Certain Bipolar Neutrosophic Competition Graphs

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J. Indones. Math. Soc.

Vol. 24, No. 1 (2018), pp. 1–25.

Muhammad Akram¹ and Maryam Nasir²

1,2 Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan

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 ofp-competition bipolar neutrosophic graphs. We then define generalization of bipolar neutrosophic competition graphs calledm-step bipolar neutrosophic competition graphs. More- over, we present some related concepts of bipolar neutrosophic graphs. Finally, we describe an application ofm-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 peneli- tian. 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 bipolarp-kompetisi. Kemudian kami mendefinisikan perumuman dari graf kompetisi neutrosopik bipolar yang dise- but dengan graf kompetisi neutrosopik bipolarm-langkah. Lebih jauh, kami menya- jikan beberapa konsep yang terkait dengan graf neutrosopik bipolar. Akhirnya, kami menggambarkan sebuah aplikasi dari graf kompetisi neutrosopik bipolarm-langkah.

Kata kunci: Graf neotrosopik bipolarp-kompetisi, graf kompetisi neutrosopik bipolarm-langkah, algoritma.

2000 Mathematics Subject Classification: 03E72, 68R10, 68R05.

Received: 28-02-2017, revised: 11-04-2017, accepted: 09-05-2017.

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M. Akram and M. Nasir 1. Introduction

The notion of competition graphs was introduced by Cohen [10] in 1968, de- pending 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 [26] and intuitionistic fuzzy set theory [6] are useful models for dealing with uncertainty and incomplete information. But they may not be suf- ficient 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 [18] as a generalization of fuzzy sets and intuitionistic fuzzy sets.

However, since neutrosophic sets are identified by three functions called truth- membership (t), indeterminacy-membership (i) and falsity-membership (f) whose values are real standard or non-standard subset of unit interval ]0⁻,1⁺[. There are some difficulties in modeling of some problems in engineering and sciences. To overcome these difficulties, in 2010, concept of single-valued neutrosophic sets and its operations defined by Wanget al. [22] as a generalization of intuitionistic fuzzy sets. Ye [24, 25] has presented several novel applications of neutrosophic sets.

Deli et al. [11] extended the ideas of bipolar fuzzy sets [28] and neutrosophic sets to bipolar neutrosophic sets and studied its operations and applications in decision making problems. Smarandache [20] proposed notion of neutrosophic graph and they separated them to four main categories. Wu [23] discussed fuzzy digraphs.

The concept of fuzzy k-competition graphs and p-competition fuzzy graphs was first introduced by Samanta and Pal in [15], it was further studied in [5, 13, 16, 17].

Choet al. [9] proposed the generalization of a digraphs known asm-step competition graphs. Samantaet al. [16] 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 [17, 13]. Samantaet al. [16] also introduced the concepts of fuzzym- step neighbouthood graphs. The notion of bipolar fuzzy graphs was first introduced by Akram [1] in 2011 as a generalization of fuzzy graphs. On the other hand, Akram and Shahzadi [4] first introduced the notion of neutrosophic soft graphs and gave its applications. Akram [2] introduced the notion of single-valued neutrosophic planar graphs. Akram and Sarwar have shown that there are some flaws in Broumi et al.

[8] ’s 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 ne- cessity ofm-step bipolar neutrosophic competition graphs. In this research paper, we first introduce the concept of p-competition bipolar neutrosophic graphs. We then define generalization of bipolar neutrosophic competition graphs calledm-step bipolar neutrosophic competition graphs. Moreover, we present some related concepts of bipolar neutrosophic graphs. Finally, we describe an application ofm-step bipolar neutrosophic competition graphs.

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2. Certain Bipolar Neutrosophic Competition Graphs

Definition 2.1. [26, 27]Afuzzy setµin a universeX is a mappingµ:X→[0,1].

A fuzzy relation onX is a fuzzy setν inX×X.

Definition 2.2. [28]Abipolar fuzzy seton a non-empty set X has the form A={(x, µ^P_A(x), µ^N_A(x)) :x∈X}

where, µ^P_A : X → [0,1] and µ^N_A : X → [−1,0] are mappings. The positive membership value µ^P_A(x) represents the strength of truth or satisfaction of an element x to a certain property corresponding to bipolar fuzzy set A and µ^N_A(x) denotes the strength of satisfaction of an element xto some counter property of bipolar fuzzy set A. Ifµ^P_A(x)6= 0 andµ^N_A(x) = 0 it is the situation whenxhas only truth satisfaction degree for propertyA. Ifµ^N_A(x)6= 0 andµ^P_A(x) = 0, it is the case that xis not satisfying the property ofAbut satisfying the counter property toA. It is possible forxthat µ^P_A(x)6= 0 andµ^N_A(x)6= 0 whenxsatisfies the property ofAas well as its counter property in some part ofX.

Definition 2.3. [1]Let X be a non-empty set. A mapping B = (µ^P_B, µ^N_B) :X × X →[0,1]×[−1,0] is abipolar fuzzy relationonX such thatµ^P_B(xy)∈[0,1] and µ^N_B(xy)∈[−1,0] forx, y∈X.

Definition 2.4. [1]A bipolar fuzzy graph on X is a pair G= (A, B) where A = (µ^P_A, µ^N_A) is a bipolar fuzzy set onX and B= (µ^P_B, µ^N_B) is a bipolar fuzzy relation in X such that

µ^P_B(xy)≤µ^P_A(x)∧µ^P_A(y) andµ^N_B(xy)≥µ^N_A(x)∨µ^N_A(y) for allx, y∈X.

Definition 2.5. [21]Aneutrosophic setAon a non-empty setX is characterized by a truth-membership fuction t_A : X → [0,1], indeterminacy-membership function i_A : X → [0,1] and a falsity-membership function f_A : X → [0,1]. There is no restriction on the sum oft_A(x),i_A(x) andf_A(x) for allx∈X.

Definition 2.6. [11]Abipolar neutrosophic setAon a non-empty setXis an object of the form

A={(x, t^P_A(x), i^P_A(x), f_A^P(x), t^N_A(x), i^N_A(x), f_A^N(x)) :x∈X},

where t^P_A, i^P_A, f_A^P : X → [0,1] and t^N_A, i^N_A, f_A^N : X → [−1,0]. The positive valuest^P_A(x),i^P_A(x),f_A^P(x) denote respectively the truth, indeterminacy and false- memberships degrees of an element x∈ X, whereas, t^N_A(x), i^N_A(x), f_A^N(x) denote the implicit counter property of the truth,indeterminacy and false-memberships degrees of the element x∈X corresponding to the bipolar neutrosophic setA.

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M. Akram and M. Nasir

Definition 2.7. The heightof bipolar neutrosophic set A= (t^P_A(x), i^P_A(x), f_A^P(x), t^N_A(x), i^N_A(x),f_A^N(x)) in universe of discourse X is defined as,

h(A) = (h1(A), h2(A), h3(A), h4(A), h5(A), h6(A))

= ( sup

x∈X

t^P_A(x),sup

x∈X

i^P_A(x), inf

x∈Xf_A^P(x),sup

x∈X

t^P_A(x),sup

x∈X

i^P_A(x), inf

x∈Xf_A^P(x)), for allx∈X.

Definition 2.8. Let−→

Gbe a bipolar neutrosophic digraph thenbipolar neutrosophic out-neighbourhoodsof a vertexxis a bipolar neutrosophic set

N⁺(x) = (X_x⁺,t^(P)x ⁺,i^(P)x ⁺,fx^(P)⁺,t^(N)x ⁺,i^(Nx ⁾⁺,fx^(N⁾⁺), where,

X_x⁺={y|B₁^P−−−→

(x, y)>0,B₂^P−−−→

(x, y)>0,B₃^P−−−→

(x, y)>0, B₁^N−−−→

(x, y)<0,B^N₂ −−−→

(x, y)<0, B^N₃ −−−→

(x, y)<0}, such that t^(P)

+

x :X_x⁺→[0,1], defined byt^(P⁾

+

x (y) =B^P₁−−−→

(x, y),i^(P)

+

x :X_x⁺→[0,1], defined by i^(Px ⁾⁺(y) = B₂^P−−−→

(x, y), fx^(P⁾⁺ : X_x⁺ → [0,1], defined by fx^(P)⁺(y) = B₃^P−−−→

(x, y), t^(N)x ⁺ : X_x⁺ → [−1,0], defined by t^(N)x ⁺(y) = B₁^N−−−→

(x, y), i^(Nx ⁾⁺ : X_x⁺ → [−1,0], defined byi^(Nx ⁾⁺(y) =B₂^N−−−→

(x, y),fx^(N)⁺:X_x⁺→[−1,0], defined byfx^(N⁾⁺(y) = B₃^N−−−→

(x, y).

Definition 2.9. Let−→

Gbe a bipolar neutrosophic digraph thenbipolar neutrosophic in-neighbourhoodsof a vertexxis a bipolar neutrosophic set

N⁻(x) = (X_x⁻, t^(P)x ⁻,i^(P)x ⁻,fx^(P⁾⁻,t^(Nx ⁾⁻,i^(N)x ⁻,fx^(N⁾⁻), where,

X_x⁻={y|B₁^P−−−→

(y, x)>0,B₂^P−−−→

(y, x)>0,B₃^P−−−→

(y, x)>0, B₁^N−−−→

(y, x)<0,B₂^N−−−→

(y, x)<0, B^N₃ −−−→

(y, x)<0},

such thatt^(Px ⁾⁻ :X_x⁻ →[0,1], defined byt^(P)x ⁻(y) =B^P₁−−−→

(y, x),i^(P)x ⁻ :X_x⁻→[0,1], defined by i^(P⁾

−

x (y) = B₂^P−−−→

(y, x), f^(P⁾

−

x : X_x⁻ → [0,1], defined by f^(P⁾

−

x (y) = B₃^P−−−→

(y, x), t^(Nx ⁾⁻ : X_x⁻ → [−1,0], defined by t^(N)x ⁻(y) = B₁^N−−−→

(y, x), i^(N)x ⁻ : X_x⁻ → [−1,0], defined byi^(N⁾

−

x (y) =B₂^N−−−→

(y, x),f^(N⁾

−

x :X_x⁻→[−1,0], defined byf^(N⁾

−

x (y) = B₃^N−−−→

(y, x).

Definition 2.10. A bipolar neutrosophic competition graph of a bipolar neutrosophic graph −→

G = (A,−→

B) is an undirected bipolar neutrosophic graph C−−→ (G) = (A, R) which has the same vertex set as in −→

G and there is an edge between two

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vertices xand y if and only ifN⁺(x)∩ N⁺(y) is non-empty. The positive truth- membership, indeterminacy-membership, falsity-membership and negative truth- membership, indeterminacy-membership, falsity-membership values of the edge (x, y) are defined as,

(1) t^P_R(x, y) = (t^P_A(x)∧t^P_A(y))h1(N⁺(x)∩ N⁺(y)), (2) i^P_R(x, y) = (i^P_A(x)∧i^P_A(y))h₂(N⁺(x)∩ N⁺(y)), (3) f_R^P(x, y) = (f_A^P(x)∨f_A^P(y))h₃(N⁺(x)∩ N⁺(y)), (4) t^N_R(x, y) = (t^N_A(x)∨t^N_A(y))h4(N⁺(x)∩ N⁺(y)), (5) i^N_R(x, y) = (i^N_A(x)∨i^N_A(y))h5(N⁺(x)∩ N⁺(y)),

(6) f_R^N(x, y) = (f_A^N(x)∧f_A^N(y))h6(N⁺(x)∩ N⁺(y)), for allx, y∈X.

Example 2.11. Consider −→

G = (A, B) is a bipolar single-valued neutrosophic digraph, such that, X = {a, b, c, d}, A = {(a, 0.3, 0.8, 0.2, −0.5, −0.2, −0.1), (b, 0.8, 0.3, 0.1,−0.5,−0.4, −0.2), (c,0.4,0.5,0.6,−0.2,−0.3,−0.5), (d, 0.7, 0.3, 0.4,

−0.2,−0.3,−0.5)}, and B = {(−−−→

(a, b), 0.2, 0.1, 0.1,−0.4, −0.1, −0.2), (−−−→

(a, c), 0.3, 0.5, 0.6, −0.2,−0.2,−0.1), (−−−→

(b, d),0.6,0.2,0.2,−0.1,−0.2,−0.3), (−−−→

(d, c), 0.2, 0.2, 0.2,−0.2,−0.3,−0.5)} as shown in Fig. 1.

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 x N⁺(x)

a {(b, 0.2, 0.1, 0.1,-0.4,-0.1,-0.2), (c, 0.3, 0.5, 0.6,-0.2,-0.2,-0.1)}

b {(d, 0.6, 0.2, 0.2,-0.1,-0.2,-0.3)}

c ∅

d {(c, 0.2, 0.2, 0.2,-0.2,-0.3,-0.5)}

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Then bipolar single-valued neutrosophic competition graph of Fig. 1 is shown in Fig. 2.

Figure 2. Bipolar single-valued neutrosophic competition graph

Definition 2.12. Thesupportof a bipolar neutrosophic set A = (x, t^P_A, i^P_A, f_A^P, t^N_A,i^N_A,f_A^N) inX is the subset ofX defined by

supp(A) ={x∈X : t^P_A(x)6= 0,i^P_A(x)6= 0,f_A^P(x)6= 1, t^N_A(x)6=−1,i^N_A(x)6=−1, f_A^N(x)6= 0}

and|supp(A)|is the number of elements in the set.

Example 2.13. The support of a bipolar neutrosophic setA={(a, 0.5, 0.7, 0.2,

−0.8,−0.9,−0.3), (b, 0.1, 0.2, 1, −0.5,−0.7, −0.6), (c, 0.3, 0.5, 0.3, −0.8, −0.6,

−0.4), (d, 0, 0, 1, −1, −1, 0)} in X = {a, b, c, d} is supp(A) = {a, b, c} and

|supp(A)|= 3.

We now discussp-competition bipolar neutrosophic graphs.

Definition 2.14. Let p be a positive integer. Then p-competition bipolar neutrosophic 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 neutrosophic set of vertices as in−→

G and has a bipolar neutrosophic edge between two vertices x, y ∈X in C^p(−→

G) if and only if|supp(N⁺(x)∩ N⁺(y))| ≥p. The positive truth-membership value of edge (x,y) in C^p(−→

G) ist^P_B(x,y) =^(i−p)+1_i [t^P_A(x)∧ t^P_A(y)]h₁(N⁺(x)∩ N⁺(y)), the positive indeterminacy-membership value of edge (x,y) in C^p(−→

G) is i^P_B(x, y) = ^(i−p)+1_i [i^P_A(x)∧i^P_A(y)]h2(N⁺(x)∩ N⁺(y)), positive falsity-membership value of edge (x, y) in C^p(−→

G) is f_B^P(x, y) = ^(i−p)+1_i [f_A^P(x)∨ f_A^P(y)]h₃(N⁺(x)∩ N⁺(y)), the negative truth-membership value of edge (x, y)

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in C^p(−→

G) is t^N_B(x, y) = ^(i−p)+1_i [t^N_A(x)∨t^N_A(y)]h4(N⁺(x)∩ N⁺(y)), the negative indeterminacy-membership value of edge (x,y) inC^p(−→

G) isi^N_B(x,y) = ^(i−p)+1_i [i^N_A(x)∨

i^N_A(y)]h5(N⁺(x)∩ N⁺(y)), negative falsity-membership value of edge (x, y) in C^p(−→

G) is f_B^N(x, y) = ^(i−p)+1_i [f_A^N(x)∧ f_A^N(y)]h6(N⁺(x)∩ N⁺(y)), where i =

|supp(N⁺(x)∩ N⁺(y))|.

The 3−competition bipolar neutrosophic graph is illustrated by the following example.

G = (A, −→

B) is a bipolar neutrosophic digraph, such that, X = {x, y, z, a, b, c}, A = {(x, 0.7, 0.8, 0.5, −0.5, −0.6, −0.7), (y, 0.6, 0.7, 0.5, −0.3, −0.2, −0.7), (z, 0.6, 0.7, 0.3, −0.2, −0.3, −0.4), (a, 0.5, 0.6, 0.7,

−0.5,−0.6,−0.8), (b, 0.5, 0.6, 0.7,−0.9,−0.8,−0.7), (c, 0.5, 0.6, 0.3,−0.1,−0.2,

−0.4)}, andB={(−−−→

(x, a),0.3, 0.4, 0.6,−0.4,−0.5,−0.7), (−−−→

(x, b),0.4, 0.5, 0.4,−0.4,

−0.5,−0.5), (−−−→

(x, c),0.4, 0.5, 0.4,−0.1,−0.1,−0.6), (−−−→

(y, a),0.4, 0.5, 0.6,−0.2,−0.2,

−0.6), (−−−→

(y, b),0.4, 0.4, 0.6,−0.2,−0.2,−0.6), (−−−→

(y, c),0.4, 0.5, 0.4,−0.1,−0.2,−0.3), (−−−→

(z, b),0.4, 0.5, 0.3,−0.1,−0.2,−0.6), (−−−→

(z, c),0.4, 0.5, 0.2, −0.1,−0.2,−0.3)}, as shown in Fig. 3. ThenN⁺(x) ={(a, 0.3, 0.4, 0.6,−0.4,−0.5,−0.7), (b, 0.4, 0.5, 0.4,−0.4,−0.5,−0.5), (c, 0.4, 0.5, 0.4,−0.1,−0.1,−0.6)},N⁺(y) ={(a, 0.4, 0.5, 0.6,−0.2, −0.2,−0.6), (b, 0.4, 0.4, 0.6, −0.2,−0.2,−0.6), (c, 0.4, 0.5, 0.4, −0.1,

−0.2,−0.3)},N⁺(z) ={(b, 0.4, 0.5, 0.3,−0.1,−0.2,−0.6), (c, 0.4, 0.5, 0.2,−0.1,

−0.2,−0.3)}. So, N⁺(x)∩ N⁺(y) ={(a, 0.3, 0.4, 0.6, −0.2,−0.2,−0.7), (b, 0.4, 0.4, 0.6,−0.2,−0.2,−0.6), (c, 0.4, 0.5, 0.4,−0.1,−0.1,−0.6)}.

Figure 3. Bipolar neutrosophic digraph

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Now i = |supp(N⁺(x)∩ N⁺(y))| = 3. For p = 3, t^P_B(x, y) = 0.08, i^P_B(x, y) = 0.1166, f_B^P(x, y) = 0.066,t^N_B(x, y) =−0.04, i^N_B(x, y) =−0.033, and f_B^N(x, y) =−0.0933. As shown in Fig. 4.

Figure 4. 3−competition bipolar neutrosophic graph

Theorem 2.16. Let−→

G = (A,−→

B) be a bipolar neutrosophic digraph. If

h1(N⁺(x)∩ N⁺(y)) = 1, h2(N⁺(x)∩ N⁺(y)) = 1, h3(N⁺(x)∩ N⁺(y)) = 0, h4(N⁺(x)∩ N⁺(y)) = 1, h5(N⁺(x)∩ N⁺(y)) = 1, h6(N⁺(x)∩ N⁺(y)) = 0, inC^[ⁱ²^](−→

G), then the edge (x, y) is strong, wherei=|supp(N⁺(x)∩ N⁺(y))|. (Note that for any real numberx, [x]=greatest integer not exceedingx.)

proof. Suppose −→

G = (A, −→

B) is a bipolar neutrosophic digraph. Let the corresponding [₂ⁱ]-bipolar neutrosophic competition graph be C^[²ⁱ^](−→

G), where i =

|supp(N⁺(x)∩ N⁺(y))|. Also, assume that

h1(N⁺(x)∩ N⁺(y)) = 1, h2(N⁺(x)∩ N⁺(y)) = 1, h3(N⁺(x)∩ N⁺(y)) = 0, h4(N⁺(x)∩ N⁺(y)) = 1, h5(N⁺(x)∩ N⁺(y)) = 1, h6(N⁺(x)∩ N⁺(y)) = 0, for allx,y∈X. Now,

t^P_B(x, y) = (i−[₂ⁱ]) + 1

i [t^P_A(x)∧t^P_A(y)]×1, t^N_B(x, y) =(i−[₂ⁱ]) + 1

i [t^N_A(x)∨t^N_A(y)]×1, i^P_B(x, y) = (i−[₂ⁱ]) + 1

i [i^P_A(x)∧i^P_A(y)]×1, i^N_B(x, y) =(i−[₂ⁱ]) + 1

i [i^N_A(x)∨i^N_A(y)]×1, f_B^P(x, y) =(i−[ⁱ₂]) + 1

i [f_A^P(x)∨f_A^P(y)]×0, f_B^N(x, y) =(i−[₂ⁱ]) + 1

i [f_A^N(x)∧f_A^N(y)]×0.

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This gives the result, t^P_B(x, y)

[t^P_A(x)∧t^P_A(y)] =(i−[₂ⁱ]) + 1

i >0.5, t^N_B(x, y)

[t^N_A(x)∨t^N_A(y)] = (i−[₂ⁱ]) + 1 i <0.5, i^P_B(x, y)

[i^P_A(x)∧i^P_A(y)] = (i−[₂ⁱ]) + 1

i >0.5, i^N_B(x, y)

[i^N_A(x)∨i^N_A(y)] = (i−[₂ⁱ]) + 1 i <0.5, f_B^P(x, y)

[f_A^P(x)∨f_A^P(y)] =(i−[₂ⁱ]) + 1

i <0.5, f_B^N(x, y)

[f_A^N(x)∧f_A^N(y)] = (i−[₂ⁱ]) + 1 i <0.5.

Hence, the edge (x, y) 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:

P_x,y^m: A bipolar neutrosophic path of lengthmfromxtoy.

−

→P^m_x,y: A directed bipolar neutrosophic path of lengthmfromxtoy.

N_m⁺(x): m-step bipolar neutrosophic out-neighbourhood of vertexx.

N_m⁻(x): m-step bipolar neutrosophic in-neighbourhood of vertexx.

Nm(x): m-step bipolar neutrosophic neighbourhood of vertexx.

N_m(G): m-step bipolar neutrosophic neighbourhood graph of the bipolar neutrosophic graphG.

Cm

−−→

(G): m-step bipolar neutrosophic competition graph of the bipolar neutrosophic digraph−→

G.

Definition 2.17. Suppose−→

G = (A,−→

B) is a bipolar neutrosophic digraph. Them- step bipolar neutrosophic digraphof−→

G is denoted by−→

G_m= (A,B), where bipolar neutrosophic set of vertices of −→

G is same with bipolar neutrosophic set of vertices of−→

G_mand has an edge betweenxandyin−→

G_mif and only if there exists a bipolar neutrosophic directed path−→

P^m_x,y in−→ G.

Definition 2.18. Thebipolar neutrosophic m-step out-neighbourhood of vertexx of a bipolar neutrosophic digraph−→

G = (A,−→

B) is bipolar neutrosophic set N_m⁺(x) = (X_x⁺, t^(P)x ⁺,i^(P)x ⁺,fx^(P⁾⁺,t^(N)x ⁺,i^(Nx ⁾⁺,fx^(N⁾⁺), where

X_x⁺ = {y| there exists a directed bipolar neutrosophic path of length m from x to y, −→

P^m_x,y}, t^(Px ⁾⁺ : X_x⁺ → [0, 1], i^(P)x ⁺ : X_z⁺ → [0, 1], fx^(P)⁺ : X_z⁺ → [0, 1]

t^(Nx ⁾⁺ :X_x⁺ →[−1, 0],i^(N)x ⁺:X_z⁺→[−1, 0],fx^(N)⁺:X_z⁺→[−1, 0] are defined by t^(P)x ⁺= min{t^P−−−−−→

(x₁,x₂), (x₁,x₂) is an edge of−→

P^m_x,y},i^(P)x ⁺ = min{i^P−−−−−→

(x₁,x₂), (x₁, x2) is an edge of −→

P^m_x,y}, fx^(P)⁺ = max{f^P−−−−−→

(x1, x2), (x1, x2) is an edge of−→ P^m_x,y}, t^(Nx ⁾⁺ = max{t^P−−−−−→

(x₁,x₂), (x₁, x₂) is an edge of −→

P^m_x,y}, i^(Nx ⁾⁺ = max{i^N−−−−−→

(x₁,x₂),

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M. Akram and M. Nasir (x1,x2) is an edge of−→

P^m_x,y},f^(N⁾

+

x = min{f^N−−−−−→

(x1, x2), (x1,x2) is an edge of−→ P^m_x,y}, respectively.

Example 2.19. Consider−→

G = (A,−→

B) is a bipolar neutrosophic digraph, such that X ={x,y,a,b,c,d}, as shown in Fig. 5. Then, 2-step out-neighbourhood of vertices x, andy is calculated as,N₂⁺(x) ={(b,0.2,0.2,0.5,−0.2,−0.3,−0.3), (d,0.2, 0.2, 0.5,−0.2,−0.3,−0.3)},N₂⁺(y) ={(b,0.1, 0.3,0.2,−0.2,−0.3,−0.6), (d,0.3, 0.5, 0.6,−0.2,−0.3,−0.5)}.

Definition 2.20. Thebipolar neutrosophicm-step in-neighbourhoodof vertexxof a bipolar neutrosophic digraph−→

G = (A,−→

B) is bipolar neutrosophic set N_m⁻(x) = (X_x⁻, t^(P)

−

x ,i^(P)

−

x ,f^(P)

−

x ,t^(N⁾

−

x ,i^(N)

−

x ,f^(N⁾

−

x ), where

X_x⁻ = {y| there exists a directed bipolar neutrosophic path of length m from y to x, −→

P^m_y,x}, t^(P)

−

x : X_x⁻ → [0, 1], i^(P)

−

x : X_z⁻ → [0, 1], f^(P)

−

x : X_z⁻ → [0, 1]

t^(N⁾

−

x :X_x⁻→[−1, 0],i^(N)

−

x :X_z⁻→[−1, 0],f^(N)

−

x :X_z⁻→[−1, 0] are defined by t^(P)x ⁻ = min{t^P−−−−−→

(x₁, x₂), (x₁,x₂) is an edge of−→

P^m_y,x},i^(P)x ⁻ = min{i^P−−−−−→

(x₁,x₂), (x₁, x2) is an edge of −→

P^m_y,x}, f^(P)

−

x = max{f^P−−−−−→

(x1,x2), (x1, x2) is an edge of−→ P^m_y,x}, t^(Nx ⁾⁻ = max{t^N−−−−−→

(x₁,x₂), (x₁, x₂) is an edge of −→

P^m_y,x}, i^(Nx ⁾⁻ = max{i^N−−−−−→

(x₁,x₂), (x1,x2) is an edge of−→

P^m_y,x},f^(N⁾

−

x = min{f^N−−−−−→

(x1,x2), (x1,x2) is an edge of−→ P^m_y,x}, respectively.

G = (A, −→

B) is a bipolar neutrosophic digraph, such that,X ={a,b,c, d, e,f}, as shown in Fig. 6. Then, 2-step in-neighbourhood of verticesa, andbis calculated as,N₂⁻(a) ={(f, 0.1, 0.1, 0.5,−0.1,−0.2,−0.6), (e,

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0.3, 0.1, 0.7,−0.1,−0.2,−0.4)},N₂⁻(b) ={(f, 0.1, 0.3, 0.6,−0.3,−0.4,−0.7), (e, 0.4, 0.3, 0.6,−0.3,−0.4,−0.5)}.

Definition 2.22. Suppose −→

G = (A, −→

B) is a bipolar neutrosophic digraph. The m-step bipolar neutrosophic competition graph of bipolar neutrosophic digraph−→ G is denoted by Cm(−→

G) = (A, B) which has same bipolar neutrosophic set of vertices as in −→

G and has an edge between two vertices x, y ∈ X in Cm(−→

G) if and only if (N_m⁺(x)∩ N_m⁺(y)) is a non-empty bipolar neutrosophic set in −→

G. The positive truth-membership value of edge (x, y) in Cm(−→

G) is t^P_B(x, y) = [t^P_A(x)∧ t^P_A(z)]h1(N_m⁺(x)∩N_m⁺(y)), the positive indeterminacy-membership value of edge (x, y) inCm(−→

G) isi^P_B(x,y) = [i^P_A(x)∧i^P_A(y)]h2(N_m⁺(x)∩ N_m⁺(y)), the positive falsity- membership value of edge (x,y) inCm(−→

G) isf_B^P(x,y) = [f_A^P(x)∨f_A^P(y)]h3(N_m⁺(x)∩

N_m⁺(y)), the negative truth-membership value of edge (x, y) in Cm(−→

G) is t^N_B(x, y) = [t^N_A(x)∨t^N_A(z)]h4(N_m⁺(x)∩ N_m⁺(y)), the negative indeterminacy-membership value of edge (x, y) in Cm(−→

G) is i^N_B(x, y) = [i^N_A(x)∨i^N_A(y)]h5(N_m⁺(x)∩ N_m⁺(y)), the negative falsity-membership value of edge (x, y) in C_m(−→

G) is f_B^N(x, y) = [f_A^N(x)∧f_A^N(y)]h₆(N_m⁺(x)∩ N_m⁺(y)).

The 2−step bipolar neutrosophic competition graph is illustrated by the following example.

G = (A, −→

B) is a bipolar neutrosophic digraph, such that X ={x, y,a, b, c, d}, as shown in Fig. 7. Then,N₂⁺(x) ={(b, 0.2, 0.2, 0.5,

−0.2,−0.3,−0.3), (d, 0.2, 0.2, 0.5,−0.2,−0.3,−0.3)},N₂⁺(y) ={(b, 0.1, 0.3, 0.2,

−0.2,−0.3,−0.6), (d,0.3, 0.5, 0.6,−0.2,−0.3,−0.5)}, there non-empty intersection

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is calculated asN₂⁺(x)∩ N₂⁺(y) ={(b, 0.1, 0.2, 0.5,−0.2,−0.3,−0.6), (d,0.2, 0.2, 0.6,−0.2,−0.3,−0.5)}.

Thus,t^P_B(x, y) = 0.08, i^P_B(x,y) = 0.04, f_B^P(x, y) = 0.35, t^N_B(x, y) =−0.06, i^N_B(x, y) =−0.06, and f_B^N(x, y) = −0.25. Then its corresponding 2-step bipolar neutrosophic competition graph is shown in Fig. 8.

Figure 8. 2-step bipolar neutrosophic competition graph

If a predator x attacks one prey y, then the linkage is shown by an edge −−−→

(x, y) in a bipolar neutrosophic digraph. But, if predator needs help of many other mediators x1, x2, . . .xm−1, then linkage among them is shown by bipolar neutrosophic 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.

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Definition 2.24. 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. Also, let −→

B₁^P(u₁, v₁), −→

B₁^P(u₂, v₂),. . . , −→

B^P₁(u_l, v_l) be the minimum positive truth- membership values,−→

B^P₂(u1, v1),−→

B₂^P(u2, v2),. . . ,−→

B₂^P(ul, vl) be the minimum positive indeterminacy-membership values, −→

B₃^P(u1, v1), −→

B₃^P(u2, v2),. . . , −→

B₃^P(ul, vl) be the maximum positive false-membership values,−−→

B₁^N(u₁, v₁),−−→

B^N₁ (u₂, v₂),. . . ,−−→ B₁^N(u_l, v_l) be the maximum negative truth-membership values, −−→

B^N₂ (u₁, v₁), −−→

B^N₂ (u₂, v₂),. . . ,

−−→

B₂^N(ul, vl) be the maximum negative indeterminacy-membership values,−−→

B₃^N(u1, v1),

−−→

B₃^N(u2, v2),. . . , −−→

B₃^N(ul, vl) be the minimum negative false-membership values, of edges of the paths−→

P^m_x₁_,w, −→

P^m_x₂_,w,. . . ,−→

P^m_x_l_,w, respectively. Them-step preyw∈X isstrong preyif

−→

B₁^P(ui, vi)>0.5, −→

B₂^P(ui, vi)>0.5, −→

B₃^P(ui, vi)<0.5,

−−→

B₁^N(ui, vi)<0.5, −−→

B₂^N(ui, vi)<0.5, −−→

B₃^N(ui, vi)<0.5, for alli= 1,2, . . . , l.

The strength of the prey wcan be measured by the mappingS :X →[0,1], such that:

S(w) = 1 l

( _l X

i=1

[−→

B₁^P(u_i, v_i)] +

l

X

i=1

[−→

B₂^P(u_i, v_i)] +

l

X

i=1

[−→

B₃^P(u_i, v_i)]

−

l

X

i=1

[−−→

B^N₁ (ui, vi)]−

l

X

i=1

[−−→

B₂^N(ui, vi)]−

l

X

i=1

[−−→ B^N₃ (ui, vi)]

) .

Example 2.25. Consider bipolar neutrosophic digraph −→

G = (A, −→

B) as shown in Fig. 7, the strength of the preyb is equal to

[0.2 + 0.2] + [0.2 + 0.3] + [0.5 + 0.3]−[−0.3−0.2]−[−0.3−0.5]−[−0.3−0.2]

2 = 1.75

>0.5.

Hence,bis strong 2-step prey.

Theorem 2.26. If a prey w of −→

G = (A, −→

B) is strong, then the strength of w, S(w)>0.5.

proof. Let −→

G = (A, −→

B) be a bipolar neutrosophic digraph. Let w be a common vertex of m-step out-neighbourhoods of verticesx1, x2,. . . , xl, i.e., there exists the paths−→

P^m_x

1,w,−→ P^m_x

2,w,. . . ,−→ P^m_x

l,w, in−→

G. Also, let−→

B^P₁(u₁, v₁),−→

B₁^P(u₂, v₂),. . . ,

−→

B₁^P(ul, vl) be the minimum positive truth-membership values,−→

B₂^P(u1,v1),−→

B₂^P(u2,v2), . . .,−→

B₂^P(ul,vl) be the minimum positive indeterminacy-membership values,−→

B₃^P(u1, v₁),−→

B₃^P(u₂,v₂),. . . ,−→

B₃^P(u_l,v_l) be the maximum positive false-membership values,

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−−→

B₁^N(u₁,v₁),−−→

B^N₁ (u₂,v₂),. . . ,−−→

B₁^N(u_l, v_l) be the maximum negative truth-membership values,−−→

B₂^N(u1, v1),−−→

B^N₂ (u2, v2),. . . ,−−→

B₂^N(ul, vl) be the maximum negative indeterminacy- membership values, −−→

B₃^N(u1, v1), −−→

B^N₃ (u2, v2),. . . , −−→

B₃^N(ul, vl) be the minimum negative false-membership values, of edges of the paths→−

P^m_x₁_,w, −→

P^m_x₂_,w,. . . ,−→

P^m_x_l_,w, respectively.

Ifwis strong, each edge (ui,vi),i= 1,2, . . . , lis strong. So,

−→

B₁^P(ui, vi)>0.5,

−→

B₂^P(ui, vi)>0.5,

−→

B₃^P(ui, vi)<0.5,

−−→

B₁^N(u_i, v_i)<0.5, −−→

B₂^N(u_i, v_i)<0.5, −−→

B₃^N(u_i, v_i)<0.5, for alli= 1,2, . . . , l.

Now,

S(w)> 0.5 + 0.5 +. . .(l times) + 0.5

l >0.5.

This proves the result.

Remark: The converse of the above theorem is not true, i.e. ifS(w)>0.5, then all preys may not be strong. This can be explained as:

LetS(w)>0.5 for a prey win−→ G. So, S(w) = 1

l ( _l

X

i=1

[

−→

B₁^P(ui, vi)] +

l

X

i=1

[

−→

B₂^P(ui, vi)] +

l

X

i=1

[

−→

B₃^P(ui, vi)]

−

l

X

i=1

[−−→

B^N₁ (u_i, v_i)]−

l

X

i=1

[−−→

B₂^N(u_i, v_i)]−

l

X

i=1

[−−→ B^N₃ (u_i, v_i)]

) . Hence,

( _l X

i=1

[−→

B₁^P(u_i, v_i)] +

l

X

i=1

[−→

B^P₂(u_i, v_i)] +

l

X

i=1

[−→

B₃^P(u_i, v_i)]

−

l

X

i=1

[−−→

B₁^N(ui, vi)]−

l

X

i=1

[−−→

B₂^N(ui, vi)]−

l

X

i=1

[−−→ B₃^N(ui, vi)]

)

> l 2. This result does not necessarily imply that

−→

B₁^P(ui, vi)>0.5, −→

B₂^P(ui, vi)>0.5, −→

B₃^P(ui, vi)<0.5,

−−→

B₁^N(ui, vi)<0.5, −−→

B₂^N(ui, vi)<0.5, −−→

B₃^N(ui, vi)<0.5, for alli= 1,2, . . . , l.

Since, all edges of the directed paths−→ P^m_x₁_,w,−→

P^m_x₂_,w,. . . ,−→

P^m_x_l_,w, are not strong. So, the converse of the above statement is not true i.e., ifS(w)>0.5, the preywof−→ G may not be strong.

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Theorem 2.27. If all preys of−→

G = (A,−→

B) are strong, then all edges ofCm(−→ G) = (A,B) are strong.

proof. Let−→

G= (A,−→

B) be a bipolar neutrosophic digraph and all preys of it are strong. Let Cm(−→

G) = (A,B), where,

t^P_B(x, y) = [t^P_A(x)∧t^P_A(z)]h1(N_m⁺(x)∩ N_m⁺(y)), i^P_B(x, y) = [i^P_A(x)∧i^P_A(y)]h₂(N_m⁺(x)∩ N_m⁺(y)), f_B^P(x, y) = [f_A^P(x)∨f_A^P(y)]h3(N_m⁺(x)∩ N_m⁺(y)),

t^N_B(x, y) = [t^N_A(x)∨t^N_A(z)]h4(N_m⁺(x)∩ N_m⁺(y)), i^N_B(x, y) = [i^N_A(x)∨i^N_A(y)]h5(N_m⁺(x)∩ N_m⁺(y)), f_B^N(x, y) = [f_A^N(x)∧f_A^N(y)]h6(N_m⁺(x)∩ N_m⁺(y)), for all edges (x, y) inCm(−→

G) = (A,B). Then there arises two cases:

Case 1.: LetN_m⁺(x)∩ N_m⁺(y) be null set. Then there does not exits any edge betweenxandy in Cm(−→

G).

Case 2.: LetN_m⁺(x)∩ N_m⁺(y) be non-empty. Now, clearly

h₁(N_m⁺(x)∩ N_m⁺(y))>0.5, h₂(N_m⁺(x)∩ N_m⁺(y))>0.5, h₃(N_m⁺(x)∩ N_m⁺(y))<0.5, h4(N_m⁺(x)∩ N_m⁺(y))<0.5, h5(N_m⁺(x)∩ N_m⁺(y))<0.5, h6(N_m⁺(x)∩ N_m⁺(y))<0.5,

in −→

G as all preys are strong. So, the edge (x,y),x, y∈X inCm(−→ G) have the memberships values

t^P_B(x, y) = [t^P_A(x)∧t^P_A(z)]h₁(N_m⁺(x)∩ N_m⁺(y)), i^P_B(x, y) = [i^P_A(x)∧i^P_A(y)]h2(N_m⁺(x)∩ N_m⁺(y)), f_B^P(x, y) = [f_A^P(x)∨f_A^P(y)]h₃(N_m⁺(x)∩ N_m⁺(y)),

t^N_B(x, y) = [t^N_A(x)∨t^N_A(z)]h4(N_m⁺(x)∩ N_m⁺(y)), i^N_B(x, y) = [i^N_A(x)∨i^N_A(y)]h5(N_m⁺(x)∩ N_m⁺(y)), f_B^N(x, y) = [f_A^N(x)∧f_A^N(y)]h₆(N_m⁺(x)∩ N_m⁺(y)), 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.

Theorem 2.28. If −→

G is a bipolar neutrosophic digraph and −→

G_m is the m-step bipolar neutrosophic digraph of−→

G, thenC(−→

G_m) =Cm(−→ G).

proof. Let −→

G = (A,−→

B) be a bipolar neutrosophic digraph and −→

G_m = (A,−→ J) is the m-step bipolar neutrosophic digraph of −→

G. Also, let C(−→

G_m) = (A, J) and C_m(−→

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