (N .
O N .
) = 1, h6 (N .
O N .
) = 0, for all x, y OO X.
Now, .
Oe [ 2i ]) 1 P .
Oe [ 2i ]) 1 N .
A .
O tP .
] .
, .
A .
O tN A .
] y 1, .
Oe [ 2i ]) 1 P .
Oe [ 2i ]) 1 N .
A .
O iP .
A .
O iN B .
, .
= A .
] y 1, iB .
, .
= A .
] y 1, .
Oe [ 2i ]) 1 P .
Oe [ 2i ]) 1 N fBP .
, .
= .
A .
O fAP .
] y 0, fBN .
, .
= .
A .
O fAN .
] y 0.
B .
, .
= Certain bipolar neutrosophic competition graphs This gives the result, .
Oe [ 2i ]) 1 B .
, .
> 0.
A .
O tA .
] .
Oe [ 2i ]) 1 B .
, .
> 0.
P A .
O iA .
] .
Oe [ 2i ]) 1 fBP .
, .
< 0.
A .
O fA .
] .
Oe [ 2i ]) 1 B .
, .
< 0.
A .
O tA .
] .
Oe [ 2i ]) 1 B .
, .
< 0.
N A .
O iA .
] .
Oe [ 2i ]) 1 fBN .
, .
< 0.
A .
O fA .
] Hence, the edge .
, .
is strong.
This proves the result.
We now define another extension of bipolar neutrosophic competition graph known as m-step bipolar neutrosophic competition graph.
In this paper, we will use the following notations:
: A bipolar neutrosophic path of length m from x to y.
Px,y Ie Oem P x,y : A directed bipolar neutrosophic path of length m from x to y.
: m-step bipolar neutrosophic out-neighbourhood of vertex x.
Oe .
: m-step bipolar neutrosophic in-neighbourhood of vertex x.
Nm .
: m-step bipolar neutrosophic neighbourhood of vertex x.
Nm (G): m-step bipolar neutrosophic neighbourhood graph of the bipolar neutrosophic graph G.
OeOeIe Cm (G): m-step bipolar neutrosophic competition graph of the bipolar neutrosophic Ie Oe digraph G .
Ie Oe Ie Oe Definition 2.
Suppose G = (A.
B ) is a bipolar neutrosophic digraph.
The mIe Oe Ie Oe step bipolar neutrosophic digraph of G is denoted by G m = (A.
B), where bipolar Ie Oe neutrosophic set of vertices of G is same with bipolar neutrosophic set of vertices Ie Oe Ie Oe of G m and has an edge between x and y in G m if and only if there exists a bipolar Ie Oem Ie Oe neutrosophic directed path P x,y in G .
Definition 2.
The bipolar neutrosophic m-step out-neighbourhood of vertex x of a bipolar neutrosophic digraph G = (A.
B ) is bipolar neutrosophic set
(P )
= (Xx , tx
(P )
, ix
(P )
, fx
(N )
, tx
(N )
, ix
(N )
, fx Xx = .
there exists a directed bipolar neutrosophic path of length m from x
(P )
(P )
(P )
to y.
P m : Xx Ie .
, .
, ix : Xz Ie .
, .
, fx : Xz Ie .
, .
x,y }, tx
(N )
(N )
(N )
: Xx Ie [Oe1, .
, ix : Xz Ie [Oe1, .
, fx : Xz Ie [Oe1, .
are defined by Oe
OeOeOeOeOeIe Ie
(P )
(P )
= min.
P .
1 , x2 ), .
1 , x2 ) is an edge of P m = min.
P .
1 , x2 ), .
1 , x,y }, ix
OeOeOeOeOeIe Ie
(P )
x2 ) is an edge of P m = max.
P .
1 , x2 ), .
1 , x2 ) is an edge of P m x,y }, fx x,y }.
OeOeOeOeOeIe OeOeOeOeOeIe Ie
Oem
(N )
(N )
= max.
1 , x2 ), .
1 , x2 ) is an edge of P x,y }, ix = max.
1 , x2 ).
Akram and M.
Nasir
OeOeOeOeOeIe Ie
(N )
1 , x2 ) is an edge of P m = min.
N .
1 , x2 ), .
1 , x2 ) is an edge of P m x,y }, fx x,y }.
Ie Oe Ie Oe Example 2.
Consider G = (A.
B ) is a bipolar neutrosophic digraph, such that X = .
, y, a, b, c, .
, as shown in Fig.
Then, 2-step out-neighbourhood of vertices x, and y is calculated as.
N2 .
= {.
, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
2, 0.
Oe0.
Oe0.
Oe0.
N2 .
= {.
, 0.
1, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
3, 0.
Oe0.
Oe0.
Oe0.
Figure 5.
Bipolar neutrosophic digraph Definition 2.
The bipolar neutrosophic m-step in-neighbourhood of vertex x of Ie Oe Ie Oe a bipolar neutrosophic digraph G = (A.
B ) is bipolar neutrosophic set (P )Oe Oe .
= (XxOe , tx (P )Oe , ix (P )Oe , fx (N )Oe , tx (N )Oe , ix (N )Oe , fx XxOe = .
there exists a directed bipolar neutrosophic path of length m from y Ie Oe (P )Oe (P )Oe (P )Oe : XzOe Ie .
, .
: XzOe Ie .
, .
, fx : XxOe Ie .
, .
, ix to x.
P m y,x }, tx (N )Oe (N )Oe (N )Oe : XzOe Ie [Oe1, .
are defined by : XxOe Ie [Oe1, .
, ix : XzOe Ie [Oe1, .
, fx Oe OeOeOeOeOeIe Oe Oe Oe Oe Oe Ie Ie Oe (P )Oe (P ) = min.
P .
1 , x2 ), .
1 , = min.
P .
1 , x2 ), .
1 , x2 ) is an edge of P m y,x }, ix OeOeOeOeOeIe Ie Oe Ie Oe (P )Oe x2 ) is an edge of P m = max.
P .
1 , x2 ), .
1 , x2 ) is an edge of P m y,x }, fx y,x }.
OeOeOeOeOeIe OeOeOeOeOeIe Ie Oem (N )Oe (N )Oe = max.
1 , x2 ), = max.
1 , x2 ), .
1 , x2 ) is an edge of P y,x }, ix OeOeOeOeOeIe Ie Oem Ie Oe (N )Oe .
1 , x2 ) is an edge of P y,x }, fx = min.
1 , x2 ), .
1 , x2 ) is an edge of P m y,x }.
Ie Oe Ie Oe Example 2.
Consider G = (A.
B ) is a bipolar neutrosophic digraph, such that.
X = .
, b, c, d, e, f }, as shown in Fig.
Then, 2-step in-neighbourhood of vertices a, and b is calculated as.
N2Oe .
= {.
, 0.
1, 0.
1, 0.
Oe0.
Oe0.
Oe0.
, .
Certain bipolar neutrosophic competition graphs 3, 0.
1, 0.
Oe0.
Oe0.
Oe0.
N2Oe .
= {.
, 0.
1, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, .
, 4, 0.
3, 0.
Oe0.
Oe0.
Oe0.
Figure 6.
Bipolar neutrosophic digraph Ie Oe Ie Oe Definition 2.
Suppose G = (A.
B ) is a bipolar neutrosophic digraph.
The Ie Oe m-step bipolar neutrosophic competition graph of bipolar neutrosophic digraph G Ie Oe is denoted by Cm ( G ) = (A.
B) which has same bipolar neutrosophic set of verIe Oe Ie Oe tices as in G and has an edge between two vertices x, y OO X in Cm ( G ) if and Ie Oe .
) is a non-empty bipolar neutrosophic set in G .
The .
O Nm only if (Nm Ie Oe positive truth-membership value of edge .
, .
in Cm ( G ) is tP B .
, .
= .
A .
O tA .
]h1 (Nm .
ONm .
), the positive indeterminacy-membership value of edge .
Ie Oe .
in Cm ( G ) is iP B .
, .
= .
A .
O iA .
]h2 (Nm .
O Nm .
), the positive falsityIe Oe .
O membership value of edge .
, .
in Cm ( G ) is fBP .
, .
= .
AP .
OfAP .
]h3 (Nm Ie Oe Nm .
), the negative truth-membership value of edge .
, .
in Cm ( G ) is tN B .
, .
= .
N .
O .
]h .
O .
), indeterminacy-membership Ie Oe value of edge .
, .
in Cm ( G ) is iN B .
, .
= .
A .
O iA .
]h5 (Nm .
O Nm .
Ie Oe the negative falsity-membership value of edge .
, .
in Cm ( G ) is fBN .
, .
= .
O Nm .
AN .
O fAN .
]h6 (Nm The 2Oestep bipolar neutrosophic competition graph is illustrated by the following Ie Oe Ie Oe Example 2.
Consider G = (A.
B ) is a bipolar neutrosophic digraph, such that X = .
, y, a, b, c, .
, as shown in Fig.
Then.
N2 .
= {.
, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
N2 .
= {.
, 0.
1, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
3, 0.
5, 0.
Oe0.
Oe0.
Oe0.
}, there non-empty intersection Akram and M.
Nasir Figure 7.
Bipolar neutrosophic digraph is calculated as N2 .
O N2 .
= {.
, 0.
1, 0.
2, 0.
Oe0.
Oe0.
Oe0.
, .
, 0.
2, 0.
Oe0.
Oe0.
Oe0.
Thus, tP B .
, .
= 0.
08, iB .
, .
= 0.
04, fB .
, .
= 0.
35, tB .
, .
= Oe0.
iB .
, .
= Oe0.
06, and fB .
, .
= Oe0.
Then its corresponding 2-step bipolar neutrosophic competition graph is shown in Fig.
Figure 8.
2-step bipolar neutrosophic competition graph OeOeOeIe If a predator x attacks one prey y, then the linkage is shown by an edge .
, .
in a bipolar neutrosophic digraph.
But, if predator needs help of many other mediators x1 , x2 , .
xmOe1 , then linkage among them is shown by bipolar neuIe Oe trosophic directed path P m x,y in a bipolar neutrosophic digraph.
So, m-step prey in a bipolar neutrosophic digraph is represented by a vertex which is the m-step out-neighbourhood of some vertices.
Now, the strength of a bipolar neutrosophic competition graphs is defined below.
Certain bipolar neutrosophic competition graphs Ie Oe Ie Oe Definition 2.
Let G = (A.
B ) be a bipolar neutrosophic digraph.
Let w be a common vertex of m-step out-neighbourhoods of vertices x1 , x2 ,.
, xl .
OeIe OeIe OeIe Also, let B1P .
1 , v1 ).
B1P .
2 , v2 ),.
B1P .
l , vl ) be the minimum positive truthOeIe OeIe OeIe membership values.
B2P .
1 , v1 ).
B2P .
2 , v2 ),.
B2P .
l , vl ) be the minimum positive OeIe OeIe OeIe indeterminacy-membership values.
B3P .
1 , v1 ).
B3P .
2 , v2 ),.
B3P .
l , vl ) be the OeOeIe OeOeIe OeOeIe maximum positive false-membership values.
B1N .
1 , v1 ).
B1N .
2 , v2 ),.
B1N .
l , vl ) OeOeIe OeOeIe be the maximum negative truth-membership values.
B2N .
1 , v1 ).
B2N .
2 , v2 ),.
OeOeIe OeOeIe B2N .
l , vl ) be the maximum negative indeterminacy-membership values.
B3N .
1 , v1 ).
OeOeIe OeOeIe B3N .
2 , v2 ),.
B3N .
l , vl ) be the minimum negative false-membership values, of Ie Oe Ie Oem Ie Oem edges of the paths P m x1 ,w .
P x2 ,w ,.
P xl ,w , respectively.
The m-step prey w OO X is strong prey if OeIe OeIe OeIe B1P .
i , vi ) > 0.
B2P .
i , vi ) > 0.
B3P .
i , vi ) < 0.
OeOeIe OeOeIe OeOeIe B1N .
i , vi ) < 0.
B2N .
i , vi ) < 0.
B3N .
i , vi ) < 0.
5, for all i = 1, 2, .
, l.
The strength of the prey w can be measured by the mapping S : X Ie .
, .
, such ( l OeIe OeIe 1 X OeIe [B1P .
i , vi )] [B2P .
i , vi )] [B3P .
i , vi )] S.
= l i=1 X OeOeIe X OeOeIe X OeOeIe Oe [B1 .
i , vi )] Oe [B2 .
i , vi )] Oe [B3 .
i , vi )] .
Ie Oe Ie Oe Example 2.
Consider bipolar neutrosophic digraph G = (A.
B ) as shown in Fig.
7, the strength of the prey b is equal to .
Oe [Oe0.
3 Oe 0.
Oe [Oe0.
3 Oe 0.
Oe [Oe0.
3 Oe 0.
= 1.
> 0.
Hence, b is strong 2-step prey.
Ie Oe Ie Oe Theorem 2.
If a prey w of G = (A.
B ) is strong, then the strength of w.
> 0.
Ie Oe Ie Oe Let G = (A.
B ) be a bipolar neutrosophic digraph.
Let w be a common vertex of m-step out-neighbourhoods of vertices x1 , x2 ,.
, xl , i.
, there exOeIe OeIe Ie Oe Ie Oem Ie Oem Ie Oe ists the paths P m x1 ,w .
P x2 ,w ,.
P xl ,w , in G .
Also, let B1 .
1 , v1 ).
B1 .
2 , v2 ),.
OeIe Oe Ie OeIe B1P .
l , vl ) be the minimum positive truth-membership values.
B2P .
1 ,v1 ).
B2P .
2 ,v2 ).
OeIe OeIe .
B2P .
l ,vl ) be the minimum positive indeterminacy-membership values.
B3P .
OeIe OeIe v1 ).
B3P .
2 , v2 ),.
B3P .
l , vl ) be the maximum positive false-membership values.
Akram and M.
Nasir OeOeIe OeOeIe OeOeIe B1N .
1 , v1 ).
B1N .
2 ,v2 ),.
B1N .
l , vl ) be the maximum negative truth-membership OeOeIe OeOeIe OeOeIe values.
B2N .
1 , v1 ).
B2N .
2 , v2 ),.
B2N .
l , vl ) be the maximum negative indeterminacyOeOeIe Oe Oe Ie OeOeIe membership values.
B3N .
1 , v1 ).
B3N .
2 , v2 ),.
B3N .
l , vl ) be the minimum negIe Oe Ie Oem Ie Oem ative false-membership values, of edges of the paths P m x1 ,w .
P x2 ,w ,.
P xl ,w , respectively.
If w is strong, each edge .
i , vi ), i = 1, 2, .
, l is strong.
So.
OeIe OeIe OeIe B1P .
i , vi ) > 0.
B2P .
i , vi ) > 0.
B3P .
i , vi ) < 0.
OeOeIe OeOeIe OeOeIe B1N .
i , vi ) < 0.
B2N .
i , vi ) < 0.
B3N .
i , vi ) < 0.
5, for all i = 1, 2, .
, l.
Now.
> 5 0.
> 0.
This proves the result.
Remark: The converse of the above theorem is not true, i.
if S.
> 0.
then all preys may not be strong.
This can be explained as:
Ie Oe Let S.
> 0.
5 for a prey w in G .
So, ( l OeIe OeIe 1 X OeIe [B3P .
i , vi )] [B1P .
i , vi )] [B2P .
i , vi )] S.
= l i=1 OeOeIe Oe Oe Ie Oe Oe Ie Oe [B1N .
i , vi )] Oe [B2N .
i , vi )] Oe [B3N .
i , vi )] .
Hence.
OeIe OeIe OeIe [B3P .
i , vi )] [B2P .
i , vi )] [B1P .
i , vi )] Oe OeOeIe [B1N .
i , vi )] Oe OeOeIe OeOeIe [B2 .
i , vi )] Oe [B3 .
i , vi )] > .
This result does not necessarily imply that OeIe OeIe OeIe B1P .
i , vi ) > 0.
B2P .
i , vi ) > 0.
B3P .
i , vi ) < 0.
OeOeIe OeOeIe OeOeIe B1N .
i , vi ) < 0.
B2N .
i , vi ) < 0.
B3N .
i , vi ) < 0.
5, for all i = 1, 2, .
, l.
Ie Oe Ie Oem Ie Oem Since, all edges of the directed paths P m x1 ,w .
P x2 ,w ,.
P xl ,w , are not strong.
So.
Ie Oe the converse of the above statement is not true i.
, if S.
> 0.
5, the prey w of G may not be strong.
Certain bipolar neutrosophic competition graphs Ie Oe Ie Oe Ie Oe Theorem 2.
If all preys of G = (A.
B ) are strong, then all edges of Cm ( G ) = (A.
B) are strong.
Ie Oe Ie Oe Let G = (A.
B ) be a bipolar neutrosophic digraph and all preys of it are Ie Oe Let Cm ( G ) = (A.
B), where.
B .
, .
= .
A .
O tA .
]h1 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h2 (Nm .
O Nm .
), fBP .
, .
= .
AP .
O fAP .
]h3 (Nm .
O Nm .
B .
, .
= .
A .
O tA .
]h4 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h5 (Nm .
O Nm .
), fBN .
, .
= .
AN .
O fAN .
]h6 (Nm .
O Nm .
Ie Oe for all edges .
, .
in Cm ( G ) = (A.
B).
Then there arises two cases:
Case 1.
: Let Nm .
O Nm .
be null set.
Then there does not exits any edge Ie Oe between x and y in Cm ( G ).
be non-empty.
Now, clearly .
O Nm Case 2.
: Let Nm h1 (Nm .
O Nm .
) > 0.
h2 (Nm .
O Nm .
) > 0.
h3 (Nm .
O Nm .
) < 0.
h4 (Nm .
O Nm .
) < 0.
5, h5 (Nm .
O Nm .
) < 0.
5, h6 (Nm .
O Nm .
) < 0.
Ie Oe Ie Oe in G as all preys are strong.
So, the edge .
, .
, x, y OO X in Cm ( G ) have the memberships values B .
, .
= .
A .
O tA .
]h1 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h2 (Nm .
O Nm .
), fBP .
, .
= .
AP .
O fAP .
]h3 (Nm .
O Nm .
B .
, .
= .
A .
O tA .
]h4 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h5 (Nm .
O Nm .
), fBN .
, .
= .
AN .
O fAN .
]h6 (Nm .
O Nm .
), and hence, all the edges are strong.
A relation is established between m-step bipolar neutrosophic competition graph of a bipolar neutrosophic digraph and bipolar neutrosophic competition graph of m-step bipolar neutrosophic digraph.
Ie Oe OeIe Theorem 2.
If G is a bipolar neutrosophic digraph and Gm is the m-step Ie Oe Ie Oe Ie Oe bipolar neutrosophic digraph of G , then C( G m ) = Cm ( G ).
Ie Oe Ie Oe OeIe Ie Oe Let G = (A.
B ) be a bipolar neutrosophic digraph and Gm = (A.
J ) Ie Oe Ie Oe is the m-step bipolar neutrosophic digraph of G .
Also, let C( G m ) = (A.
J) and Ie Oe Cm ( G ) = (A.
B).
It can be easily observed that bipolar neutrosophic vertex sets of these graphs are same.
So, we have to show that the bipolar neutrosophic edge Akram and M.
Nasir Ie Oe Ie Oe sets of C( G m ) and Cm ( G ) are equal.
Ie Oe Let .
, .
be an edge in C( G m ).
So, there exists bipolar neutrosophic directed OeOeOeOeIe OeOeOeOeIe OeOeOeOeIe OeOeOeOeIe OeOeOeIe OeOeOeIe edges .
, a1 ), .
, a1 ).
, a2 ), .
, a2 ).
, al ), .
, al ), for some positive integer l in Ie Oe Ie Oe G m .
Now, in G m .
N N N .
O N .
= {.
i , sP i , qi , ri , si , qi , ri ).
= 1, 2, .
, .
Ie Oe Ie Oe i = J .
, ai ) O J .
, ai ).
Ie Oe Ie Oe qiP = J .
, ai ) O J .
, ai ).
Ie Oe Ie Oe riP = J .
, ai ) O J .
, ai ).
Ie Oe Ie Oe i = J .
, ai ) O J .
, ai ).
Ie Oe Ie Oe qiN = J .
, ai ) O J .
, ai ).
Ie Oe Ie Oe riN = J .
, ai ) O J .
, ai ).
Let S P = max.
P i .
= 1, 2, .
, .
S N = min.
N i .
= 1, 2, .
, .
QP = max.
iP .
= 1, 2, .
, .
QN = min.
iN .
= 1, 2, .
, .
RP = min.
iP .
= 1, 2, .
, .
RN = max.
iN .
= 1, 2, .
, .
Hence.
J .
, .
= .
A .
O tA .
)h1 (N .
O N .
) = S y tA .
O tA .
J .
, .
= .
A .
O iA .
)h2 (N .
O N .
) = Q y iA .
O iA .
, fJP .
, .
= .
AP .
O fAP .
)h3 (N .
O N .
) = RP y fAP .
O fAP .
J .
, .
= .
A .
O tA .
)h4 (N .
O N .
) = S y tA .
O tA .
J .
, .
= .
A .
O iA .
)h5 (N .
O N .
) = Q y iA .
O iA .
, fJN .
, .
= .
AN .
O fAN .
)h6 (N .
O N .
) = RN y fAN .
O fAN .
OeOeOeIe Ie Oe An edge .
, ai ) exists in G m that implies there exists a bipolar neutrosophic diIe Oe Ie Oe rected path from x to ai of length m.
P m x,ai in G and Ie OeP Ie Oe Ie Oem J 1 .
, ai ) = min{ B P 1 .
, .
, .
is an edge in P x,ai }.
Ie OeP Ie Oe Ie Oem J 2 .
, ai ) = min{ B P 2 .
, .
, .
is an edge in P x,ai }.
Ie OeP Ie Oe Ie Oem J 3 .
, ai ) = max{ B P 3 .
, .
, .
is an edge in P x,ai }.
Ie OeN Ie Oe Ie Oem J 1 .
, ai ) = max{ B N 1 .
, .
, .
is an edge in P x,ai }.
Ie OeN Ie Oe Ie Oem J 2 .
, ai ) = max{ B N 2 .
, .
, .
is an edge in P x,ai }.
Ie OeN Ie Oe Ie Oem J 3 .
, ai ) = min{ B N 3 .
, .
, .
is an edge in P x,ai }.
Certain bipolar neutrosophic competition graphs Ie Oe Thus, the edge .
, .
is also available in Cm ( G ).
Also, h1 (Nm .
O Nm .
) = S P , h4 (Nm .
O Nm .
) = S N , h2 (Nm .
O Nm .
) = QP , h5 (Nm .
O Nm .
) = QN , h3 (Nm .
O Nm .
) = RP , h6 (Nm .
O Nm .
) = RN .
Ie Oe in G .
Hence, finally B .
, .
= .
A .
O tA .
]h1 (Nm .
O Nm .
) = S y tA .
O tA .
B .
, .
= .
A .
O iA .
]h2 (Nm .
O Nm .
) = Q y iA .
O iA .
, fBP .
, .
= .
AP .
O fAP .
]h3 (Nm .
O Nm .
) = RP y fAP .
O fAP .
B .
, .
= .
A .
O tA .
]h4 (Nm .
O Nm .
) = S y tA .
O tA .
B .
, .
= .
A .
O iA .
]h5 (Nm .
O Nm .
) = Q y iA .
O iA .
, fBN .
, .
= .
AN .
O fAN .
]h6 (Nm .
O Nm .
) = RN y fAN .
O fAN .
Ie Oe Ie Oe This proves that there exists an edge in Cm ( G ) for every edge in C( G m ).
Similarly.
Ie Oe Ie Oe Ie Oe for every edge in Cm ( G ) there exists an edge in C( G m ).
This proves that C( G m ) = Ie Oe Cm ( G ).
Ie Oe Ie Oe Theorem 2.
Let G = (A.
B ) be a bipolar neutrosophic digraph.
If m > |X| Ie Oe then Cm ( G ) = (A.
B) has no edge.
Ie Oe Ie Oe Ie Oe Let G = (A.
B ) be a bipolar neutrosophic digraph and Cm ( G ) = (A.
B) be the corresponding m-step bipolar neutrosophic competition graph, where.
B .
, .
= .
A .
O tA .
]h1 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h2 (Nm .
O Nm .
), fBP .
, .
= .
AP .
O fAP .
]h3 (Nm .
O Nm .
B .
, .
= .
A .
O tA .
]h4 (Nm .
O Nm .
B .
, .
= .
A .
O iA .
]h5 (Nm .
O Nm .
), fBN .
, .
= .
AN .
O fAN .
]h6 (Nm .
O Nm .
Ie Oe for all edges .
, .
in Cm ( G ).
If m > |X|, there does not exist any directed bipolar neutrosophic path of length Ie Oe m in G .
So.
Nm .
O Nm .
ia an empty set.
Hence, there does not exist any edge Ie Oe in Cm ( G ).
Now, m-step bipolar neutrosophic neighbouhood graphs are defines below.
Definition 2.
The bipolar neutrosophic m-step out-neighbourhood of vertex x Ie Oe Ie Oe of a bipolar neutrosophic digraph G = (A.
B ) is bipolar neutrosophic set Akram and M.
Nasir Nm .
= (Xx , tP x , ix , fx , tx , ix , fx ).
Xx = .
there exists a directed bipolar neutrosophic path of length m from x to y.
Px,y }, tP x : Xx Ie .
, .
, ix : Xx Ie .
, .
, fx : Xx Ie .
, .
, tx : Xx Ie [Oe1, .
, ix : Xx Ie [Oe1, .
, fx : Xx Ie [Oe1, .
, are defined by tx = min.
P .
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, iP x = min.
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, fx = max.
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, tN x = max.
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, iN x = max.
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, fx = min.
1 , x2 ), .
1 , x2 ) is an edge of Px,y }, respectively.
Definition 2.
Suppose G = (A.
B) is a bipolar neutrosophic graph.
Then m-step bipolar neutrosophic neighbouhood graph Nm (G) is defined by Nm (G) = (A.
BA) where A = (AP 1 .
A2 .
A3 .
A1 .
A2 .
A3 ).
BA = (BA1 .
BA2 .
BA3 .
BA1 .
BA2 .
BA3 ).
BA1 : X yX Ie .
, .
BA2 : X yX Ie .
, .
BA3 : X yX Ie .
, .
BA1 : X yX Ie [Oe1, .
BA2N : X y X Ie [Oe1, .
, and BA3N : X y X Ie [Oe1, .
are such that:
BA1P .
, .
= AP 1 .
O A1 .
h1 (Nm .
O Nm .
BA2P .
, .
= AP 2 .
O A2 .
h2 (Nm .
O Nm .
BA3P .
, .
= AP 3 .
O A3 .
h3 (Nm .
O Nm .
BA1N .
, .
= AN 1 .
O A1 .
h4 (Nm .
O Nm .
BA2N .
, .
= AN 2 .
O A2 .
h5 (Nm .
O Nm .
BA3N .
, .
= AN 3 .
O A3 .
h6 (Nm .
O Nm .
Definition 2.
[?]Consider a bipolar neutrosophic graph G = (A.
B), where A = (AP 1 .
A2 .
A3 .
A1 .
A2 .
A3 ), and B = (B1 .
B2 .
B3 .
B1 .
B2 .
B3 ) then, an edge .
, .
, x, y OO X is called independent strong if [A .
O AP 1 .
] < B1 .
, .
, [A .
O AP 2 .
] < B2 .
, .
, [A .
O AP 3 .
] > B3 .
, .
, [A .
O AN 1 .
] > B1 .
, .
, [A .
O AN 2 .
] > B2 .
, .
, [A .
O AN 3 .
] < B3 .
, .
Otherwise, it is called weak.
Ie Oe Ie Oe Theorem 2.
If all the edges of bipolar neutrosophic digraph G = (A.
B ) are Ie Oe independent strong, then all the edges of Cm ( G ) are independent strong.
Ie Oe Ie Oe Ie Oe Suppose G = (A.
B ) is a bipolar neutrosophic digraph and Cm ( G ) = (A.
B) is corresponding m-step bipolar neutrosophic competition graph.
Since all the Certain bipolar neutrosophic competition graphs Ie Oe edges of G are independent strong, then h1 (Nm .
O Nm .
) > 0.
h2 (Nm .
O Nm .
) > 0.
h3 (Nm .
O Nm .
) < 0.
h4 (Nm .
O Nm .
) < 0.
h5 (Nm .
O Nm .
) < 0.
h6 (N m .
O Nm .
) < 0.
Then.
B .
, .
= .
A .
O tA .
)h1 (Nm .
O Nm .
) or, tB .
, .
> 0.
A .
O tA .
) or.
B .
P A .
OtA .
) > 0.
5, iP B .
, .
= .
A .
O iA .
)h2 (Nm .
O Nm .
) or, iB .
, .
> iP .
P A .
O iA .
) or, .
P .
OiP .
) > 0.
5, fB .
, .
= .
A .
O fA .
)h3 (Nm .
O f P .
) or, fBP .
, .
< 0.
AP .
O fAP .
) or, .
P .
Of P .
) < 0.
5, tB .
, .
= tN .
N A .
OtA .
)h4 (Nm .
ONm .
) or, tB .
, .
< 0.
A .
OtA .
) or, .
N .
OtN .
) < 5, iN B .
, .
= .
A .
O iA .
)h5 (Nm .
O Nm .
) or, iB .
, .
< 0.
A .
O iN .
A .
) or, .
N .
OiN .
) < 0.
5, fB .
, .
= .
A .
O fA .
)h6 (Nm .
O Nm .
) or, f N .
fBN .
, .
< 0.
AN .
O fAN .
) or, .
N .
Of < 0.
A .
) Ie Oe Hence, the edge .
, .
is independent strong in Cm ( G ).
Since, .
, .
is taken to Ie Oe Ie Oe be arbitrary edge of Cm ( G ), thus all the edges of Cm ( G ) are independent strong.
Application Sports are very important, every society has its own special kinds of sports.
The proper end of sports is bodily health arid physical fitness.
Sports and games have now come to stay in our civilization as an essential feature of human activity, and their object is not merely fun, they also instill the sprit of discipline and teamwork.
Sports like cricket, hockey and foot ball are popular because of the sprit of team work which they inspire.
This no doubt true.
The discipline that gained in playing up sports is invaluable in later life.
It makes for a life of co-operation and team work which could be used for building up a great society and a nation.
Key components of sports are goals, rules, challenge, and interaction.
Sports generally involve mental or physical stimulation, and often both.
Many sports help develop practical skills, serve as a form of exercise, or otherwise perform an educational, simulational, or psychological role etc.
Many sports require special equipment and dedicated playing fields, leading to the involvement of a community much larger than the group of players.
A city or town may set aside such resources for the organization of sports leagues, like, tabletop games, board games, etc.
All these types of sports are called local sports.
These sports can be extended to provisional level sports.
After provisional level sports there are national sports.
A national sport is a sport or game that is considered to be an intrinsic part of the culture of a nation.
Every nation has different sports, such as, baseball is known as national sports in the United States, cricket is in England, and hockey is in Pakistan, etc.
After, national level of sports there are international level of sports.
International sport is a sport in which the participants represent different countries.
The most Akram and M.
Nasir well-known international sports event is the Olympic Games.
FIFA World Cup and the Paralympic Games.
Consider the set consisting of three countries {C1 .
C2 .
C3 } and also consider the set of players {(Abigail, 0.
9, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, (Alex, 0.
6, 0.
3, 0.
Oe0.
Oe0.
Oe0.
,(Amelia, 0.
8, 0.
7, 0.
Oe0.
Oe0.
Oe0.
, (Agatha, 0.
9, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, (Angela, 0.
9, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, (Belinda, 0.
9, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, (Ann, 0.
5, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, (Arlene, 0.
8, 0.
8, 0.
Oe0.
Oe0.
Oe0.
, (Bella, 0.
6, 0.
4, 0.
Oe0.
Oe0.
Oe0.
, (Anne, 0.
9, 0.
7, 0.
Oe0.
Oe0.
Oe0.
, (April, 0.
5, 0.
3, 0.
Oe0.
Oe0.
Oe0.
, (Abbey, 0.
5, 0.
3, 0.
Oe0.
Oe0.
Oe0.
}, which are taking part in their local, provisional, national, and international level games, as shown in Fig.
The positive degree of membership tP .
of each player represent the percentage of hardwork towards to achieve the success in particular game, iP .
and f P .
represent the indeterminacy and falsity in this The negative degree of membership tN .
represents the percentage that the player faces failure in the achievement of success in a particular game, iN .
and f N .
represent the indeterminacy and falsity in this percentage.
The positive degree of membership tP .
of each directed edge between player and local, provisional, national and international level games represent the percentage of having stamina for that level of sports in international game, iP .
and f P .
represent the indeterminacy and falsity in this percentage.
The negative degree of membership tN .
of each directed edge between player and local, provisional, national and international level games represent the percentage of having no stamina for that level of sports in international game, iN .
and f N .
represent the indeterminacy and falsity in this percentage.
Thus, 4-step bipolar neutrosophic competition graph can be used in order to find the best results.
There 4-step bipolar neutrosophic out-neighbourhoods is calculated in Table 2.
Table 2.
4-Step bipolar neutrosophic out-neighbourhoods xOOX N4 .
Abigail {(International games, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
Alex {(International games, 0.
4, 0.
2, 0.
Oe0.
Oe0.
Oe0.
Amelia {(International games, 0.
5, 0.
5, 0.
Oe0.
Oe0.
Oe0.
Therefore.
N4 (Abigai.
O N4 (Ale.
= {(International games, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
N4 (Abigai.
O N4 (Ameli.
= {(International games, 0.
2, 0.
2, 0.
Oe0.
Oe0.
Oe0.
}, and N4 (Ale.
O N4 (Ameli.
= {(International games, 0.
2, 0.
Oe0.
Oe0.
Oe0.
Further, h(N4 (Abigai.
O N4 (Ale.
) = .
2, 0.
2, 0.
2, 0.
2, 0.
, h(N4 (Abigai.
O N4 (Ameli.
) = .
2, 0.
2, 0.
6, 0.
2, 0.
2, 0.
, and h(N4 (Ameli.
O N4 (Ale.
) = .
4, 0.
2, 0.
6, 0.
4, 0.
2, 0.
Thus, we obtain 4-step bipolar neutrosophic competition graph, as shown in Fig.
Certain bipolar neutrosophic competition graphs Figure 9.
Bipolar neutrosophic digraph Table 3.
Strength of competition of applicants for international games .
, .
(Abigail.
Ale.
(Abigail.
Ameli.
(Alex.
Ameli.
T .
, .
12, 0.
06, 0.
Oe0.
Oe0.
Oe0.
16, 0.
14, 0.
Oe0.
Oe0.
Oe0.
24, 0.
06, 0.
Oe0.
Oe0.
Oe0.
, .
The strength to compete the others players with respect hardwork in order to achieve success is calculated in Table 3.
In Table 3.
T .
, .
represents the value of strength of competition between players x and y with respect to hardwork to Akram and M.
Nasir Figure 10.
4-Step bipolar neutrosophic competition graph achieve the success in particular game.
From Table 3, it is clear that the strength of competition between Alex and Amelia to achieve the success in particular game in international level is 1.
24, while strength of competition between between Abigail and Amelia is 1, and strength of competition between between Abigail and Alex is It is also clear from the Table 3, that Alex and Amelia are strongest contestants, as the strength of competition between them has the largest value than the other contestants.
We now elaborate this method with the help of an algorithm.
Certain bipolar neutrosophic competition graphs Algorithm Step 1.
: Input the positive truth, indeterminacy and falsity-memberships values and negative truth, indeterminacy and falsity-memberships values for set of r applicants.
Step 2.
: If for any two distinct vertices xi and xj , tP .
i xj ) > 0, iP .
i xj ) > 0, f P .
i xj ) > 0, tN .
i xj ) < 0, iN .
i xj ) < 0, f N .
i xj ) < 0, then .
j , tP .
i xj ), iP .
i xj ), f P .
i xj ), tN .
i xj ), iN .
i xj ), f N .
i xj )) OO Nm .
i ).
Step 3.
: Repeat step 2 for all vertices xi and xj to calculate m-step bipolar neutrosophic-out-neighbourhoods Nm .
i ).
Step 4.
: Calculate Nm .
i ) O Nm .
j ) for each pair of distinct vertices xi and Step 5.
: Calculate h[Nm .
i ) O Nm .
j )].
Step 6.
: If Nm .
i ) O Nm .
j ) 6= OI then draw an edge xi xj .
Step 7.
: Repeat step 6 for all pair of distinct vertices.
Step 8.
: Assign membership values to each edge xi xj using the conditions tP .
i xj ) = .
i O xj )h1 [Nm .
i ) O Nm .
j )] tN .
i xj ) = .
i O xj )h4 [Nm .
i ) O Nm .
j )] iP .
i xj ) = .
i O xj )h2 [Nm .
i ) O Nm .
j )] iN .
i xj ) = .
i O xj )h5 [Nm .
i ) O Nm .
j )] f P .
i xj ) = .
i O xj )h3 [Nm .
i ) O Nm .
j )] f N .
i xj ) = .
i O xj )h6 [Nm .
i ) O Nm .
j )].
Step 10.
: Calculate S.
, .
, the strength of competition between players x and y.
, .
= tP .
, .
Oe .
P .
, .
f P .
, .
) 1 tN .
, .
Oe .
N .
, .
f N .
, .
Step 11.
: Maximum value of S.
, .
gives that x and y are strongest players than the others.
Concluding Remarks Graph theory is an enjoyable playground for the research of proof techniques in discrete mathematics.
There are many applications of graph theory in different We have introduced the concepts of the bipolar neutrosophic competition We have described an application of m-step bipolar neutrosophic competition graphs in different level of games with the help of an algorithm.
We aim to extend our research work to .
Bipolar fuzzy rough graphs.
Bipolar fuzzy rough hypergraphs, .
Bipolar fuzzy rough neutrosophic graphs, and .
Decision support systems based on bipolar neutrosophic graphs.
Acknowledgement.
The authors are highly thankful to Executive Editor and the referees for their valuable comments and suggestions.
Akram and M.
Nasir
References