CERTAIN BIPOLAR NEUTROSOPHIC COMPETITION GRAPHS

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1 J. Indones. Math. Soc. Vol. 24, o. 1 (2018), pp CERTAI IPOLAR EUTROSOPHIC COMPETITIO GRAPHS Muhammad Akram 1 and Maryam asr 2 1,2 Department of Mathematcs, Unversty of the Punjab, ew Campus, Lahore, Pakstan makrammath@yahoo.com Abstract. polarty plays an mportant role n many research domans. A bpolar fuzzy model s a very mportant model n whch postve nformaton represents what s possble or preferred, whle negatve nformaton represents what s forbdden or surely false. In ths research paper, we frst ntroduce the concept of p-competton bpolar neutrosophc graphs. We then defne generalzaton of bpolar neutrosophc competton graphs called m-step bpolar neutrosophc competton graphs. Moreover, we present some related concepts of bpolar neutrosophc graphs. Fnally, we descrbe an applcaton of m-step bpolar neutrosophc competton graphs. Key words and Phrases: p-competton bpolar neutrosophc graphs, m-step bpolar neutrosophc competton graphs, Algorthm. Abstrak. polart memankan peran pentng dalam berbaga macam topk peneltan. Sebuah model Fuzzy bpolar adalah sebuah model yang sangat pentng dalam hal nformas postf menyatakan apa yang mungkn dplh, sedangkan nformas negatf menyatakan apa yang dlarang atau past salah. Pada paper n, pertama kal kam memperkenalkan konsep graf neotrosopk bpolar p-kompets. Kemudan kam mendefnskan perumuman dar graf kompets neutrosopk bpolar yang dsebut dengan graf kompets neutrosopk bpolar m-langkah. Lebh jauh, kam menyajkan beberapa konsep yang terkat dengan graf neutrosopk bpolar. Akhrnya, kam menggambarkan sebuah aplkas dar graf kompets neutrosopk bpolar m-langkah. Kata kunc: Graf neotrosopk bpolar p-kompets, graf kompets neutrosopk bpolar m-langkah, algortma Mathematcs Subject Classfcaton: 03E72, 68R10, 68R05. Receved: , revsed: , accepted:

2 2 M. Akram and M. asr 1. Introducton The noton of competton graphs was ntroduced by Cohen [10] n 1968, dependng upon a problem n ecology. The competton graphs have many utlzatons n solvng daly lfe problems, ncludng channel assgnment, modelng of complex economc, phytogenetc tree reconstructon, codng and energy systems. Fuzzy set theory [26] and ntutonstc fuzzy set theory [6] are useful models for dealng wth uncertanty and ncomplete nformaton. ut they may not be suffcent n modelng of ndetermnate and nconsstent nformaton encountered n real world. In order to cope wth ths ssue, neutrosophc set theory was proposed by Smarandache [18] as a generalzaton of fuzzy sets and ntutonstc fuzzy sets. However, snce neutrosophc sets are dentfed by three functons called truthmembershp (t), ndetermnacy-membershp () and falsty-membershp (f) whose values are real standard or non-standard subset of unt nterval ]0, 1 + [. There are some dffcultes n modelng of some problems n engneerng and scences. To overcome these dffcultes, n 2010, concept of sngle-valued neutrosophc sets and ts operatons defned by Wang et al. [22] as a generalzaton of ntutonstc fuzzy sets. Ye [24, 25] has presented several novel applcatons of neutrosophc sets. Del et al. [11] extended the deas of bpolar fuzzy sets [28] and neutrosophc sets to bpolar neutrosophc sets and studed ts operatons and applcatons n decson makng problems. Smarandache [20] proposed noton of neutrosophc graph and they separated them to four man categores. Wu [23] dscussed fuzzy dgraphs. The concept of fuzzy k-competton graphs and p-competton fuzzy graphs was frst ntroduced by Samanta and Pal n [15], t was further studed n [5, 13, 16, 17]. Cho et al. [9] proposed the generalzaton of a dgraphs known as m-step competton graphs. Samanta et al. [16] ntroduced the generalzaton of fuzzy competton graphs, called m-step fuzzy competton graphs. On the other hand, the concepts of bpolar fuzzy competton graphs and ntutonstc fuzzy competton graphs are dscussed n [17, 13]. Samanta et al. [16] also ntroduced the concepts of fuzzy m- step neghbouthood graphs. The noton of bpolar fuzzy graphs was frst ntroduced by Akram [1] n 2011 as a generalzaton of fuzzy graphs. On the other hand, Akram and Shahzad [4] frst ntroduced the noton of neutrosophc soft graphs and gave ts applcatons. Akram [2] ntroduced the noton of sngle-valued neutrosophc planar graphs. Akram and Sarwar have shown that there are some flaws n roum et al. [8] s defnton, whch cannot be appled n network models. All the predator-prey relatons cannot only be represented by bpolar neutrosophc competton graphs. For example, n a food web, speces may have a chan consstng of same number of preys by whch they can reach to ther common preys. Ths dea motvates the necessty of m-step bpolar neutrosophc competton graphs. In ths research paper, we frst ntroduce the concept of p-competton bpolar neutrosophc graphs. We then defne generalzaton of bpolar neutrosophc competton graphs called m-step bpolar neutrosophc competton graphs. Moreover, we present some related concepts of bpolar neutrosophc graphs. Fnally, we descrbe an applcaton of m-step bpolar neutrosophc competton graphs.

3 Certan bpolar neutrosophc competton graphs 3 2. Certan polar eutrosophc Competton Graphs Defnton 2.1. [26, 27]A fuzzy set µ n a unverse X s a mappng µ : X [0, 1]. A fuzzy relaton on X s a fuzzy set ν n X X. Defnton 2.2. [28]A bpolar fuzzy set on a non-empty set X has the form A = {(x, µ P A (x), µ A (x)) : x X} where, µ P A : X [0, 1] and µ A : X [ 1, 0] are mappngs. The postve membershp value µ P A (x) represents the strength of truth or satsfacton of an element x to a certan property correspondng to bpolar fuzzy set A and µ A (x) denotes the strength of satsfacton of an element x to some counter property of bpolar fuzzy set A. If µ P A (x) 0 and µ A (x) = 0 t s the stuaton when x has only truth satsfacton degree for property A. If µ A (x) 0 and µp A (x) = 0, t s the case that x s not satsfyng the property of A but satsfyng the counter property to A. It s possble for x that µ P A (x) 0 and µ A (x) 0 when x satsfes the property of A as well as ts counter property n some part of X. Defnton 2.3. [1]Let X be a non-empty set. A mappng = (µ P, µ ) : X X [0, 1] [ 1, 0] s a bpolar fuzzy relaton on X such that µ P (xy) [0, 1] and (xy) [ 1, 0] for x, y X. µ Defnton 2.4. [1]A bpolar fuzzy graph on X s a par G = (A, ) where A = (µ P A, µ A ) s a bpolar fuzzy set on X and = (µp, µ ) s a bpolar fuzzy relaton n X such that µ P (xy) µp A (x) µp A (y) and µ (xy) µ A (x) µ A (y) for all x, y X. Defnton 2.5. [21]A neutrosophc set A on a non-empty set X s characterzed by a truth-membershp fucton t A : X [0, 1], ndetermnacy-membershp functon A : X [0, 1] and a falsty-membershp functon f A : X [0, 1]. There s no restrcton on the sum of t A (x), A (x) and f A (x) for all x X. Defnton 2.6. [11]A bpolar neutrosophc set A on a non-empty set X s an object of the form A = {(x, t P A (x), P A (x), f A P (x), t A (x), A (x), f A (x)) : x X}, where t P A, P A, f A P : X [0, 1] and t A, A, f A : X [ 1, 0]. The postve values t P A (x), P A (x), f A P (x) denote respectvely the truth, ndetermnacy and falsemembershps degrees of an element x X, whereas, t A (x), A (x), f A (x) denote the mplct counter property of the truth,ndetermnacy and false-membershps degrees of the element x X correspondng to the bpolar neutrosophc set A.

4 4 M. Akram and M. asr Defnton 2.7. The heght of bpolar neutrosophc set A = (t P A (x), P A (x), f A P (x), t A (x), A (x), f A (x)) n unverse of dscourse X s defned as, h(a) = (h 1 (A), h 2 (A), h 3 (A), h 4 (A), h 5 (A), h 6 (A)) for all x X. = (sup t P A(x), sup P A(x), nf f A P (x), sup t P A(x), sup P A(x), nf f A P (x)), x X x X x X x X x X x X Defnton 2.8. Let G be a bpolar neutrosophc dgraph then bpolar neutrosophc out-neghbourhoods of a vertex x s a bpolar neutrosophc set where, X + x = {y P 1 + (x) = (X x + (P )+ (P )+, t x, x, f (x, y) > 0, 2 P 3 (P )+ x, t ()+ x (x, y) > 0, 3 P (x, y) > 0, 1 (x, y) < 0},, ()+ x, f x ()+ ), (x, y) < 0, 2 (x, y) < 0, (P )+ such that t x : X x + (P )+ [0, 1], defned by t x (y) = 1 P (P )+ (x, y), x : X x + [0, 1], (P )+ defned by x (y) = 2 P (P )+ (x, y), f x : X x + (P )+ [0, 1], defned by f x (y) = 3 P (x, y), t ()+ x : X x + [ 1, 0], defned by t x ()+ (y) = 1 (x, y), x ()+ : X x + [ 1, 0], defned by ()+ x (y) = 2 (x, y), f x ()+ : X x + [ 1, 0], defned by f x ()+ (y) = (x, y). 3 Defnton 2.9. Let G be a bpolar neutrosophc dgraph then bpolar neutrosophc n-neghbourhoods of a vertex x s a bpolar neutrosophc set where, X x = {y P 1 (P ) (x) = (Xx (P ), t x, x, f (y, x) > 0, 2 P 3 (P ) x, t () x (y, x) > 0, 3 P (y, x) > 0, 1 (y, x) < 0},, x (), f x () ), (y, x) < 0, 2 (y, x) < 0, (P ) such that t x : Xx (P ) [0, 1], defned by t x (y) = 1 P (P ) (y, x), x : Xx [0, 1], (P ) defned by x (y) = 2 P (P ) (y, x), f x : Xx (P ) [0, 1], defned by f x (y) = 3 P (y, x), t () x : Xx [ 1, 0], defned by t x () (y) = 1 (y, x), () x : Xx [ 1, 0], defned by () x (y) = 2 (y, x), f x () : Xx [ 1, 0], defned by f x () (y) = (y, x). 3 Defnton A bpolar neutrosophc competton graph of a bpolar neutrosophc graph G = (A, ) s an undrected bpolar neutrosophc graph C (G) = (A, R) whch has the same vertex set as n G and there s an edge between two

5 Certan bpolar neutrosophc competton graphs 5 vertces x and y f and only f + (x) + (y) s non-empty. The postve truthmembershp, ndetermnacy-membershp, falsty-membershp and negatve truthmembershp, ndetermnacy-membershp, falsty-membershp values of the edge (x, y) are defned as, (1) t P R (x, y) = (tp A (x) tp A (y))h 1( + (x) + (y)), (2) P R (x, y) = (P A (x) P A (y))h 2( + (x) + (y)), (3) f P R (x, y) = (f P A (x) f P A (y))h 3( + (x) + (y)), (4) t R (x, y) = (t A (x) t A (y))h 4( + (x) + (y)), (5) R (x, y) = ( A (x) A (y))h 5( + (x) + (y)), (6) f R (x, y) = (f A (x) f A (y))h 6( + (x) + (y)), for all x, y X. Example Consder G = (A, ) s a bpolar sngle-valued neutrosophc dgraph, 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 = {( (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 n Fg. 1. Fgure 1. polar sngle-valued neutrosophc dgraph y drect calculatons we have Table 1 representng bpolar sngle-valued neutrosophc out-neghbourhoods. Table 1. polar sngle-valued neutrosophc out-neghbourhoods x + (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)}

6 6 M. Akram and M. asr Then bpolar sngle-valued neutrosophc competton graph of Fg. 1 s shown n Fg. 2. Fgure 2. polar sngle-valued neutrosophc competton graph Defnton The support of a bpolar neutrosophc set A = (x, t P A, P A, f A P, t A, A, f A ) n X s the subset of X defned by supp(a) = {x X : t P A (x) 0, P A (x) 0, f A P (x) 1, t A (x) 1, A (x) 1, (x) 0} and supp(a) s the number of elements n the set. f A Example The support of a bpolar neutrosophc set A = {(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)} n X = {a, b, c, d} s supp(a) = {a, b, c} and supp(a) = 3. We now dscuss p-competton bpolar neutrosophc graphs. Defnton Let p be a postve nteger. Then p-competton bpolar neutrosophc graph C p ( G) of the bpolar neutrosophc dgraph G = (A, ) s an undrected bpolar neutrosophc graph G = (A, ) whch has same bpolar neutrosophc set of vertces as n G and has a bpolar neutrosophc edge between two vertces x, y X n C p ( G) f and only f supp( + (x) + (y)) p. The postve truth-membershp value of edge (x, y) n C p ( G) s t P ( p)+1 (x, y) = [t P A (x) t P A (y)]h 1( + (x) + (y)), the postve ndetermnacy-membershp value of edge (x, y) n C p ( G) s P ( p)+1 (x, y) = [ P A (x) P A (y)]h 2( + (x) + (y)), postve falsty-membershp value of edge (x, y) n C p ( G) s f P ( p)+1 (x, y) = [fa P (x) fa P (y)]h 3( + (x) + (y)), the negatve truth-membershp value of edge (x, y)

7 Certan bpolar neutrosophc competton graphs 7 n C p ( G) s t ( p)+1 (x, y) = [t A (x) t A (y)]h 4( + (x) + (y)), the negatve ndetermnacy-membershp value of edge (x, y) n C p ( G) s ( p)+1 (x, y) = [ A (x) A (y)]h 5( + (x) + (y)), negatve falsty-membershp value of edge (x, y) n C p ( G) s f ( p)+1 (x, y) = [fa (x) f A (y)]h 6( + (x) + (y)), where = supp( + (x) + (y)). The 3 competton bpolar neutrosophc graph s llustrated by the followng example. Example Consder G = (A, ) s a bpolar neutrosophc dgraph, 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)}, and = {( (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 n Fg. 3. Then + (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)}, + (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)}, + (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, + (x) + (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)}. Fgure 3. polar neutrosophc dgraph

8 8 M. Akram and M. asr ow = supp( + (x) + (y)) = 3. For p = 3, t P (x, y) = 0.08, P (x, y) = , f P (x, y) = 0.066, t (x, y) = 0.04, (x, y) = 0.033, and f (x, y) = As shown n Fg. 4. Fgure 4. 3 competton bpolar neutrosophc graph Theorem Let G = (A, ) be a bpolar neutrosophc dgraph. If h 1 ( + (x) + (y)) = 1, h 2 ( + (x) + (y)) = 1, h 3 ( + (x) + (y)) = 0, h 4 ( + (x) + (y)) = 1, h 5 ( + (x) + (y)) = 1, h 6 ( + (x) + (y)) = 0, n C [ 2 ] ( G), then the edge (x, y) s strong, where = supp( + (x) + (y)). (ote that for any real number x, [x]=greatest nteger not exceedng x.) proof. Suppose G = (A, ) s a bpolar neutrosophc dgraph. Let the correspondng [ 2 ]-bpolar neutrosophc competton graph be C[ 2 ] ( G), where = supp( + (x) + (y)). Also, assume that h 1 ( + (x) + (y)) = 1, h 2 ( + (x) + (y)) = 1, h 3 ( + (x) + (y)) = 0, h 4 ( + (x) + (y)) = 1, h 5 ( + (x) + (y)) = 1, h 6 ( + (x) + (y)) = 0, for all x, y X. ow, t P (x, y) = ( [ 2 ]) + 1 P (x, y) = ( [ 2 ]) + 1 f P (x, y) = ( [ 2 ]) + 1 [t P A(x) t P A(y)] 1, t (x, y) = ( [ 2 ]) + 1 [t A (x) t A (y)] 1, [ P A(x) P A(y)] 1, (x, y) = ( [ 2 ]) + 1 [ A (x) A (y)] 1, [fa P (x) fa P (y)] 0, f (x, y) = ( [ 2 ]) + 1 [fa (x) fa (y)] 0.

9 Ths gves the result, Certan bpolar neutrosophc competton graphs 9 t P (x, y) [t P A (x) tp A (y)] = ( [ 2 ]) + 1 > 0.5, P (x, y) [ P A (x) P A (y)] = ( [ 2 ]) + 1 > 0.5, f P (x, y) [fa P (x) f A P (y)] = ( [ 2 ]) + 1 < 0.5, Hence, the edge (x, y) s strong. Ths proves the result. t (x, y) [t A (x) t A (y)] = ( [ 2 ]) + 1 < 0.5, (x, y) [ A (x) A (y)] = ( [ 2 ]) + 1 < 0.5, f (x, y) [fa (x) f A (y)] = ( [ 2 ]) + 1 < 0.5. We now defne another extenson of bpolar neutrosophc competton graph known as m-step bpolar neutrosophc competton graph. In ths paper, we wll use the followng notatons: Px,y: m A bpolar neutrosophc path of length m from x to y. P m x,y : A drected bpolar neutrosophc path of length m from x to y. m(x): + m-step bpolar neutrosophc out-neghbourhood of vertex x. m(x): m-step bpolar neutrosophc n-neghbourhood of vertex x. m (x): m-step bpolar neutrosophc neghbourhood of vertex x. m (G): m-step bpolar neutrosophc neghbourhood graph of the bpolar neutrosophc graph G. C m(g): m-step bpolar neutrosophc competton graph of the bpolar neutrosophc dgraph G. Defnton Suppose G = (A, ) s a bpolar neutrosophc dgraph. The m- step bpolar neutrosophc dgraph of G s denoted by G m = (A, ), where bpolar neutrosophc set of vertces of G s same wth bpolar neutrosophc set of vertces of G m and has an edge between x and y n G m f and only f there exsts a bpolar neutrosophc drected path P m x,y n G. Defnton The bpolar neutrosophc m-step out-neghbourhood of vertex x of a bpolar neutrosophc dgraph G = (A, ) s bpolar neutrosophc set X + x (P )+ m(x) + = (X x + (P )+, t x, x, f (P )+ x, t ()+ x, x ()+, f x ()+ ), where = {y there exsts a drected bpolar neutrosophc path of length m from x (P )+ (P )+ to y, P m x,y}, t x : X x + [0, 1], x : X z + [0, 1], f x : X z + [0, 1] t ()+ (P )+ t x x : X x + [ 1, 0], ()+ x : X z + [ 1, 0], f x ()+ : X z + [ 1, 0] are defned by = mn{t P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m (P )+ x,y}, = mn{ P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m (P )+ x,y}, f x t ()+ x = max{t P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m x,y}, x ()+ x (P )+ = max{f P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m x,y}, = max{ (x 1, x 2 ),

10 10 M. Akram and M. asr (x 1, x 2 ) s an edge of P m x,y}, f x ()+ respectvely. = mn{f (x 1, x 2 ), (x 1, x 2 ) s an edge of P m x,y}, Example Consder G = (A, ) s a bpolar neutrosophc dgraph, such that X = {x, y, a, b, c, d}, as shown n Fg. 5. Then, 2-step out-neghbourhood of vertces x, and y s calculated as, 2 + (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)}, 2 + (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)}. Fgure 5. polar neutrosophc dgraph Defnton The bpolar neutrosophc m-step n-neghbourhood of vertex x of a bpolar neutrosophc dgraph G = (A, ) s bpolar neutrosophc set X x m(x) = (Xx (P ) (P ), t x, x, f (P ) x, t () x, x (), f x () ), where = {y there exsts a drected bpolar neutrosophc path of length m from y (P ) (P ) to x, P m y,x}, t x : Xx [0, 1], x : Xz [0, 1], f x : Xz [0, 1] t () (P ) t x x : Xx [ 1, 0], () x : Xz [ 1, 0], f x () : Xz [ 1, 0] are defned by = mn{t P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m (P ) y,x}, = mn{ P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m (P ) y,x}, f x = max{t (x 1, x 2 ), (x 1, x 2 ) s an edge of P m y,x}, x () t () x (x 1, x 2 ) s an edge of P m y,x}, f x () respectvely. x (P ) = max{f P (x 1, x 2 ), (x 1, x 2 ) s an edge of P m y,x}, = max{ (x 1, x 2 ), = mn{f (x 1, x 2 ), (x 1, x 2 ) s an edge of P m y,x}, Example Consder G = (A, ) s a bpolar neutrosophc dgraph, such that, X = {a, b, c, d, e, f}, as shown n Fg. 6. Then, 2-step n-neghbourhood of vertces a, and b s calculated as, 2 (a) = {(f, 0.1, 0.1, 0.5, 0.1, 0.2, 0.6), (e,

11 Certan bpolar neutrosophc competton graphs , 0.1, 0.7, 0.1, 0.2, 0.4)}, 2 0.4, 0.3, 0.6, 0.3, 0.4, 0.5)}. (b) = {(f, 0.1, 0.3, 0.6, 0.3, 0.4, 0.7), (e, Fgure 6. polar neutrosophc dgraph Defnton Suppose G = (A, ) s a bpolar neutrosophc dgraph. The m-step bpolar neutrosophc competton graph of bpolar neutrosophc dgraph G s denoted by C m ( G) = (A, ) whch has same bpolar neutrosophc set of vertces as n G and has an edge between two vertces x, y X n C m ( G) f and only f ( m(x) + m(y)) + s a non-empty bpolar neutrosophc set n G. The postve truth-membershp value of edge (x, y) n C m ( G) s t P (x, y) = [tp A (x) t P A (z)]h 1( m(x) + m(y)), + the postve ndetermnacy-membershp value of edge (x, y) n C m ( G) s P (x, y) = [P A (x) P A (y)]h 2( m(x) + m(y)), + the postve falstymembershp value of edge (x, y) n C m ( G) s f P (x, y) = [f A P (x) f A P (y)]h 3( m(x) + m(y)), + the negatve truth-membershp value of edge (x, y) n C m ( G) s t (x, y) = [t A (x) t A (z)]h 4( m(x) + m(y)), + the negatve ndetermnacy-membershp value of edge (x, y) n C m ( G) s (x, y) = [ A (x) A (y)]h 5( m(x) + m(y)), + the negatve falsty-membershp value of edge (x, y) n C m ( G) s f (x, y) = [fa (x) f A (y)]h 6( m(x) + m(y)). + The 2 step bpolar neutrosophc competton graph s llustrated by the followng example. Example Consder G = (A, ) s a bpolar neutrosophc dgraph, such that X = {x, y, a, b, c, d}, as shown n Fg. 7. Then, 2 + (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)}, 2 + (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 ntersecton

12 12 M. Akram and M. asr Fgure 7. polar neutrosophc dgraph s calculated as 2 + (x) 2 + (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 (x, y) = 0.08, P (x, y) = 0.04, f P (x, y) = 0.35, t (x, y) = 0.06, (x, y) = 0.06, and f (x, y) = Then ts correspondng 2-step bpolar neutrosophc competton graph s shown n Fg. 8. Fgure 8. 2-step bpolar neutrosophc competton graph If a predator x attacks one prey y, then the lnkage s shown by an edge (x, y) n a bpolar neutrosophc dgraph. ut, f predator needs help of many other medators x 1, x 2,... x m 1, then lnkage among them s shown by bpolar neutrosophc drected path P m x,y n a bpolar neutrosophc dgraph. So, m-step prey n a bpolar neutrosophc dgraph s represented by a vertex whch s the m-step out-neghbourhood of some vertces. ow, the strength of a bpolar neutrosophc competton graphs s defned below.

13 Certan bpolar neutrosophc competton graphs 13 Defnton Let G = (A, ) be a bpolar neutrosophc dgraph. Let w be a common vertex of m-step out-neghbourhoods of vertces x 1, x 2,..., x l. Also, let 1 P (u 1, v 1 ), 1 P (u 2, v 2 ),..., 1 P (u l, v l ) be the mnmum postve truthmembershp values, P 2 (u 1, v 1 ), P 2 (u 2, v 2 ),..., P 2 (u l, v l ) be the mnmum postve ndetermnacy-membershp values, P 3 (u 1, v 1 ), P 3 (u 2, v 2 ),..., P 3 (u l, v l ) be the maxmum postve false-membershp values, 1 (u 1, v 1 ), 1 (u 2, v 2 ),..., 1 (u l, v l ) be the maxmum negatve truth-membershp values, 2 (u 1, v 1 ), 2 (u 2, v 2 ),..., 2 (u l, v l ) be the maxmum negatve ndetermnacy-membershp values, 3 (u 1, v 1 ), 3 (u 2, v 2 ),..., 3 (u l, v l ) be the mnmum negatve false-membershp values, of edges of the paths P m x 1,w, P m x 2,w,..., P m x l,w, respectvely. The m-step prey w X s strong prey f 1 P (u, v ) > 0.5, 1 (u, v ) < 0.5, 2 P (u, v ) > 0.5, 2 (u, v ) < 0.5, 3 P (u, v ) < 0.5, 3 (u, v ) < 0.5, for all = 1, 2,..., l. The strength of the prey w can be measured by the mappng S : X [0, 1], such that: { S(w) = 1 [ 1 P (u, v )] + [ 2 P (u, v )] + [ 3 P (u, v )] l =1 =1 =1 [ 1 (u, v )] [ 2 (u, v )] [ } 3 (u, v )]. =1 =1 =1 Example Consder bpolar neutrosophc dgraph G = (A, ) as shown n Fg. 7, the strength of the prey b s equal to [ ] + [ ] + [ ] [ ] [ ] [ ] 2 Hence, b s strong 2-step prey. = 1.75 > 0.5. Theorem If a prey w of G = (A, ) s strong, then the strength of w, S(w) > 0.5. proof. Let G = (A, ) be a bpolar neutrosophc dgraph. Let w be a common vertex of m-step out-neghbourhoods of vertces x 1, x 2,..., x l,.e., there exsts the paths P m x 1,w, P m x 2,w,..., P m x l,w, n G. Also, let 1 P (u 1, v 1 ), 1 P (u 2, v 2 ),..., 1 P (u l, v l ) be the mnmum postve truth-membershp values, 2 P (u 1,v 1 ), 2 P (u 2,v 2 ),..., P 2 (u l,v l ) be the mnmum postve ndetermnacy-membershp values, P 3 (u 1, v 1 ), P 3 (u 2, v 2 ),..., P 3 (u l, v l ) be the maxmum postve false-membershp values,

14 14 M. Akram and M. asr 1 (u 1, v 1 ), 1 (u 2,v 2 ),..., 1 (u l, v l ) be the maxmum negatve truth-membershp values, 2 (u 1, v 1 ), 2 (u 2, v 2 ),..., 2 (u l, v l ) be the maxmum negatve ndetermnacy- membershp values, 3 (u 1, v 1 ), 3 (u 2, v 2 ),..., 3 (u l, v l ) be the mnmum negatve false-membershp values, of edges of the paths P m x 1,w, P m x 2,w,..., P m x l,w, respectvely. If w s strong, each edge (u, v ), = 1, 2,..., l s strong. So, 1 P (u, v ) > 0.5, 1 (u, v ) < 0.5, ow, 2 P (u, v ) > 0.5, 2 (u, v ) < 0.5, S(w) > Ths proves the result. 3 P (u, v ) < 0.5, 3 (u, v ) < 0.5, for all = 1, 2,..., l (l tmes) l > 0.5. Remark: The converse of the above theorem s not true,.e. f S(w) > 0.5, then all preys may not be strong. Ths can be explaned as: Let S(w) > 0.5 for a prey w n G. So, Hence, S(w) = 1 l =1 { =1 [ 1 P (u, v )] + [ 1 (u, v )] =1 [ 2 P (u, v )] + =1 [ 2 (u, v )] =1 { [ 1 P (u, v )] + [ 2 P (u, v )] + =1 [ 1 (u, v )] =1 [ 3 P (u, v )] =1 [ } 3 (u, v )]. =1 [ 3 P (u, v )] =1 [ 2 (u, v )] =1 [ } 3 (u, v )] > l 2. Ths result does not necessarly mply that 1 P (u, v ) > 0.5, 2 P (u, v ) > 0.5, 3 P (u, v ) < 0.5, 1 (u, v ) < 0.5, 2 (u, v ) < 0.5, 3 (u, v ) < 0.5, for all = 1, 2,..., l. Snce, all edges of the drected paths P m x 1,w, P m x 2,w,..., P m x l,w, are not strong. So, the converse of the above statement s not true.e., f S(w) > 0.5, the prey w of G may not be strong. =1

15 Certan bpolar neutrosophc competton graphs 15 Theorem If all preys of G = (A, ) are strong, then all edges of C m ( G) = (A, ) are strong. proof. Let G = (A, ) be a bpolar neutrosophc dgraph and all preys of t are strong. Let C m ( G) = (A, ), where, t P (x, y) = [t P A(x) t P A(z)]h 1 ( + m(x) + m(y)), P (x, y) = [ P A(x) P A(y)]h 2 ( + m(x) + m(y)), f P (x, y) = [f P A (x) f P A (y)]h 3 ( + m(x) + m(y)), t (x, y) = [t A (x) t A (z)]h 4 ( + m(x) + m(y)), (x, y) = [ A (x) A (y)]h 5 ( + m(x) + m(y)), f (x, y) = [f A (x) f A (y)]h 6 ( + m(x) + m(y)), for all edges (x, y) n C m ( G) = (A, ). Then there arses two cases: Case 1.: Let + m(x) + m(y) be null set. Then there does not exts any edge between x and y n C m ( G). Case 2.: Let + m(x) + m(y) be non-empty. ow, clearly h 1 ( + m(x) + m(y)) > 0.5, h 2 ( + m(x) + m(y)) > 0.5, h 3 ( + m(x) + m(y)) < 0.5, h 4 ( + m(x) + m(y)) < 0.5, h 5 ( + m(x) + m(y)) < 0.5, h 6 ( + m(x) + m(y)) < 0.5, n G as all preys are strong. So, the edge (x, y), x, y X n C m ( G) have the membershps values t P (x, y) = [t P A(x) t P A(z)]h 1 ( + m(x) + m(y)), P (x, y) = [ P A(x) P A(y)]h 2 ( + m(x) + m(y)), f P (x, y) = [f P A (x) f P A (y)]h 3 ( + m(x) + m(y)), t (x, y) = [t A (x) t A (z)]h 4 ( + m(x) + m(y)), (x, y) = [ A (x) A (y)]h 5 ( + m(x) + m(y)), f (x, y) = [f A (x) f A (y)]h 6 ( + m(x) + m(y)), and hence, all the edges are strong. A relaton s establshed between m-step bpolar neutrosophc competton graph of a bpolar neutrosophc dgraph and bpolar neutrosophc competton graph of m-step bpolar neutrosophc dgraph. Theorem If G s a bpolar neutrosophc dgraph and G m s the m-step bpolar neutrosophc dgraph of G, then C( G m ) = C m ( G). proof. Let G = (A, ) be a bpolar neutrosophc dgraph and G m = (A, J ) s the m-step bpolar neutrosophc dgraph of G. Also, let C( G m ) = (A, J) and C m ( G) = (A, ). It can be easly observed that bpolar neutrosophc vertex sets of these graphs are same. So, we have to show that the bpolar neutrosophc edge

16 16 M. Akram and M. asr sets of C( G m ) and C m ( G) are equal. Let (x, y) be an edge n C( G m ). So, there exsts bpolar neutrosophc drected edges (x, a 1 ), (y, a 1 ); (x, a 2 ), (y, a 2 );... ; (x, a l ), (y, a l ), for some postve nteger l n G m. ow, n G m, + (x) + (y) = {(a, s P, q P, r P, s, q, r ) = 1, 2,..., l}, where, s P = J (x, a ) J (y, a ), s = J (x, a ) J (y, a ), q P = J (x, a ) J (y, a ), q = J (x, a ) J (y, a ), r P = J (x, a ) J (y, a ), r = J (x, a ) J (y, a ). Let S P = max{s P = 1, 2,..., l}, Q P = max{q P = 1, 2,..., l}, R P = mn{r P = 1, 2,..., l}, S = mn{s = 1, 2,..., l}, Q = mn{q = 1, 2,..., l}, R = max{r = 1, 2,..., l}. Hence, t P J (x, y) = (t P A(x) t P A(y))h 1 ( + (x) + (y)) = S P t P A(x) t P A(y), P J (x, y) = ( P A(x) P A(y))h 2 ( + (x) + (y)) = Q P P A(x) P A(y), f P J (x, y) = (f P A (x) f P A (y))h 3 ( + (x) + (y)) = R P f P A (x) f P A (y), t J (x, y) = (t A (x) t A (y))h 4 ( + (x) + (y)) = S t A (x) t A (y), J (x, y) = ( A (x) A (y))h 5 ( + (x) + (y)) = Q A (x) A (y), f J (x, y) = (f A (x) f A (y))h 6 ( + (x) + (y)) = R f A (x) f A (y). An edge (x, a ) exsts n G m that mples there exsts a bpolar neutrosophc drected path from x to a of length m, P m x,a n G and J P 1 (x, a ) = mn{ P 1 (u, v) (u, v) s an edge n P m x,a }, J P 2 (x, a ) = mn{ P 2 (u, v) (u, v) s an edge n P m x,a }, J P 3 (x, a ) = max{ P 3 (u, v) (u, v) s an edge n P m x,a }, J 1 (x, a ) = max{ 1 (u, v) (u, v) s an edge n P m x,a }, J 2 (x, a ) = max{ 2 (u, v) (u, v) s an edge n P m x,a }, J 3 (x, a ) = mn{ 3 (u, v) (u, v) s an edge n P m x,a }.

17 Certan bpolar neutrosophc competton graphs 17 Thus, the edge (x, y) s also avalable n C m ( G). Also, n G. Hence, fnally h 1 ( + m(x) + m(y)) = S P, h 4 ( + m(x) + m(y)) = S, h 2 ( + m(x) + m(y)) = Q P, h 5 ( + m(x) + m(y)) = Q, h 3 ( + m(x) + m(y)) = R P, h 6 ( + m(x) + m(y)) = R, t P (x, y) = [t P A(x) t P A(z)]h 1 ( + m(x) + m(y)) = S P t P A(x) t P A(z), P (x, y) = [ P A(x) P A(y)]h 2 ( + m(x) + m(y)) = Q P P A(x) P A(y), f P (x, y) = [f P A (x) f P A (y)]h 3 ( + m(x) + m(y)) = R P f P A (x) f P A (y), t (x, y) = [t A (x) t A (z)]h 4 ( + m(x) + m(y)) = S t A (x) t A (z), (x, y) = [ A (x) A (y)]h 5 ( + m(x) + m(y)) = Q A (x) A (y), f (x, y) = [f A (x) f A (y)]h 6 ( + m(x) + m(y)) = R f A (x) f A (y). Ths proves that there exsts an edge n C m ( G) for every edge n C( G m ). Smlarly, for every edge n C m ( G) there exsts an edge n C( G m ). Ths proves that C( G m ) = C m ( G). Theorem Let G = (A, ) be a bpolar neutrosophc dgraph. If m > X then C m ( G) = (A, ) has no edge. proof. Let G = (A, ) be a bpolar neutrosophc dgraph and C m ( G) = (A, ) be the correspondng m-step bpolar neutrosophc competton graph, where, t P (x, y) = [t P A(x) t P A(z)]h 1 ( + m(x) + m(y)), P (x, y) = [ P A(x) P A(y)]h 2 ( + m(x) + m(y)), f P (x, y) = [f P A (x) f P A (y)]h 3 ( + m(x) + m(y)), t (x, y) = [t A (x) t A (z)]h 4 ( + m(x) + m(y)), (x, y) = [ A (x) A (y)]h 5 ( + m(x) + m(y)), f (x, y) = [f A (x) f A (y)]h 6 ( + m(x) + m(y)), for all edges (x, y) n C m ( G). If m > X, there does not exst any drected bpolar neutrosophc path of length m n G. So, + m(x) + m(y) a an empty set. Hence, there does not exst any edge n C m ( G). ow, m-step bpolar neutrosophc neghbouhood graphs are defnes below. Defnton The bpolar neutrosophc m-step out-neghbourhood of vertex x of a bpolar neutrosophc dgraph G = (A, ) s bpolar neutrosophc set

18 18 M. Akram and M. asr m (x) = (X x, t P x, P x, f P x, t x, x, f x ), where X x = {y there exsts a drected bpolar neutrosophc path of length m from x to y, Px,y}, m t P x : X x [0, 1], P x : X x [0, 1], fx P : X x [0, 1], t x : X x [ 1, 0], x : X x [ 1, 0], fx : X x [ 1, 0], are defned by t P x = mn{t P (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m P x = mn{ P (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m fx P = max{f P (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m t x = max{t (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m x = max{ (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m fx = mn{f (x 1, x 2 ), (x 1, x 2 ) s an edge of Px,y}, m respectvely. Defnton Suppose G = (A, ) s a bpolar neutrosophc graph. Then m-step bpolar neutrosophc neghbouhood graph m (G) s defned by m (G) = (A, ) where A = (A P 1, A P 2, A P 3, A 1, A 2, A 3 ), = ( P 1, P 2, P 3, 1, 2, 3 ), 1 P : X X [0, 1], 2 P : X X [0, 1], 3 P : X X [0, 1], 1 : X X [ 1, 0], 2 : X X [ 1, 0], and 3 : X X [ 1, 0] are such that: P 1 (x, y) = A P 1 (x) A P 1 (y)h 1 ( m (x) m (y)), P 2 (x, y) = A P 2 (x) A P 2 (y)h 2 ( m (x) m (y)), P 3 (x, y) = A P 3 (x) A P 3 (y)h 3 ( m (x) m (y)), 1 (x, y) = A 1 (x) A 1 (y)h 4 ( m (x) m (y)), 2 (x, y) = A 2 (x) A 2 (y)h 5 ( m (x) m (y)), 3 (x, y) = A 3 (x) A 3 (y)h 6 ( m (x) m (y)), respectvely. Defnton [?]Consder a bpolar neutrosophc graph G = (A, ), where A = (A P 1, A P 2, A P 3, A 1, A 2, A 3 ), and = ( P 1, P 2, P 3, 1, 2, 3 ) then, an edge (x, y), x, y X s called ndependent strong f 1 2 [AP 1 (x) A P 1 (y)] < 1 P (x, y), 1 2 [AP 2 (x) A P 2 (y)] < 2 P (x, y), 1 2 [AP 3 (x) A P 3 (y)] > 3 P (x, y), Otherwse, t s called weak. 1 2 [A 1 (x) A 1 (y)] > 1 (x, y), 1 2 [A 2 (x) A 2 (y)] > 2 (x, y), 1 2 [A 3 (x) A 3 (y)] < 3 (x, y). Theorem If all the edges of bpolar neutrosophc dgraph G = (A, ) are ndependent strong, then all the edges of C m ( G) are ndependent strong. proof. Suppose G = (A, ) s a bpolar neutrosophc dgraph and C m ( G) = (A, ) s correspondng m-step bpolar neutrosophc competton graph. Snce all the

19 Certan bpolar neutrosophc competton graphs 19 edges of G are ndependent strong, then h 1 ( + m(x) + m(y)) > 0.5, h 2 ( + m(x) + m(y)) > 0.5, h 3 ( + m(w) + m(z)) < 0.5, h 4 ( + m(x) + m(y)) < 0.5, h 5 ( + m(x) + m(y)) < 0.5, h 6 ( + m(w) + m(z)) < 0.5. Then, t P (x, y) = (tp A (x) tp A (y))h 1( m(x) + m(y)) + or, t P (x, y) > 0.5(tP A (x) tp A (y)) or, t P (x,y) (t P A (x) tp A (y)) > 0.5, P (x, y) = (P A (x) P A (y))h 2( m(x) + m(y)) + or, P (x, y) > 0.5( P A (x) P A (y)) or, P (x,y) ( P A (x) P A (y)) > 0.5, f P (x, y) = (f A P (x) f A P (y))h 3( m(x) + m(y)) + or, f P (x, y) < 0.5(f A P (x) f A P f (y)) or, P (x,y) (fa P (x) f A P (y)) < 0.5, t (x, y) = (t A (x) t A (y))h 4( m(x) + m(y)) + or, t (x, y) < 0.5(t A (x) t A (y)) or, t (x,y) (t A (x) t A (y)) < 0.5, (x, y) = ( A (x) A (y))h 5( m(x) + m(y)) + or, (x, y) < 0.5( A (x) A (y)) or, (x,y) ( A (x) A (y)) < 0.5, f (x, y) = (f A (x) f A (y))h 6( + m(x) + m(y)) or, f (x, y) < 0.5(f A (x) f A f (y)) or, (x,y) (f A (x) f A (y)) < 0.5. Hence, the edge (x, y) s ndependent strong n C m ( G). Snce, (x, y) s taken to be arbtrary edge of C m ( G), thus all the edges of C m ( G) are ndependent strong. 3. Applcaton Sports are very mportant, every socety has ts own specal knds of sports. The proper end of sports s bodly health ard physcal ftness. Sports and games have now come to stay n our cvlzaton as an essental feature of human actvty, and ther object s not merely fun, they also nstll the sprt of dscplne and teamwork. Sports lke crcket, hockey and foot ball are popular because of the sprt of team work whch they nspre. Ths no doubt true. The dscplne that ganed n playng up sports s nvaluable n later lfe. It makes for a lfe of co-operaton and team work whch could be used for buldng up a great socety and a naton. Key components of sports are goals, rules, challenge, and nteracton. Sports generally nvolve mental or physcal stmulaton, and often both. Many sports help develop practcal sklls, serve as a form of exercse, or otherwse perform an educatonal, smulatonal, or psychologcal role etc. Many sports requre specal equpment and dedcated playng felds, leadng to the nvolvement of a communty much larger than the group of players. A cty or town may set asde such resources for the organzaton of sports leagues, lke, tabletop games, board games, etc. All these types of sports are called local sports. These sports can be extended to provsonal level sports. After provsonal level sports there are natonal sports. A natonal sport s a sport or game that s consdered to be an ntrnsc part of the culture of a naton. Every naton has dfferent sports, such as, baseball s known as natonal sports n the Unted States, crcket s n England, and hockey s n Pakstan, etc. After, natonal level of sports there are nternatonal level of sports. Internatonal sport s a sport n whch the partcpants represent dfferent countres. The most

20 20 M. Akram and M. asr well-known nternatonal sports event s the Olympc Games, FIFA World Cup and the Paralympc Games. Consder the set consstng of three countres {C 1, C 2, C 3 } and also consder the set of players {(Abgal, 0.9, 0.8, 0.5, 0.6, 0.5, 0.2), (Alex, 0.6, 0.3, 0.4, 0.2, 0.4, 0.4),(Amela, 0.8, 0.7, 0.2, 0.7, 0.8, 0.5), (Agatha, 0.9, 0.8, 0.5, 0.6, 0.5, 0.2), (Angela, 0.9, 0.8, 0.5, 0.6, 0.5, 0.2), (elnda, 0.9, 0.8, 0.5, 0.6, 0.5, 0.2), (Ann, 0.5, 0.3, 0.5, 0.5, 0.3, 0.2), (Arlene, 0.8, 0.8, 0.9, 0.8, 0.9, 0.8), (ella, 0.6, 0.4, 0.9, 0.6, 0.7, 0.5), (Anne, 0.9, 0.7, 0.8, 0.8, 0.8, 0.8), (Aprl, 0.5, 0.3, 0.5, 0.5, 0.3, 0.2), (Abbey, 0.5, 0.3, 0.5, 0.5, 0.3, 0.2)}, whch are takng part n ther local, provsonal, natonal, and nternatonal level games, as shown n Fg. 9. The postve degree of membershp t P (x) of each player represent the percentage of hardwork towards to acheve the success n partcular game, P (x) and f P (x) represent the ndetermnacy and falsty n ths percentage. The negatve degree of membershp t (x) represents the percentage that the player faces falure n the achevement of success n a partcular game, (x) and f (x) represent the ndetermnacy and falsty n ths percentage. The postve degree of membershp t P (x) of each drected edge between player and local, provsonal, natonal and nternatonal level games represent the percentage of havng stamna for that level of sports n nternatonal game, P (x) and f P (x) represent the ndetermnacy and falsty n ths percentage. The negatve degree of membershp t (x) of each drected edge between player and local, provsonal, natonal and nternatonal level games represent the percentage of havng no stamna for that level of sports n nternatonal game, (x) and f (x) represent the ndetermnacy and falsty n ths percentage. Thus, 4-step bpolar neutrosophc competton graph can be used n order to fnd the best results. There 4-step bpolar neutrosophc out-neghbourhoods s calculated n Table 2. Table 2. 4-Step bpolar neutrosophc out-neghbourhoods x X 4 + (x) Abgal {(Internatonal games, 0.2, 0.2, 0.6, 0.1, 0.3, 0.4)} Alex {(Internatonal games, 0.4, 0.2, 0.6, 0.1, 0.2, 0.7)} Amela {(Internatonal games, 0.5, 0.5, 0.6, 0.2, 0.2, 0.8)} Therefore, 4 + (Abgal) 4 + (Alex) = {(Internatonal games, 0.2, 0.2, 0.6, 0.1, 0.2, 0.7)}, 4 + (Abgal) 4 + (Amela) = {(Internatonal games, 0.2, 0.2, 0.6, 0.1, 0.2, 0.8)}, and 4 + (Alex) 4 + (Amela) = {(Internatonal games, 0.4, 0.2, 0.6, 0.1, 0.2, 0.8)}. Further, h( 4 + (Abgal) 4 + (Alex)) = (0.2, 0.2, 0.6, 0.2, 0.2, 0.6), h( 4 + (Abgal) 4 + (Amela)) = (0.2, 0.2, 0.6, 0.2, 0.2, 0.6), and h( 4 + (Amela) 4 + (Alex)) = (0.4, 0.2, 0.6, 0.4, 0.2, 0.6). Thus, we obtan 4-step bpolar neutrosophc competton graph, as shown n Fg. 10.

21 Certan bpolar neutrosophc competton graphs 21 Fgure 9. polar neutrosophc dgraph Table 3. Strength of competton of applcants for nternatonal games (x, y) (Abgal, Alex) (Abgal, Amela) (Alex, Amela) T (x, y) (0.12, 0.06, 0.30, 0.04, 0.08, 0.24) (0.16, 0.14, 0.30, 0.12, 0.10, 0.30) (0.24, 0.06, 0.24, 0.08, 0.08, 0.30) S(x, y) The strength to compete the others players wth respect hardwork n order to acheve success s calculated n Table 3. In Table 3, T (x, y) represents the value of strength of competton between players x and y wth respect to hardwork to

22 22 M. Akram and M. asr Fgure Step bpolar neutrosophc competton graph acheve the success n partcular game. From Table 3, t s clear that the strength of competton between Alex and Amela to acheve the success n partcular game n nternatonal level s 1.24, whle strength of competton between between Abgal and Amela s 1, and strength of competton between between Abgal and Alex s It s also clear from the Table 3, that Alex and Amela are strongest contestants, as the strength of competton between them has the largest value than the other contestants. We now elaborate ths method wth the help of an algorthm.

23 Certan bpolar neutrosophc competton graphs 23 Algorthm Step 1.: Input the postve truth, ndetermnacy and falsty-membershps values and negatve truth, ndetermnacy and falsty-membershps values for set of r applcants. Step 2.: If for any two dstnct vertces x and x j, t P (x x j ) > 0, P (x x j ) > 0, f P (x x j ) > 0, t (x x j ) < 0, (x x j ) < 0, f (x x j ) < 0, then (x j, t P (x x j ), P (x x j ), f P (x x j ), t (x x j ), (x x j ), f (x x j )) + m(x ). Step 3.: Repeat step 2 for all vertces x and x j to calculate m-step bpolar neutrosophc-out-neghbourhoods + m(x ). Step 4.: Calculate + m(x ) + m(x j ) for each par of dstnct vertces x and x j. Step 5.: Calculate h[ + m(x ) + m(x j )]. Step 6.: If + m(x ) + m(x j ) then draw an edge x x j. Step 7.: Repeat step 6 for all par of dstnct vertces. Step 8.: Assgn membershp values to each edge x x j usng the condtons t P (x x j ) = (x x j )h 1 [ + m(x ) + m(x j )] t (x x j ) = (x x j )h 4 [ + m(x ) + m(x j )] P (x x j ) = (x x j )h 2 [ + m(x ) + m(x j )] (x x j ) = (x x j )h 5 [ + m(x ) + m(x j )] f P (x x j ) = (x x j )h 3 [ + m(x ) + m(x j )] f (x x j ) = (x x j )h 6 [ + m(x ) + m(x j )]. Step 10.: Calculate S(x, y), the strength of competton between players x and y. S(x, y) = t P (x, y) ( P (x, y) + f P (x, y)) t (x, y) ( (x, y) + f (x, y)). Step 11.: Maxmum value of S(x, y) gves that x and y are strongest players than the others. 4. Concludng Remarks Graph theory s an enjoyable playground for the research of proof technques n dscrete mathematcs. There are many applcatons of graph theory n dfferent felds. We have ntroduced the concepts of the bpolar neutrosophc competton graphs. We have descrbed an applcaton of m-step bpolar neutrosophc competton graphs n dfferent level of games wth the help of an algorthm. We am to extend our research work to (1) polar fuzzy rough graphs; (2) polar fuzzy rough hypergraphs, (3) polar fuzzy rough neutrosophc graphs, and (4) Decson support systems based on bpolar neutrosophc graphs. Acknowledgement. The authors are hghly thankful to Executve Edtor and the referees for ther valuable comments and suggestons.

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