Received: 29 November 2018; Accepted: 12 January 2019; Published: 16 January 2019

Transcription

1 mthemtis Artile q-rung Orthopir Fuzzy Competition Grphs with Applition in the Soil Eosystem Amn Hi 1 Muhmmd Akrm 1 * nd Adeel Frooq 2 1 Deprtment of Mthemtis University of the Punj New Cmpus Lhore Pkistn; mnhi96@gmil.om 2 Deprtment of Mthemtis COMSATS University Islmd Lhore Cmpus Pkistn; deelfrooq@uilhore.edu.pk * Correspondene: m.krm@puit.edu.pk Reeived: 29 Novemer 2018; Aepted: 12 Jnury 2019; Pulished: 16 Jnury 2019 Astrt: The q-rung orthopir fuzzy set is powerful tool for depiting fuzziness nd unertinty s ompred to the Pythgoren fuzzy model. The im of this pper is to present q-rung orthopir fuzzy ompetition grphs (q-rofcgs) nd their generliztions inluding q-rung orthopir fuzzy k-ompetition grphs p-ompetition q-rung orthopir fuzzy grphs nd m-step q-rung orthopir fuzzy ompetition grphs with severl importnt properties. The study proposes the novel onepts of q-rung orthopir fuzzy liques nd tringulted q-rung orthopir fuzzy grphs with rel-life hrteriztions. In prtiulr the present work evolves the notion of ompetition numer nd m-step ompetition numer of q-rung piture fuzzy grphs with lgorithms nd explores their ounds in onnetion with the size of the smllest q-rung orthopir fuzzy edge lique over. In ddition n pplition is illustrted in the soil eosystem with n lgorithm to highlight the ontriutions of this reserh rtile in prtil pplitions. Keywords: q-rung orthopir fuzzy ompetition grphs; q-rung orthopir fuzzy lique; ompetition numer; m-step ompetition numer; soil eosystem 1. Introdution Configurtions of node onnetions tke ple in wide diversity of pplitions. They my depit physil networks suh s eletri iruits rodwys nd orgni moleules. They re lso employed in depiting fewer intertions s might rise in eosystems dtses soiologil reltionships or in the flow of ontrol in omputer progrm. Any mthemtil ojet onerning points nd onnetions etween them is lled grph. The genesis of grph theory n e tred k to Euler s work on the Königserg ridges prolem in Aristotle verlly outlined the first direted grph in mnner to orgnize logil rguments. In 1968 Cohen [1] formlly introdued ompetition grphs in ssoition with prolem in eology. The ompetition grph is n undireted grph of digrph D where the digrph orresponds to the food we of group of predtor nd prey speies in n eosystem. In this regrd the digrph is usully yli. Cohen defined the ompetition grph of digrph D = (V E ) s n undireted grph C( D ) with the sme vertex set V nd hs n edge etween two distint verties u v V if there exist vertex x V nd rs ux vx E in G. Thus the nlogy of Cohen is sed on the ft tht if two speies hve ommon prey they will strive for the prey. After Cohen s prologue on the ompetition grph vrious vritions of it re deteted in the literture see [2 7]. In 2000 Cho et l. [8] introdued nother generliztion nmed the m-step ompetition grph of direted grph. Besides in eosystems ompetition grphs hve mny pplitions in vrious fields suh s modeling mrket strutures in the field of eonomis ommunitions over noisy hnnel energy systems soil intertions hnnel ssignments et. After the primry motivtion of food we models Mthemtis ; doi: /mth

2 Mthemtis of 33 of eosystems lot of work hve een done on ompetition grphs where it is ssumed tht verties nd edges re preisely defined. However this ssumption is not suffiient to desrie ompetition in mny rel-world senrios suh s in n eosystem where speies my e wek strong vegetrin non-vegetrin et. nd prey my e hrmful digestive tsty et. Due to the vgueness in desription of speies nd prey nd their reltionships it is nturl to design fuzzy ompetition grph model. Zdeh proposed the notion of fuzzy sets in his monumentl pper [9] in 1965 to model unertinty or vgue ides y nominting degree of memership to eh entity rnging etween 0 nd 1. A fuzzy grph originted y Kufmnn [10] in 1973 from the Zdeh s fuzzy reltions [11] n well express the unertinty of plenty of networks. Rosenfeld [12] developed severl theortil onepts of fuzzy grphs in In 1983 intuitionisti fuzzy sets (IFSs) primrily proposed y Atnssov [13] offered mny signifint dvntges in representing humn knowledge y denoting fuzzy memership not only with single vlue ut pirs of mutully orthogonl fuzzy sets lled orthopirs whih llow the inorportion of unertinty. Sine IFSs onfine the seletion of orthopirs to oming only from tringulr region s shown in Figure 1 Pythgoren fuzzy sets (PFSs) proposed y Yger [14] s new extension of IFSs hve emerged s n effiient tool for onduting unertinty more properly in humn nlysis s one n see in [14 17]. Although oth IFSs nd PFSs mke use of orthopirs to nrrte ssessment ojets they still hve visile differenes. The memership funtion µ : X [0 1] nd non-memership funtion ν : X [0 1] of IFSs re required to stisfy the onstrint ondition µ(x) + ν(x) 1. However these two funtions in PFSs re needed to stisfy the ondition µ(x) 2 + ν(x) 2 1 whih shows tht PFSs hve expnded spe to ssign orthopirs s ompred to IFSs displyed in Figure 1. Certin notions of Pythgoren fuzzy grphs hve een disussed in [18 21] q=2 q=3 q=4 0.6 q=1 ν Inresing q Figure 1. Spes of eptle q-rung orthopirs. Figure 2.1: Spes of eptle q-rung orthopirs A q-rung orthopir fuzzy set (q-rofs) originlly proposed y Yger [22] in 2017 is new generliztion of orthopir fuzzy sets (e.g. IFSs PFSs) whih further relx the onstrint of orthopir 7. Let P memership = (µ grdes P (u) νwith P (u)) µ(x) q + ν(x) e q 1 q-rung (q 1) [23]. orthopir As q inreses fuzzy it is esyset. to see tht Then the the rdi representtion spe of llowle orthopir memership grde inreses. Figure 1 displys spes rd(p) is of defined the most widely s eptle orthopirs for different q rungs. Ali [24] lulted the re of spes with dmissile orthopirs upcrd(p) to 10-rungs. Consider = ( P n exmple µ P in the field ν ) of eonomis: in mrket struture huge numer of firms ompete ginst eh other with differentited produts with respet to rnding or qulity whih in nture re vgue words. Sine IFSs hve the pility to explore P µ = u X µ P (u) µ

3 Mthemtis of 33 oth spets of miguous words for exmple it ssigns n orthopir memership grde to qulity i.e. support for qulity nd support for not-qulity of n ojet with the ondition tht their sum is ounded y 1. This onstrint lerly limits the seletion of orthopirs. Moreover in grph theortil onepts it is strightforwrd to oserve the dpttions of predtors towrds their prey nd fight of prey ginst predtors. The IFS n trnslte the unertinty ssoited with oth phses of speies t the sme time with the restrition tht their sum is less thn or equl to one. To relx this ondition nd enhne one s pility to express this knowledge more preisely this pper defines q-rofcgs to del with ompetitions in mny fields. Fuzzy ompetition grphs firstly defined y Smnt nd Pl [25] with its generliztions [25 27] express the prtilness of speies nd prey regrding their extent of ompetition. A lot of work hs een done on fuzzy ompetition grphs. Reently Shoo nd Pl [28] introdued the oneption of intuitionisti fuzzy ompetition grphs to extend the pility to model humn knowledge. Nsir et l. [29] disussed opertions on intuitionisti fuzzy ompetition grphs. Moreover Al-Shehrie nd Akrm [30] disussed ipolr fuzzy ompetition grphs. Srwr nd Akrm further studied this onept in [3132] y defining some opertions. Akrm nd Nsir [33] disussed intervl-vlued neutrosophi ompetition grphs. Akrm nd Srwr [34] nlyzed m-polr fuzzy ompetition grphs. Reently Srwr et l. [35] introdued fuzzy ompetition hyper grphs. Furthermore Sun et l. [36] defined fuzzy liques. Furthermore Sun et l. [36] defined fuzzy liques. In the present study n ttempt is mde to desrie the novel onept of q-rung orthopir fuzzy ompetition grphs. We extend intuitionisti fuzzy k-ompetition grphs p-ompetition intuitionisti fuzzy grphs nd m-step intuitionisti fuzzy ompetition grphs nd otin nlogous results under q-rung orthopir fuzzy environment. In prtiulr we propose novel hrteriztion towrds q-rung orthopir fuzzy liques nd tringulted q-rung orthopir fuzzy grphs with severl relted results. On the other hnd in the literture of ompetition grphs muh ttention hs een foused on finding the ompetition numer of grph. Roerts [37] showed tht fter dding suffiient numers of isolted verties to n yli digrph it leds to ompetition grph. He defined ompetition numer s the smllest suh possile numer. This prmeter hs een extensively studied y mny reserhers; see [38 40]. Previous work hs underlined hrteriztion not only of the ompetition grphs ut lso m-step ompetition grphs. Cho et l. [8] introdued m-step ompetition numer in this regrd whih is nlogous to the notion of ompetition numer y Roerts [37]. Certin ounds on ompetition numer ws disussed in [384041]. The ontriution or this reserh rtile is not only restrited to q-rofcgs ut it introdues the onept of ompetition numer nd m-step ompetition numer of q-rung orthopir fuzzy grphs long with two lgorithms. The results show their onnetion with the size of smllest q-rung orthopir fuzzy edge lique over s ounds. Finlly this work suggests novel pproh towrds the soil eosystem y exploring the strength of ompetition of teri with n lgorithm. 2. Preliminries This setion presents rief review of ompetition grphs fuzzy ompetition grphs nd q-rung orthopir fuzzy sets. Menwhile we define rdinlity support nd height of q-rung orthopir fuzzy sets whih will e used for further developments. A grph G = (V E) onsists of two sets V nd E. The elements of V nd E re lled verties (or nodes) nd edges respetively where eh edge hs set of one or two verties ssoited with it. A digrph (or direted grph) D is grph eh of whose edges is direted usully denoted y D = (V E ) where E is the set of rs uv for u v V. A wlk in grph G is n lternting sequene of verties nd edges W = v 0 e 1 v 1 e 2... e n v n suh tht for j = n the verties v j 1 nd v j re the end points of edge e j. If moreover the edge e j is direted from v j 1 to v j then W is direted wlk. A tril in grph is wlk suh tht no edge ours more thn one. A pth in grph is tril suh tht no internl vertex is repeted. A yle is losed pth of length t lest one. The out-neighorhood nd in-neighorhood [42] of vertex u in D n e defined y N + (u) = v V u} : uv E } nd N (u) = v V u} : vu E } respetively.

4 Mthemtis of 33 Definition 1 ([1]). The ompetition grph C( D ) of digrph D = (V E ) is n undireted grph G = (V E) whih hs the sme vertex set V nd hs n edge etween two distint verties u v V if there exist vertex x V nd rs ux vx E in G. A lique in grph G is mximl set of mutully djent verties of G. The lique numer denoted y w(g) is the numer of verties in lrgest lique of G. An edge lique over of grph G is ny fmily θ(g) = C 1 C 2... C k } of omplete sugrphs of G suh tht every edge of G is in t lest one of E(C 1 ) E(C 2 )... E(C k ). Competition numer defined y Roerts in [37] sttes tht if G is ny grph we n otin ompetition grph y dding mny isolted verties. Add one isolted vertex x α to G orresponding to every edge α in G. Construt food we F with vertex set V(F) = V(G) x α : α E(G)} with n r from end points u nd v of edge α to vertex x α. Then F is food we for the grph G I r. Thus the smllest r suh tht G I r is ompetition grph is lled ompetition numer k(g). Sometimes verties nd edges of grphs re not preisely defined. A fuzzy grph n well express suh unertinty. A fuzzy grph [10] on non-empty set X is pir G = (σ µ) where σ : V [0 1] nd µ : V V [0 1] suh tht for ll u v V µ(u v) σ(u) σ(v) where σ(u) nd µ(u v) represent the memership vlues of the vertex u nd edge uv in G respetively. A fuzzy digrph [43] on non-empty set X is pir D = (A B ) where µ A : V [0 1] nd µ B : V V [0 1] suh tht for ll u v V µ B ( uv) µ A (u) µ A (v) where µ A (u) nd µ B ( uv) represent the memership vlues of the vertex u nd edge uv in D respetively. When we del with prolem in eology speies nd prey my e fuzzy in nture nd the reltionship etween them n e designed y fuzzy ompetition grphs. A lot of work hs een done on fuzzy ompetition grphs nd its vritions whih re designed s motivted y the fuzzy food we. A fuzzy out-neighorhood [25] of vertex v of direted fuzzy grph D = (P Q ) is fuzzy set N + (v) = (X + v µ + P v ) where X + v = u : µ Q ( vu) 0} nd µ + P v : X v + [0 1] defined y µ + P v (u) = µ ( vu). A fuzzy in-neighorhood [25] of vertex v of Q direted fuzzy grph D is fuzzy set N (v) = (Xv µ P v ) where Xv = u : µ ( uv) Q 0} nd µ P v : Xv [0 1] defined y µ P v (u) = µ Q ( uv). Definition 2 ([25]). Let D = (P Q ) e direted fuzzy grph. The fuzzy ompetition grph C( D ) of fuzzy digrph D is n undireted fuzzy grph G = (P R) whih hs sme fuzzy vertex set s in D nd hs fuzzy edge etween two verties u v P in C( D ) if nd only if N + (u) N + (v) = φ in D nd the memership grde of edge uv in C( D ) is µ R (uv) = (µ P (u) µ P (v)) h(n + (u) N + (v)). Definition 3 ([25]). Let k e non-negtive rel numer nd D = (P Q ) e direted fuzzy grph. The fuzzy k-ompetition grph C k ( D ) of fuzzy digrph D is n undireted fuzzy grph G = (P R) whih hs sme fuzzy vertex set s in D nd hs fuzzy edge etween two verties u v P in C k ( D ) if nd only if N + (u) N + (v) > k The memership grde of edge uv in C k ( D ) is where k = N + (u) N + (v). µ R (uv) = k k k (µ P (u) µ P (v)) h(n + (u) N + (v)) Definition 4 ([25]). Let p e positive integer nd D = (P Q ) e direted fuzzy grph. The p-ompetition fuzzy grph C p ( D ) of fuzzy digrph D is n undireted fuzzy grph G = (P R) whih hs sme fuzzy vertex set s in D nd hs fuzzy edge etween two verties u v P in C p ( D ) if nd only if supp(n + (u) N + (v)) p. The memership grde of edge uv in C p ( D ) is

5 Mthemtis of 33 µ R (uv) = where n = supp(n + (u) N + (v)). (n p) + 1 (µ n P (u) µ P (v)) h(n + (u) N + (v)) Before introduing m-step fuzzy ompetition grph we define n m-step fuzzy digrph. An m-step fuzzy digrph D m = (A C ) of fuzzy digrph D = (A B ) hs sme fuzzy vertex set A nd hs fuzzy edge etween u v X if there exist fuzzy direted pth P m uv in D of length m suh tht µ C ( uv) = minµ C ( ) is n edge o f P m uv} where m is positive integer. An m-step fuzzy out-neighorhood [27] of vertex v of direted fuzzy grph D n m-step fuzzy set N + m(v) = (X v + µ + P v ) where X v + = u : P m vu exists} nd µ + P v : X v + [0 1] defined y µ + P v (u) = minµ ( xy) : Q m xy is n edge in P vu}. An m-step fuzzy in-neighorhood [27] of vertex v of direted fuzzy grph D is n m-step fuzzy set N m (v) = (Xv µ P v ) where Xv = u : P m uv exists} nd µ P v : Xv [0 1] defined y µ P v (u) = minµ Q ( ) : is n edge in P m uv}. Definition 5 ([27]). Let D = (P Q ) e direted fuzzy grph. An m-step fuzzy ompetition grph C m ( D ) of fuzzy digrph D is n undireted fuzzy grph G = (P R) whih hs sme fuzzy vertex set s in D nd hs fuzzy edge etween two verties u v P in C m ( D ) if nd only if N + m(u) N + m(v) = φ in D nd the memership grde of edge uv in C m ( D ) is µ R (uv) = (µ P (u) µ P (v)) h(n + m(u) N + m(v)). Definition 6 ([22]). Let X e universe of disourse q-rung orthopir fuzzy set(q-rofs) P on X is given y P = u µ P (u) ν P (u) u X} hrterized y memership funtion µ P : X [0 1] nd non-memership funtion ν P : X [0 1] suh tht 0 µ q P (u) + νq P (u) 1 for ll u X. Moreover π P(u) = 1 q µ q P (u) νq P (u) is lled q-rung orthopir fuzzy index or indeterminy degree of u to the set P. Definition 7. Let P = (µ P (u) ν P (u)) e q-rung orthopir fuzzy set. Then the rdinlity of P is denoted y Crd(P) is defined s Crd(P) = ( P µ P ν ) suh tht P µ = µ P (u) u X P ν = ν P (u). u X Definition 8. Let P = (µ P (u) ν P (u)) e q-rung orthopir fuzzy set. Then the support of P is denoted y Supp(P) is defined s Supp(P) = Supp µ (P) Supp ν (P) suh tht Supp µ (P) = u µ P (u) > 0} Supp ν (P) = u ν P (u) > 0}.

6 y Supp(P) is defined s suh tht Supp(P) = Supp µ (P) Supp ν (P) Supp µ (P) = u µ P (u) > 0} Mthemtis of 33 Supp ν (P) = u ν P (u) > 0}. Definition 9. Let P = (µ P (u) ν P (u)) e q-rung orthopir fuzzy set. Then the height of P is denoted y Definition h(p) 2.9. is Let defined P = s (µ P (u) ν P (u)) e q-rung orthopir fuzzy set. Then the height of P is denoted y h(p) is defined s h(p) = (h µ (P) h ν (P)) h(p) = (h µ (P) h ν (P)) suh tht suh tht h µ (P) sup µ P (u) h µ (P) = u X sup µ P (u) h ν (P) = inf u X ν P(u). u X h ν (P) = inf ν P(u). u X The geometril interprettion is shown in Figure 2. 1 ν hµ hν µ 0 X Figure 2. Geometri interprettion of height of q-rofs (q-rung orthopir fuzzy set). Figure 2.2: Geometri interprettion of height of q-rofs 3. q-rung Orthopir Fuzzy Grphs The q-rung orthopir fuzzy sets (q-rofss) enhne the pility of deision-mkers in ssigning 5 orthopirs y their own hoie. We now define q-rung orthopir fuzzy grphs whih n e extensively used in mny prtil prolems. Definition 10. Let X e non-empty set. A mpping P = (µ P ν P ) : X X [0 1] is lled q-rung orthopir fuzzy reltion on X suh tht µ P ν P [0 1] for ll u v X. Definition 11. Let A = (µ A ν A ) nd B = (µ B ν B ) e q-rung orthopir fuzzy sets on non-empty set X. If B = (µ B ν B ) is q-rung orthopir fuzzy reltion on X then B is lled q-rung orthopir fuzzy reltion on A if µ B (uv) µ A (u) µ A (v) ν B (uv) ν A (u) ν A (v) f or ll u v X nd 0 µ q B (uv) + νq B (uv) 1 for ll u v X. Definition 12. A q-rung orthopir fuzzy grph on non-empty set X is pir G = (A B) with A q-rung orthopir fuzzy set on X nd B q-rung orthopir fuzzy reltion on X suh tht µ B (uv) µ A (u) µ A (v) ν B (uv) ν A (u) ν A (v) f or ll u v X nd 0 µ q B (uv) + νq B (uv) 1 for ll u v X where µ B : X X [0 1] nd ν B : X X [0 1] represents the memership nd non-memership funtions of B respetively. Exmple 1. Figure 3 represents 3-rung orthopir fuzzy grph G = (A B) where ( ) ( ) ( A = nd B = ) ( ). 0.6

7 Figure 3.1 represents 3-rung orthopir fuzzy grph G = (A B) where ) ( ) ( Mthemtis nd B = ) ( of ( ) ( ) ( ) ( ) ( ) ( ) Figure 3. A 3-rung orthopir fuzzy grph G. Figure 3.1: A fuzzy grph G. A q-rung orthopir fuzzy digrph on non-empty set X is pir D = (A r fuzzy set on X nd µ B ( uv) µ A (u) µ A (v) B q-rung ν ( uv) B orthopir ν A (u) ν A (v) f or fuzzy ll u v reltion X on X suh tht µ nd 0 µ q ( uv) + ν q ( uv) 1 for µ ll u A (u) v X where µ B µ A (v) : X B ν B ( B X [0 1] nd ν : X B X [0 1] represents the memership nd non-memership funtions of uv) ν A (u) ν A (v) B respetively. for ll u v X ) + ν q ( uv) hs sme vertex 1 for set A nd ll hs u q-rof v edgex etween where u v X ifµ there exist B B : X q-rof Xdireted pth[0 P m uv1] in Dnd ν B : X emership nd non-memership funtions of B respetively. ν ( uv) = mxν ( ). An m-step q-rung orthopir fuzzy digrph D m = (A C C is n edge o f P m uv} C ) of q-rof digrp set A nddefinition hs 15. q-rof Let D = (A edge C ) e etween q-rung orthopir u fuzzy v digrph. X if Thethere underlyingexist q-rung orthopir q-rof fuzzy direted tht grph of D is q-rofg U( D ) = (A C ) with µ C ( uv) = minµ C ( sme vertex ) set A nd hs q-rung orthopir is n edge of fuzzy edge etween two distint verties u v X suh tht P m uv} tive integer. Definition 13. A q-rung orthopir fuzzy digrph on non-empty set X is pir D = (A B ) with A q-rung orthopir fuzzy set on X nd B q-rung orthopir fuzzy reltion on X suh tht Definition 14. An m-step q-rung orthopir fuzzy digrph D m = (A C ) of q-rof digrph D = (A B ) of length m suh tht where m is positive integer. µ C ( uv) = minµ C ( ) is n edge o f P m uv} (µ ( uv) C µ ( vu) ν ( uv) C C ν ( vu)) C if uv C vu C (µ C (uv) ν C (uv)) = (µ ( vu) ν ( vu)) C C if vu C uv C. ν C ( uv) = mxν C ( ) is n edge of P m (µ ( uv) ν ( uv)) if uv C C C vu C uv} Definition 16. Let G = (µ A (u) ν B (u)) e q-rung orthopir fuzzy grph. An edge uv of G is lled strong if nd wek otherwise. µ B (u v) 1 2 (µ A (u) µ A (v)) ν B (u v) 1 2 (ν A (u) ν A (v)) 6 Definition 17. Let G = (µ A (u) ν B (u)) e q-rung orthopir fuzzy grph. The strength of q-rung orthopir fuzzy edge uv n e mesured s I uv = ((I uv ) µ (I uv ) ν )

8 ht I uv = ((I uv ) µ (I uv ) ν ) Mthemtis of 33 suh tht (I uv ) µ = (I uv ) ν = (I uv ) µ = µ B (uv) µ A (u) µ A (v) ν B (uv) ν A (u) ν A (v). µ B (uv) µ A (u) µ A (v) ν B (uv) ν A (u) ν A (v). ition 3.9. A q-rung orthopir fuzzy grph H = (P Q) is lled q-rung orthopir fuzzy sugr (I uv ) ν = B) if µp (u) µ A (u) Definition 18. A q-rung orthopir fuzzy grph H = (P Q) is lled q-rung orthopir fuzzy sugrph of G = (A B) if ν P (u) ν P (u) for ll u X µ P (u) µ A (u) µq (uv) ν P (u) µ ν P (u) f or ll u X nd B (uv) ν Q (uv) ν B (uv) for ll u v X µ nd Q (uv) µ B (uv) ν Q (uv) ν B (uv) f or ll u v X ht 0 µ Q (uv) + ν Q (uv) 1 for ll u v X. ver q-rung orthopir fuzzy sugrph H = (P Q) is sid to e spnning q-rung orthopir fuzzy su (A B) if µp (u) = µ A (u) suh tht 0 µ Q (uv) + ν Q (uv) 1 for ll u v X. Moreover q-rung orthopir fuzzy sugrph H = (P Q) is sid to e spnning q-rung orthopir fuzzy sugrph of G = (A B) if µ P (u) = µ A (u) ν P (u) = ν ν P (u) = P (u) for ll u X. (u) f or ll u X. ple 3.2. Consider Exmple 2. 3-rung Consider orthopir 3-rung orthopir fuzzy fuzzygrph G = = (A (A B) s B) displyed s displyed in Figure 3. infigure Figure Figure ents 3-rung orthopir represents 3-rung fuzzy orthopir sugrph fuzzy sugrph of G ofnd G ndfigure 43.2() represents represents spnning 3-rung spnning orthopir fuzzy 3-rung orthopir ph of G. sugrph of G. ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) () Figure 4. () 3-Rung Orthopir Fuzzy Sugrph; () Spnning 3-Rung Orthopir Fuzzy Sugrph. Figure 3.2 Definition 19. In q-rung orthopir fuzzy grph G = (A B) q-rung orthopir fuzzy suset P of A is lled q-rung orthopir fuzzy lique if q-rung orthopir fuzzy sugrph of G indued y P is omplete. The size of lrgest q-rof lique is lled lique numer of G. Exmple 3. Figure 5 represents 3-rung orthopir fuzzy grph G = (A B) where ( ) ( ) A = d d nd ( 0.85 B = d 0.6 d ) ( d 0.7 d ). 0.5 Tke P = ( ) ( ) ( )} suh tht eh pir of verties is joined y n edge in G. A 3-rung orthopir fuzzy sugrph H = (P Q) of G indued y P is given in Figure 6. 7

9 3.3. FigureB 3.3= represents rung 0.65 orthopir fuzzy 0.7 grph 0.7 G 0.55 = (A 0.7 B) where. 0.5 ( ) ( ) A = d d nd ( 0.85 B = d 0.6 d ) ( d 0.7 d ). 0.5 ( ) Mthemtis of 33 We see tht H = (P Q) is omplete 3-rung orthopir fuzzy grph. Hene 3-rung orthopir fuzzy suset P is 3-rung orthopir fuzzy lique. ( ) ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( )} suh ( ) tht eh pir of verties is joined y n edg rthopir fuzzy sugrph H = (P Q) of G indued y P is given in Figure 3.4. Figure 5. A 3-rung orthopir fuzzy grph G. Figure 3.3: A fuzzy grph G ( ) ( ) ( )} suh tht eh pir of verties is joined y n edge orthopir fuzzy sugrph H = (P Q) of G indued y P is given in Figure 3.4. ( ) ( ) d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) tht H = Exmple (P Q) 4. Aiswholesler omplete must rrnge 3-rung stll orthopir in n exhiitionfuzzy to displygrph. lredy existing Hene rnds3-rung produts. orthopir fuz He must fe ompetition for higher mnufturing qulity produts. Mny ompnies hve turned to ng orthopir fuzzy Figure lique. 3.4: A 3-rung orthopir fuzzy indued sugrph H e disply nigpplition differene etweenof qulities q-rung of produts orthopir of some ompnies fuzzy(i.e. lique. more thn 5%) then he nnot hoose those e tht H = ompnies. (P Q) Clerly is omplete the produt qulity 3-rung of theorthopir stll is investigted fuzzy y tking grph. into ount Hene the lowest 3-rung qulityorthopir of fuzz rung 3.4. orthopir A wholesler their produts. fuzzyhve lique. to rrnge stll in n exhiition to disply lredy existing r To understnd the ide of q-rung orthopir fuzzy liques tke ompnies s verties. If the produts of two e hs to fe ompetition for higher mnufturing qulity produts. Mny ompnies h we disply n ompnies pplition re displyed ofinq-rung sme stll orthopir then there is nfuzzy edge etween lique. them. The informtion to orgnize suh onl ttisto stllreover n e summrized theiry qulity 6-rung orthopir imge. fuzzythere grph G = re (A B) five given well in Figure known 7 where ompnies support for of differe s 3.4. givenainwholesler Tle memership 1. nd hve The non-memership to wholesler of verties rrnge hve for orresponding stlltoinselet stll n exhiition some inditeof their to them signifint disply ininrese order nd lredy to not existing disply es r inrese in produt qulity respetively. there He hs is to ig fedifferene ompetition etween for higher qulities mnufturing of produts qulity of some produts. ompnies Mny (i.e. ompnies more thn The memership grdes of edges show the extent of oth ompnies to e linked with sme stll with respet hv hoose tionl ttis those to high ompnies. to qulity reover produts. their Clerly Thequlity 6-rungthe orthopir imge. produt fuzzythere set qulity re five of stll wellis known investigted ompnies y of tking differen int qulity s givenof intheir Tle produts. 1. The wholesler hve to selet some of them in order to disply est P = ( ) ( ) ( )} f there is ig differene etween qulities of produts of some ompnies (i.e. more thn 5 t hoose those is 6-rung ompnies. orthopir fuzzy Clerly lique of G s the 6-rung produt orthopir qulity fuzzy grph ofindued stll y ispinvestigted omplete sugrph y tking into of G. Whih shows tht the wholesler n disply the produts of ompnies t qulity of their produts. 1 2 nd 5 s they ll re linked ( ) ( ) Figure 3.3: A 3-rung orthopir fuzzy grph G ( ) ( ) ( ) ( ) ( ) ( ) Figure 6. A 3-rung orthopir fuzzy indued sugrph H. Figure 3.4: A 3-rung orthopir fuzzy indued sugrph H Next we disply n pplition of q-rung orthopir fuzzy lique. promotionl ttis to reover their qulity imge. There re five well-known ompnies of different qulity produts s given in Tle 1. The wholesler must selet some of them to disply est produt qulity. If there is with eh other nd their produt qulities re mthing to some extent. Hene the orresponding produt qulity of stll is ( ).

10 ( ) 5 73% Mthemtis of 33 the ide of q-rung orthopir fuzzy liques tke ompnies s verties. If t Tle 1. Qulity of Produts. lying in sme stll then there is n edge etween them. The informtio Compny Produt Qulity mrized y 6-rung orthopir fuzzy grph G = (A B) given in Figure 3.5 on-memership of verties for orresponding stll indite their signifin t qulity respetively. 1 70% 2 77% 3 65% 4 60% 5 73% ( ) 1 ( ) ( ) 5 ( ) ( ) 2 ( ) ( ) 3 4 ( ) ( ) Figure 7. G = (A B). Figure 3.5: G = (A B) Definition 20. Let G = (A B) e q-rung orthopir fuzzy grph. The olletion of q-rung orthopir fuzzy liques whih over ll the edges of G = (A B) is lled q-rung orthopir fuzzy edge lique over(q-rung orthopir FECC) of G. ip grdes of The edges size of the show smllestthe q-rungextent orthopir FECC ofisoth denoted yompnies θ e (G ). to e linked with sm duts. The 6-rung orthopir fuzzy set Exmple 5. Consider 4-rung orthopir fuzzy grph G = (A B) where ( ) ( A = d 0.85 e d 0.75 e ( B = d 0.65 d 0.6 e 0.4 de 0.4 ) nd P = ( ) (0.4 2 ) ) ( )} ) ( d 0.2 d 0.82 e 0.99 de 0.97 ir fuzzy lique s shown inof Figure G 8. s the 6-rung orthopir fuzzy grph indued y P is Some 4-ROF liques of G re given elow tht the wholesler n disply the produts of ompnies 1 2 nd 5 s P nd their produt qulities 1 = ( ) ( ) ( )} re mthing to some extent. Hene the orr P 1 = ( ) ( ) (d )} ). P 1 = ( ) (d ) (e )}. We see tht 4-ROF sugrphs H Let G = (A B) e q-rung 1 = (P orthopir 1 Q 1 ) H 2 = (P fuzzy 2 Q 2 ) nd H grph. 3 = (P 3 Q The 3 ) of G indued y P 1 P 2 nd P 3 displyed in Figure 9 re omplete. Also H 1 H 2 nd H 3 indued y P 1 P 2 nd olletion P 3 overs of q-ru r ll the ll edges of G. of Thus G the olletion = (A B) is lled q-rung orthopir fuzzy edge F = P 1 P 2 P 3 } of G. is the 4-rung orthopir fuzzy edge lique over. Moreover it is the smllest 4-ROF edge lique over s the size llest q-rung of ll 4-ROF orthopir liques other thn FECC P 1 P 2 ndisp 3 denoted must e less thny 3. Hene θ e (G θ e (G).) = 3. nsider 4-rung orthopir fuzzy grph G = (A B) where ( ) ( ) A = d 0.85 e d 0.75 e ( ) ( 0.99 nd

11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) Mthemtis of 33 ( ) d (0.8 ( ) 0.9) (0.85 ( ) 0.87) ( ) ( ) ( ) ( ) ( ) e ( ) Figure 3.6: A 4-rung orthopir fuzzy grph G Some 4-ROF liques of G re given elow ( ) ( ) ( ) d ( ) ( ) ( ) e ( ) P 1 = ( ) ( ) ( )} P 1 = ( ) ( ) (d )} P 1 = ( ) (d ) (e )}. Figure 8. A 4-rung orthopir fuzzy grph G. Figure 3.6: A fuzzy grph G ( ) ( ) ( ) me 4-ROF liques of G re given elow ( ) ( ) ( ) ( ) P 1 = ( ) ( ) ( )} P 1 = ( ) ( ) (d )} ( ) ( ) P 1 = ( ) (d ) (e )}. ( ) ( ) d ( ) ( ) d ( ) ( ) e ( ) () H 1 () H 2 () H 3 ( ) ( ) ( ) Figure 3.7: 4-rung orthopir fuzzy indued sugrphs Figure 9. 4-rung orthopir fuzzy indued sugrphs. () H 1 ; () H 2 ; () H 3. ( ) ( ) ( ) 4. q-rung Orthopir Fuzzy Competition Grphs is the 4-rung orthopir fuzzy edge lique over. Moreover it is the smllest 4-ROF edge lique over s the size ( ) of ll 4-ROF liques In relisti other senrios thn P 1 sometimes P 2 nd the P 3 fuzzy must verties e less nd thn edges 3. of Hene fuzzy ompetition θ e (G ) = 3. grphs my d d not e enough ( ) ( ) ( ) ( ) Definition Let to explore D = (A vrious types of speies or prey. For exmple nimls B) q-rof yli digrph defined on hve suh dpttions tht enhne their ility for eing suessful predtors like uilt D = (A () H 1 () H 2 () B ). The injetive H 3 mpping π : A A } is lled vertex leling of for speed; use jws D suh tht if shrp teeth nd lws to th nd kill prey; nd n mouflge to hide themselves from prey. DThese hsqulities q-rof re r uv then vgue in nture. Thus there my exist π(u) < π(v) for ll q-roffigure verties 3.7: u v4-rung in some predtors whih use more thn one dpttion towrds D. Inorthopir other words fuzzy everyindued q-rof r sugrphs goes from lower integer to higher it is prey for instne lion n prey either with shrp teeth or lws or even jws i.e. the ility integer. of lion towrds its trget to prey with lws s well s without lws is non-zero. On the ontrry severl prey speies fight ginst predtors through hemils ommunl defense or y ejeting toxi sustnes. To overome suh ses we need orthopirs of fuzzy sets. Shoo nd Pl [28] disussed 4 q-rung Orthopir Fuzzy Competition Grphs this sitution for Atnssov s IFSs with restrition µ + ν 1 on support for memership (µ) nd support for non-memership (ν) whih llows the orthopirs to e in the tringulr region shown in In relisti senrios sometimes the fuzzy verties nd edges of fuzzy ompetition grphs my not enough to Figure 1. Sine q-rung orthopir fuzzy explore vrious types of speies or preys for F sets exmple = relx Pthe nimls 1 Pondition 2 P with hve 3 } µ suh q + ν dpttions q 1 (for suffiiently lrge tht enhne their ility q) results inrese in the re of permissile orthopirs. They n trnslte the unertinty ssoited for eing suessful with oth predtors phses of speies like uilt t the for sme speed; time use in jws more omprehensive shrp teeth nd mnner. lws This to motivtes th nd the kill prey; nd ility of mouflge neessity of toq-rung hide orthopir themselves fuzzy from ompetition preys. grphs. Whih re vgue in nture. Thus there my exist some predtors whih use more thn one dpttion towrds it s preys for instne Lion n prey either with shrp teeth or lws Definition or even22. jws A q-rung i.e. the orthopir ility fuzzy of out-neighorhood Lion towrds of its vertex trgetv to of prey direted with q-rofg lws Ds is well the s without q-rofs N + (v) = (X v + µ + P v ν P + v ) where X v + = u : µ ( vu) Q 0 or ν ( vu) Q 0} nd µ + P v : X v + [0 1] ( ) ( ) Definition 21. Let D = (A B ) e q-rof yli digrph defined on D We see tht 4-ROF sugrphs H 1 = (P 1 Q 1 ) H 2 = (P 2 Q 2 ) nd H = (A 3 = (P B 3 Q ). The injetive 3 ) of G indued y P 1 P 2 mpping π : A A nd P 3 displyed in Figure 3.7 re omplete. } is lled vertex leling of D suh tht if D hs q-rof r uv then Also H 1 H 2 nd H 3 indued y P 1 P 2 nd P 3 overs ll edges π(u) < π(v) for ll q-rof verties u v in D. In other words every q-rof r goes from lower integer to of G. Thus the olletion higher integer. e ( ) F = P 1 P 2 P 3 } see tht 4-ROF sugrphs H 1 = (P 1 Q 1 ) H 2 = (P 2 Q 2 ) nd H 3 = (P 3 Q 3 ) of G indued y P 3 displyed in Figure 3.7 re omplete. Also H 1 H 2 nd H 3 indued y P 1 P 2 nd P 3 overs l Thus the olletion 4-rung orthopir fuzzy edge lique over. Moreover it is the smllest 4-ROF edge lique over s t 4-ROF liques other thn P 1 P 2 nd P 3 must e less thn 3. Hene θ e (G ) = 3. ition Let D = (A B) e q-rof yli digrph defined on D = (A B ). The in ng π : A A } is lled vertex leling of D suh tht if defined y µ + D hs q-rof r P v (u) = µ ( vu) nd uv π(v) for ll q-rof verties u v in ν Q P + v : X v + [0 1] defined y ν P + v (u) = ν ( vu). A q-rung orthopir Q 10 D. In other words every q-rof r goes from lower integer to r. ( ) ( ) ( )

12 ( ) ( ) Mthemtis of 33 fuzzy in-neighorhood of vertex v of direted q-rofg D is the q-rofs N (v) = (Xv µ P v νp v ) where Xv = u : µ ( uv) Q 0 or ν ( uv) Q 0} nd µ P v : Xv [0 1] defined y µ P v (u) = µ ( uv) nd Q νp v : Xv [0 1] defined y νp v (u) = ν ( uv). Q Definition 23. Let D = (P Q ) e direted q-rung orthopir fuzzy grph. The q-rung orthopir fuzzy ompetition grph C( D ) of q-rung orthopir fuzzy digrph D is n undireted q-rofg G = (P R) whih hs sme q-rof vertex set s in D nd hs q-rof edge etween two verties u v P in C( D ) if nd only if N + (u) N + (v) = φ in D nd the support for memership nd support for non-memership of edge uv in C( D ) is suh tht µ q R (uv) + νq R (uv) 1 for ll u v X. µ R (uv) = (µ P (u) µ P (v)) h µ (N + (u) N + (v)) ν R (uv) = (ν P (u) ν P (v)) h ν (N + (u) N + (v)). Exmple 6. Let D = (P Q ) e 3-rung orthopir fuzzy digrph given in Figure 10 where ( ) ( ) P = d 0.83 e d 0.73 e nd 0.95 ( ) ( d d e de ) d d e de Q = The out-neighorhoods of q-rof verties re N + () = ( ) ( )} N + () = ( ) (d ) (e )} N + () = (d )} N + (d) = (e )} N + (e) = φ. By using Definition 23 we get the orresponding 3-ROFCG C( D ) = (A B) of D = (P Q ) s displyed in Figure 10. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) ( ) e ( ) ( ) d ( ) ( ) ( ) e ( ) () D () C( D ) Figure 10. Competition grph Figure of direted 4.1 grph. () D ; () C( D ) Theorem 1. Theorem 4.1. Let Let D = (P D = (P Q ) e direted q-rung orthopir fuzzy grph. If N + (u) N Q) e direted q-rung orthopir fuzzy grph. If N + + (v) ontins (u) N + only (v) ontins only one element one element of of D then the q-rof edge of C( D then the q-rof edge of C( D ) is independent strong if nd only if N + (u) N + D ) is independent strong if nd only if N + (v) µ > 0.5 (u) N + (v) µ > 0.5 nd N + nd N + (u) N + (u) N + (v) ν < 0.5. (v) ν < 0.5. Theorem 2. If ll the edges of q-rung orthopir fuzzy grph D = Theorem 4.2. If ll the edges of q-rung orthopir fuzzy grph (P Q ) re D = (P independent strong then µ R (uv) Q) re independent strong then µ R (uv) µ P µ P (u) µ P (v) > (u) µ 0.5 P (v) nd > 0.5 ν nd ν R (uv) R (uv) ν P (u) ν ν P (u) ν P (v) < P (v) 0.5 < for 0.5 ll for ll q-rof q-rof edges edges uv uv in in C( C( D ) = (P R). D ) = (P R). Definition 4.3. Let k e non-negtive rel numer nd D = (P Q) e direted q-rung orthopir fuzzy grph. The q-rung orthopir fuzzy k-ompetition grph C k ( D ) of q-rung orthopir fuzzy digrph D is n undireted q-rofg G = (P R) whih hs sme q-rof vertex set s in D nd hs q-rof edge etween two

13 Theorem 4.2. If ll the edges of q-rung orthopir fuzzy grph D = (P Q) re independent strong then µ R (uv) µ P (u) µ P (v) > 0.5 nd ν R (uv) ν P (u) ν P (v) < 0.5 for ll q-rof edges uv in C( D ) = (P R). Definition 4.3. Mthemtis Let2019 k e 7 91 non-negtive rel numer nd D = (P Q) e direted q-rung 13 of 33orthopir fuzzy grph. The q-rung orthopir fuzzy k-ompetition grph C k ( D ) of q-rung orthopir fuzzy digrph D is n undireted q-rofg Definition G 24. = (P Let k R) e non-negtive whih hs rel sme numer q-rof nd vertex D = (P Q ) e direted q-rung orthopir fuzzy grph. The q-rung orthopir fuzzy k-ompetition grph C k ( set s in D nd hs q-rof D ) of q-rung orthopir fuzzy digrph edge etween two verties u v P in C k ( D ) if nd only if N + (u) N + (v) µ > k nd D is n undireted q-rofg G = (P R) whih hs sme q-rof vertex set s in N + (u) N + (v) ν > k. The support for D nd hs q-rof edge etween two memership nd support for non-memership verties u v P in C k ( of edge uv in C D ) if nd only if N + (u) N + k ( D ) is (v) µ > nd N + (u) N + (v) ν > k. The support for memership nd support for non-memership of edge uv in C k ( D ) is µ R (uv) = k 1 k k 1 (µ P (u) µ P (v)) h µ (N + (u) N + (v)) ν R (uv) µr = (uv) k 2 k = k 1 k k k (µ 1 P (u) µ h µ (N + (u) N + 2 (ν P (u) ν P (v)) h ν (N + (u) (v)) N + (v)) ν R (uv) = k 2 k k (ν 2 P (u) ν P (v)) h ν (N + (u) N + (v)) suh tht µ q R (uv) + νq R (uv) 1 for ll u v X where k 1 = N + (u) N + (v) µ k 2 = N + (u) N + (v) ν. suh tht µ q R (uv) + νq R (uv) 1 for ll u v X where k 1 = N + (u) N + (v) µ k 2 = N + (u) N + (v) ν. Exmple 4.2. Consider 3-rung orthopir fuzzy digrph Exmple 7. Consider 3-rung orthopir fuzzy digrph D = D = (P (P Q) given in exmple 4.1 displyed in Figure 4.2(). C Q ) given in Exmple 6 displyed in 0.4 ( D ) hs Figure 10. C 0.4 ( only two 3-ROF edges sine D ) hs only two 3-ROF edges sine N + () N + () N + () N + µ = 0.6 > 0.4 nd N + () N + () () µ = 0.6 > 0.4 nd N + () N + ν = 0.89 > 0.4 () ν = 0.89 > 0.4 N + () N + () µ = 0.6 > 0.4 nd N + () N + () ν = 0.82 > 0.4. N + () N + () µ = 0.6 > 0.4 nd N + () N + () ν = 0.82 > 0.4. Thus y using Definition Thus y using?? Definition we get the 24 orresponding we get the orresponding 3-rung 3-rung orthopir fuzzy 0.4-ompetition grph grph C 0.4 ( D ) = (A B) of D C= 0.4 ((P D ) = Q) (A s B) shown of D = (P in Figure Q ) s shown 4.2(). in Figure 11. ( ) ( ) ( ) ( ) e ( ) d ( ) ( ) Figure : C C 0.4 ( 0.4 ( D ). D ) Theorem 3. Let D = (P Q ) e direted q-rung orthopir fuzzy grph. If h µ (N + (u) N + (v)) = h ν (N + (u) N + (v)) = 1 N + (u) N + (v) µ > 2k nd N + (u) N + (v) ν < 2k then the q-rof edge is independent strong in C k ( D ). 12 Definition 25. Let p e positive integer nd D = (P Q ) e direted q-rung orthopir fuzzy grph. The p-ompetition q-rung orthopir fuzzy grph C p ( D ) of q-rung orthopir fuzzy digrph D is n undireted q-rofg G = (P R) whih hs sme q-rof vertex set s in D nd hs q-rof edge etween two verties u v P in C p ( D ) if nd only if supp(n + (u) N + (v)) p. The support for memership nd support for non-memership of edge uv in C p ( D ) is µ R (uv) = (n p)+1 n (µ P (u) µ P (v)) h µ (N + (u) N + (v)) ν R (uv) = (n p)+1 n (ν P (u) ν P (v)) h ν (N + (u) N + (v)) suh tht µ q R (uv) + νq R (uv) 1 for ll u v X where n = supp(n + (u) N + (v)). Exmple 8. Let D = (P Q ) e 5-rung orthopir fuzzy digrph given in Figure 12 where ( ) ( ) P = d 0.79 e 0.5 f d 0.89 e 0.99 f nd 0.9

14 R n P P ν suh tht µ q R (uv) + νq R (uv) 1 for ll u v X where n = supp(n + (u) N + (v)). Exmple 4.3. Let D = (P Q) e 5-rung orthopir fuzzy digrph given in Figure 4.3() where Mthemtis ( 91 ) ( ) 14 of 33 P = d 0.79 e 0.5 f d 0.89 e 0.99 f nd 0.9 ( ) ) Q d d e de f d f f e = d d d e de f df fe 0.45 (( ) ) d d d e de f d f f e d d d e de f df fe C 2 ( D ) C 2 hs ( D ) only hs only two two 5-ROF 5-ROF edges edges nd ndd sine supp(n + () N + + ()) = ()) = supp(n supp(n + () () N + + (d)) = (d)) = Thus y using Thus Definition y using Definition 4.4 we get 25 the we get orresponding the orresponding 2-ompetition 2-ompetition 5-rung 5-rung orthopir orthopir fuzzy fuzzy grph grphc 2 ( D ) = (A B) C 2 ( of D D ) = = (A(P B) Q) of D s = shown (P Q in) s Figure shown4.3(). in Figure 12. ( ) ( ) f e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) () D ( ) ( ) ( ) ( ) ( ) ( ) f e ( ) ( ) ( ) ( ) ( ) d ( ) () C p ( D ) Figure 12. p-competition q-rungfigure orthopir 4.3fuzzy grph. () D ; () C p ( D ) Theorem Theorem Let Let D = (P Q) Q ) e direted q-rung q-rung orthopir orthopir fuzzy fuzzy grph. grph. If h µ (N If h + µ (u) (N + N (u) + (v)) N = + 1(v)) nd= 1 nd h ν (N + h (u) N + (v)) = 0 in C n 2 ( ν (N + (u) N + (v)) = 0 in C n 2 ( D ) ) then the q-rof edge edge is independent is independent strong strong where where n = supp(n n = supp(n + (u) + (u) N + (v)). N + (v)). Definition Definition A A q-rung orthopir fuzzy m-step out-neighorhood out-neighourhood of of vertex vertex v of direted q-rofg D is the q-rof m-step set N m + (v) (X+ µ+ + v ) X v + = u : v of direted q-rofg D is the q-rof m-step set N m(v) + = P m vu exists} nd µ+ P v : X v + [0 1] v + µ + P v ν P + v ) where X v + = u : P m vu exists} nd µ + P v : X v + [0 1] defined y µ + P v (u) = minµ ( xy) : xy is n edge in P m Q vu} nd ν P + v : X v + [0 1] defined y ν P + v (u) = 13 mxν ( xy) : xy is n edge in P m Q vu}. A q-rung orthopir fuzzy m-step in-neighorhood of vertex v of direted q-rofg D is the q-rof m-step set Nm (v) = (Xv µ P v νp v ) where Xv = u : P m uv exists} nd µ P v : Xv [0 1] defined y µ P v (u) = minµ ( ) : is n edge in P m Q uv} nd νp v : Xv [0 1] defined y νp v (u) = mxν ( ) : is n edge in P m Q uv}. Definition 27. Let D = (P Q ) e direted q-rung orthopir fuzzy grph. The m-step q-rung orthopir fuzzy ompetition grph C m ( D ) of q-rung orthopir fuzzy digrph D is n undireted q-rofg G = (P R) whih hs sme q-rof vertex set s in D nd hs q-rof edge etween two verties u v P in C m ( D ) if nd only if N + m(u) N + m(v) = φ in D nd the support for memership nd support for non-memership of edge uv in C m ( D ) is suh tht µ q R (uv) + νq R (uv) 1 for ll u v X. µ R (uv) = (µ P (u) µ P (v)) h µ (N + m(u) N + m(v)) ν R (uv) = (ν P (u) ν P (v)) h ν (N + m(u) N + m(v)) Exmple 9. Let D = (P Q ) e 5-rung orthopir fuzzy digrph given in Figure 13 where

15 µ R (uv) = (µ P (u) µ P (v)) h µ (N m (u) N m (v)) ν R (uv) = (ν P (u) ν P (v)) h ν (N + m (u) N + m (v)) ( ) suh tht µ q R (uv) + νq R (uv) 1 for ll u v X. Exmple Mthemtis 4.4. Let 2019 D 7 91 = (P Q) e 5-rung orthopir fuzzy digrph given in Figure 4.4() where 15 of 33 ( ) ( ) ( P = ) ( ) P = d d e e f 0.5 f d 0.87 d 0.89 e 0.89 e f f nd 0.9 nd ( 0.9 ( ) d d d e df ) Q = d d d e d f f e Q = fe 0.45 ( 0.45 ( ) ) d d d e d f f e d d d e df fe C C 2 ( 2 ( D ) hs D only ) hs two only 5-ROF two 5-ROF edges edges nd nd d d sine sine N N () () N () = (f )} φ () ( f )} = φ N 2 + N 2 + () () N (d) = (e )} φ. (d) = (e )} = φ. Thus y using Definition 4.6 we get the orresponding 2-step 5-rung orthopir fuzzy ompetition grph Thus y using Definition 27 we get the orresponding 2-step 5-rung orthopir fuzzy ompetition grph C 2 ( D ) = (A C 2 ( B) of D = D ) = (A B) of (P Q) D = (P s shown in Figure 4.4(). Q ) s shown in Figure 13. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e ( ) ( ) ( ) ( ) e f ( ) ( ) ( ) ( ) ( ) d ( ) ( ) f ( ) ( ) d ( ) () D () C 2 ( D ) Figure 13. m-step q-rung orthopirfigure fuzzy ompetition 4.4 grph. () D ; () C 2 ( D ) Theorem Theorem If If ll ll q-rof edges edges of of q-rung q-rung orthopir orthopir fuzzy digrph fuzzy digrph D = (P D Q = ) re (P independent Q) re independent strong then strong ll the q-rof edges of C then ll the q-rof edges of C m ( m ( D ) re independent strong. D ) re independent strong. Theorem 6. Let D = (P Q ) e q-rung orthopir fuzzy digrph defined on D = (P Q ). If m > P then C m ( D ) hs no edges. 14 Theorem 7. If D e q-rung orthopir fuzzy digrph nd D m is the m-step q-rung orthopir fuzzy digrph of D then Cm ( D ) = C( D m ). Definition 28. Let D = (P Q ) e direted q-rung orthopir fuzzy grph. Let z e ommon q- ROF prey of m-step out-neighorhoods of q- ROF verties x 1 x 2... x k. Also let µ ( u Q 1 v 1 ) µ ( u Q 2 v 2 )... µ ( u Q k v k ) e minimum support for memership of q- ROF edges of the pths P m x 1 z P m x 2 z... P m x k z respetively nd ν ( u Q 1 v 1 ) ν ( u Q 2 v 2 )... ν ( u Q k v k ) e mximum support for non-memership of q- ROF edges of the pths m P x 1 z P m x 2 z... P m x k z respetively. The m-step q- ROF prey z X is lled independent strong q- ROF prey if for ll 1 i k µ Q ( u i v i ) > 0.5 ν Q ( u i v i ) < 0.5. The strength of q- ROF prey z mesured y the mppings s 1 : X [0 1] nd s 2 : X [0 1] is defined y (s 1 (z) s 2 (z)) suh tht s 1 (z) = k i=1 µ Q ( u i v i ) k nd s 2 (z) = k i=1 ν Q ( u i v i ). k

16 s 1 (z) = k i=1 µ Q ( u i v i ) k nd s 2 (z) = k i=1 ν Q ( u i v i ). k 4.5. Let D Mthemtis = (P 2019 Q) 7 91 e 5-rung orthopir fuzzy digrph given in Figure 16 of We see t ROF prey of 2-step out-neighourhoods of 5-ROF verties nd d. is sid to e stron y sine there exist only two direted pths for to e ommon 5-ROF prey. Exmple 10. Let D = (P Q ) e 5-rung orthopir fuzzy digrph given in Figure 14. We see tht is ommon 5-ROF prey of 2-step out-neighorhoods of 5-ROF verties nd d. is sid to e strong 2-step 5-ROF prey sine there exist only two direted pths for to e ommon 5-ROF prey. 2 P : µ Q ( d) µ Q ( d) = 0.65 > 0.5 nd ν Q ( d) ν Q ( d) = 0.45 < 0.5 P 2 : µ Q ( d) µ Q ( d) = 0.65 > 0.5 nd ν Q ( d) ν Q ( d) = 0.45 < nd P d : µ Q ( d) µ Q ( ) = 0.60 > 0.5 nd ν Q ( d) ν Q ( ) = 0.33 < P d : µ ( d) Q µ ( ) = 0.60 Q > 0.5 nd ν ( d) Q ν ( ) = 0.33 Q < 0.5. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) Figure 14. A strong 2-step 5-rung orthopir fuzzy prey. Figure 4.5 Theorem 8. If q-rung orthopir fuzzy prey z of D is independent strong then the strength of z s 1 (z) > 0.5 nd s 2 (z) < 0.5 ut onverse my not hold If q-rung orthopir fuzzy prey z of D is independent strong then the strength of z s 1 ( For the proofs of the ove theorems reders re referred to [28 32]. 0.5 ut onverse my not hold. Theorem 9. A q-rung orthopir fuzzy grph G is sid to e q-rung orthopir fuzzy ompetition grph of some q-rung orthopir fuzzy digrph if nd only if θ e (G ) n where n = V(G ). proves of ove theorems reders re referred to [ ] A q-rung Proof. orthopir Let D = (A fuzzy B ) e grph q-rung orthopir G is sid fuzzyto digrph e ndq-rung G = C( D orthopir ) = (A P) efuzzy q-rungompetition g orthopir orthopir fuzzy digrph fuzzy ompetition if ndgrph only of Dif suh θ e (G tht ) V(G n ) = where n. Then y n Definition = V (G 23). µ P (uv) = (µ A (u) µ A (v)) h µ (N D = (A B) e q-rung orthopir fuzzy digrph + (u) N nd G + (v)) = C( D ) = (A P) e q-rung o ν P (uv) = (ν A (u) ν A (v)) h ν (N + (u) N + (v)) etition grph of D suh tht V (G ) = n. Then y Definition 4.2 Consider the q-rung orthopir fuzzy liques C µp (uv) = (µ A (u) µ A (v)) i = (H i Q h µ (N i ) of + G suh tht (u) N + (v)) ν P (uv) = (ν A (u) V(C ν A (v)) h ν (N + (u) N + i ) = u j u j v j E( D )} (v)) where N + (u) nd N + (v) re q-rung orthopir fuzzy out-neighorhoods of verties u nd v respetively. with µ H (u j ) µ A (u 15 j ) ν H (u j ) ν A (u j ) f or ll u j V(C i ). Let S i = (K i R i ) re the q-rung orthopir fuzzy sugrphs of G = (A P) indued y C i respetively. Then y Definition of q-rung orthopir fuzzy lique S i re omplete q-rung orthopir fuzzy sugrphs of G nd every edge uv of G must e in some S i i.e. there exist olletion of liques whih over ll edges of G suh tht µ R (uv) = µ P (uv) ν R (uv) = ν P (uv) f or ll u v X

17 Mthemtis of 33 lled q-rung orthopir fuzzy edge lique over. Then lerly the size of the smllest suh q-rung orthopir FECC denoted y θ e (G ) nnot exeed the numer of verties of G i.e. n. Hene θ e (G ) n. Conversely let θ e (G ) = k n nd let C 1 C 2... C k } e the olletion of q-rung orthopir fuzzy liques whih overs ll edges of G y omplete q-rung orthopir fuzzy sugrphs. Construt q-rung orthopir fuzzy digrph D = (A B ) with V( D ) = v 1 v 2... v n } suh tht for ll v i C i (1 i k) the support for memership nd non-memership of v i re sme nd f or ll v i v j C i (1 i k) µ A (v ij ) = µ A (v i ) µ A (v j ) ν A (v ij ) = ν A (v i ) ν A (v j ) where v ij re the isolted verties tht is there exist rs v i v ij nd v j v ij in D. Also v i v j E( D ) if nd only if v i C j with µ B ( v i v j ) = ν B ( v i v j ) = µ P (v i v j ) µ A (v i ) µ A (v j ) ν P (v i v j ) ν A (v i ) ν A (v j ). Then G = (A P) is lled q-rung orthopir fuzzy ompetition grph of D i.e. G = C( D ). This ompletes the proof. The notion of tringulted grphs lso rises when we del with ompetition grphs. As the terminologil kground of ompetition grph sed on predtor-prey reltionships onsider the eosystem in whih three predtors nd hve ommon prey x whih leds to the formtion of tringle in the grph. We now explore this sitution for the q-rung orthopir fuzzy grph. Definition 29. A q-rung orthopir fuzzy grph G = (A B) is sid to e tringulted if for every q-rung orthopir fuzzy yle C of length l > 3 there is q-rung orthopir fuzzy edge of G joining two non-onseutive verties of C. In other words G does not hve yle of length l > 3 s indued q-rung orthopir fuzzy sugrph. Theorem 10. Let G is q-rung orthopir fuzzy grph. If G hs q-rof lique numer 3 nd θ e (G ) V(G ) then G is the tringulted q-rof ompetition grph. Proof. Let G is q-rung orthopir fuzzy grph with lique numer 3. Let P e lrgest q-rung orthopir fuzzy lique of G of size 3 nd θ e (G ) V(G ). (1) Then P indues omplete q-rung orthopir fuzzy sugrph C of G. Clerly C is q-rof yle of length 3 eing omplete q-rung orthopir fuzzy generted sugrph. There is no q-rof yles of length l > 3 in G s it is the lrgest possile set of q-rof mutully djent verties. Sine G ontins no indued q-rof yles of length greter thn 3 G is tringulted. Also sine the size of smllest edge lique over stisfies the reltion 1 y Theorem 9 G is q-rof ompetition grph. Hene G is tringulted q-rof ompetition grph. Plese note tht if G is tringulted grph then the vertex set of ny indued yle of length 3 must e lique ut this my not true for tringulted q-rung orthopir fuzzy grphs. Theorem 11. Let G is q-rung orthopir fuzzy grph. If G is tringulted then the vertex set of ny indued q-rof yle of length 3 my not e q-rung orthopir fuzzy lique of G. Proof. Let G = (A B) e tringulted q-rofg. Then G hs q-rung orthopir fuzzy yle of length 3 s generted sugrph. Let C = (P Q) e suh q-rof yle of length 3 suh tht

18 ν P (u) ν A (u) for ll u P µq (uv) µ B (uv) ν Q (uv) ν B (uv) for ll u v P. Mthemtis of 33 µ P (u) µ A (u) ν P (u) ν A (u) f or ll u P (2) = (u µ P (u) ν P (u)) : u P } is q-rof vertex set of C. Consider omple ) indued y P. Then nd µ Q (uv) µ B (uv) ν Q (uv) ν B (uv) f or ll u v P (3) µr (uv) = µ P (u) µ P (v). ν R (uv) = ν P (u) ν P (v) for ll u v P. The set P = (u µ P (u) ν P (u)) : u P } is q-rof vertex set of C. Consider omplete q-rofg H = (P R) indued y P. Then nequlities 4.3 nd 4.4 we get µ R (uv) = µ P (u) µ P (v) ν R (uv) = ν P (u) ν P (v) f or ll u v P (4). Comining inequlities (3) nd (4) we get µ Q (uv) = µ P (u) µ P (v) µ Q (uv) = µ P (u) µ P (v) µ A (u) µ A (v). µ A (u) µ A (v). µ A (u) µ A (v). Whih shows tht µ Q (uv) my not lwys less thn or equl to µ B e q-rofg of H = (P R) my not e sugrph of tringulted q-rofg G = (A q-rung orthopir fuzzy lique of G. Also µ B (uv) µ A (u) µ A (v). Whih shows tht µ Q (uv) my not lwys less thn or equl to µ B (uv). Thus the omplete q-rofg of H = (P R) my not e sugrph of tringulted q-rofg G = (A B). Hene H my not q-rung orthopir fuzzy lique of G..6. Consider Exmple tringulted 11. Consider tringulted 3-rung 3-rung orthopir fuzzy fuzzy grph G grph = (A B) G s shown = (A in Figure B) 15. s shown in Fig ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) Figure 15. A tringulted q-rung orthopir fuzzy grph G. Figure 4.6: A orthopir fuzzy grph G Let C e the q-rof indued yle of G of length 3 s shown in Figure 16. Nme the q-rof vertex set of C s P given y the q-rof indued yle ofp G= of ( 0.75 length 0.95) ( s 0.7) shown ( )}. in Figure 4.7(). Nme the q-ro given y Figure 16 represents omplete q-rofg H = (P R) indued y P. Clerly H is not sugrph of G sine P = ( ) ( ) ( )}. µ Q () = = µ B (). Hene P is not q-rung orthopir fuzzy lique of G. One n see tht the ove result holds only when C is omplete.

19 Mthemtis of 33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () C () H 4.1. Competition Numer of q-rofgs Figure 16. () C ; () H. Figure 4.7 A si eologil priniple is tht two speies ompete if nd only if their eologil nihes () represents overlp. In omplete risp grph theory q-rofg we n otin H = orresponding (P R) digrphs indued y the ysheme P. proposed Clerly y H is not Roerts in [37] whih led him to introdue the onept of ompetition numer. The ompetition numer hs een extensively studied y mny reserhers for risp grphs. While deling with unertinties of ompetition µ Q in() mny prtil = 0.75senrios 0.5 the = ompetition µ B (). numer lso plys vitl role. However when we del with fuzziness the ext fuzzy digrphs nnot e otined from fuzzy not q-rung ompetition orthopir grphs. fuzzy lique of G. One n see tht the ove result holds on To understnd our dopted pproh to define the ompetition numer in q-rung orthopir fuzzy environment onsider n exmple in whih two persons re ompeting for n ojet where the ility to ompete is given y their memership grdes. Now the prolem is to find the extent of ompetition of oth persons towrds the ojet. petition Numer of q-rofgs Let D = (P Q ) e fuzzy direted grph. The out-neighorhoods of verties re gil priniple is tht two speiesn + ompete (u) = (x µ if ( nd ux))} Q only if their eologil nihes ove we n otin orresponding digrphs N + (v) = (x yµ the ( vx))}. Q sheme proposed y Roerts in [29 ue the onept The height of ompetition of ommon neighorhood numer. N The ompetition numer hs een extensiv + (u) N + (v) = (x µ ( ux) Q µ ( Q vx))} is hers for risp grphs. While deling with unertinities of ompetition in mny prt h(n ion numer lso plys vitl role. + (u) N But + (v)) = µ ( when we ux) Q µ ( del vx). Q with fuzziness the ext fuzzy By definition of fuzzy ompetition grph G = (A B) ed from fuzzy ompetition grphs. µ B (uv) = (µ A (u) µ A (v)) h(n + (u) N + (v)) d our dopted pproh to define the ompetition numer in q-rung orthopir f or er n exmple in whih two persons h(n + (u) N re + µ (v)) = ompeting B (uv) for n ojet where the ilit µ A (u) µ A (v) heir memership grdes. Now the prolem is to find the extent of ompetition of µ ( jet. ux) Q µ ( µ vx) = R (uv) Q µ A (u) µ A (v). (5) Q) e fuzzy We know direted tht grph. The out-neighourhoods of verties re µ Q ( ux) µ P (u) µ P (x) µ Q ( vx) µ P (v) µ P (x). (6) N + (u) = (x µ Q ( ux))} N + (v) = (x µ Q ( vx))}. f ommon neighourhood N + (u) N + (v) = (x µ Q ( ux) µ Q ( vx))} is

20 Mthemtis of 33 Comining (5) nd (6) we get µ P ( ux) µ P ( vx) µ Q ( ux) µ P (u) µ P (x) µ P ( ux) µ P ( vx) µ Q ( vx) µ P (v) µ P (x). Thus we n otin the upper ounds nd lower ounds of memership grdes for edges in orresponding digrphs for the se when two speies re ompeting for only one prey. However when two speies re ompeting for more thn one prey we nnot even find their ounds. Sine the memership grde relted to eh direted edge of D nnot e found extly we introdue the term power of ompetition onneted with eh r to define the ompetition numer of q-rofgs. Also we illustrte n lgorithm in this ontext. Theorem 12. Let G e q-rung orthopir fuzzy grph then dding suffiient numer of r isolted q-rof verties δ ui v i 1 i r to G suh tht µ P (δ ui v i ) = µ A (u i ) µ A (v i ) ν P (δ ui v i ) = ν A (u i ) ν A (v i ) produes q-rung orthopir fuzzy ompetition grph G I r of some digrph D. Proof. Let G = (A B) e q-rung orthopir fuzzy grph where A = (µ A ν A ) is q-rung orthopir fuzzy set on X nd B = (µ B ν B ) is q-rung orthopir fuzzy reltion on X. Construt digrph D = (P Q ) s follows: Let u v X e ny two q-rof verties of G suh tht µp (uv) > 0 or ν P (uv) > 0. Add q-rof vertex δ uv suh tht µ P (δ uv ) = µ A (u) µ A (v) ν P (δ uv ) = ν A (u) ν A (v). Remove the edge uv nd drw direted edges(rs) from u nd v to δ uv suh tht the µ-power of ompetition nd ν-power of ompetition of q-rof verties u nd v towrds the vertex δ uv (i.e. power of ompetition ssoited with rs uδ uv nd vδ uv ) re (P uδuv ) µ = (P vδuv ) µ = µ B(uv) µ P (δ uv ) (P uδuv ) ν = (P vδuv ) ν = ν B(uv) ν P (δ uv ). (7) Continuing the proess we get n yli digrph D whose direted edges n e reognized y (7). This digrph D gives q-rung orthopir fuzzy ompetition grph C( D ) = G I r where I r is q-rung orthopir fuzzy set of r isolted q-rof verties dded to G. This ompletes the proof. The method of onstruting the orresponding digrph D of q-rofg is illustrted in Algorithm 1. The omplexity of Algorithm 1 is O(n 2 ).

21 Mthemtis of 33 Algorithm 1. q-rung orthopir fuzzy digrph INPUT: A q-rofg G = (A B). OUTPUT: A q-rof direted grph D = (P Q ). proedure Digrph D = (P Q ) for i := 1 to n do for j := i + 1 to n do if µ B (u i u j ) > 0 or ν B (u i u j ) > 0 then dd δ ui v j suh tht µ P (δ ui v j ) = µ A (u i ) µ A (u j ) ν P (δ ui v j ) = ν A (u i ) ν A (u j ) end if end for end for for i := 1 to n do for j := i + 1 to n do while µ B (u i u j ) > 0 or ν B (u i u j ) > 0 do remove u i u j drw u i δ ui v j nd u j δ ui v j suh tht (P ) u i δ µ = (P ) ui u j u j δ µ = µ B(u i u j ) ui u µ j P (δ ui u j ) (P ) u i δ ν = (P ) ui u j u j δ ν = ν B(u i u j ) ui u ν j P (δ ui u j ) end while end for end for end proedure Remrk 1. In Algorithm 1 the only informtion out direted edges of D re their power of ompetition towrds ommon prey. Thus q-rung orthopir fuzzy out-neighorhoods of verties n e defined in similr mnner y tking into ount the power of ompetition of edges insted of their memership grdes. The Theorem 12 nturlly guide us to define ompetition numer k(g) of q-rung orthopir fuzzy grph G. Definition 30. Let G e ny q-rung orthopir fuzzy grph. The smllest possile numer of isolted q-rof verties whih when dd in G leds q-rung orthopir fuzzy ompetition grph of ertin yli digrph(s onstruted in Algorithm 1) is lled ompetition numer of G. Roerts proved in [37] tht if G = (V E) is onneted grph without tringles then k(g) E V +2. We now generlize this result for q-rung orthopir fuzzy grph. Theorem 13. If G = (A B) is onneted q-rung orthopir fuzzy grph defined on G = (A B) without ny q-rof tringle then k(g ) B A +2. Proof. Let G = (A B) e q-rung orthopir fuzzy grph defined on G = (A B). Suppose tht G I r is q-rung orthopir fuzzy ompetition grph onstruted ording to the Algorithm 1. Lel the q-rof verties of G suh tht every q-rof r goes from lower integer to higher integer in D. For eh q-rof edge uv of G there is q-rof vertex δ uv suh tht δ uv is ommon prey of u nd v in G. Moreover sine G hs no q-rof tringles the δ uv re distint. It follows tht G I r hs t lest B q-rof verties δ uv. Furthermore sine these B q-rof verties ll hve t lest two inoming q-rof rs in G therefore t lest two of the q-rof verties of G I r re not δ uv. These q-rof verties re leled 1 nd 2 where the q-rof vertex leled 1 hs no inoming q-rof r nd q-rof vertex leled 2 hs only one inoming q-rof r. Hene k(g ) + A 2 B

22 Mthemtis of 33 implies tht k(g ) B A +2. This ompletes the proof. The ove result only gives the simple lower ounds of ompetition numer of q-rung orthopir fuzzy grphs for the se when q-rofgs re tringle-free. In risp grph theory Opsut [40] improved this result y defining oth upper nd lower ounds for ny grph in onnetion with the size of smllest edge lique over. It n e generlized for q-rung orthopir fuzzy grphs s follows: Lemm 1. If G = (A B) is onneted q-rung orthopir fuzzy grph defined on G = (A B) with no q-rof tringle then the size of smllest q-rung orthopir fuzzy edge lique over of G is extly equl to the numer of edges in G. Proof. Let G = (A B) e onneted q-rung orthopir fuzzy grph defined on G = (A B). Let P i e the q-rof liques of G. Sine G hs no q-rof tringle therefore eh q-rof lique P i must hve t most two q-rof verties u i nd v i in order to get omplete q-rof indued sugrph of G. Thus the smllest q-rung orthopir fuzzy edge lique over hs q-rof liques equl to the numer of edges of G. Hene the size of smllest q-rof edge lique over is B i.e. θ e (G ) = B. Theorem 14. For ny q-rung orthopir fuzzy grph G θ e (G ) + A 2 k(g ) θ e (G ). Proof. Let G = (A B) e q-rung orthopir fuzzy grph defined on G = (A B). Suppose tht G I r is q-rung orthopir fuzzy ompetition grph onstruted ording to the Algorithm 1. Lel the integers A +k(g ) to q-rof verties of G so tht every q-rof r goes from lower integer to higher integer in D. In prtiulr the q-rof vertex leled 1 hs no inoming q-rof r nd q-rof vertex leled 2 hs only one inoming q-rof r. Consider the set P = A +k(g )} nd for eh i P the set P i = (u µ A (u) ν B (u)) : ui B}. Then sine G I r is the ompetition grph for digrph eh P i is q-rof lique of G. Moreover the sugrphs indued y eh P i must over ll q-rof edges of G. If θ e (G ) denotes the size of smllest edge lique over of G. Then θ e (G ) P = A +k(g ) 2 or k(g ) θ e (G ) A +2. (8) Whih ompletes the proof of lower ounds of the ompetition numer of G. To prove its upper ounds onsider the q-rof liques P i (1 i θ e (G )) elongs to smllest edge lique over of G. Construt digrph D on the verties q-rofg of G I r ording to Algorithm 1 where the dded isolted verties re leled i 1 i r. Then D is yli nd G I r is q-rof ompetition grph for D. So y definition of ompetition numer of q-rof grph k(g ) θ e (G ). Hene θ e (G ) + A 2 k(g ) θ e (G ). This ompletes the proof m-step Competition Numer of q-rofgs Cho et l. in [8] use the notion of m-step ompetition numer nlogous to the ompetition numer of Roerts [37]. They defined the m-step ompetition numer of G s the smllest numer r suh tht G together with r isolted verties is m-step ompetition grph of n yli digrph. Anlogous to this eminent onept we now define it s orthopir memership grdes.

23 Mthemtis of 33 Theorem 15. Let G e q-rung orthopir fuzzy grph nd m 1 is n integer then dding suffiient numer of r isolted q-rof verties δ i uv (1 i m) for eh edge of G suh tht µ P (δuv) 1 = µ P (u) µ P (v) µ P (δuv) i = µ P (u) µ P (v) µ P (δuv) 1... µ P (δ i 1 ν P (δuv) 1 = ν P (u) ν P (v) ν P (δuv) i = ν P (u) ν P (v) µ P (δuv) 1... ν P (δ i 1 uv ) uv ) produes n m-step q-rung orthopir fuzzy ompetition grph G I r of some yli digrph D for ll 2 i m. Proof. Let G = (A B) e q-rung orthopir fuzzy grph where A = (µ A ν A ) is q-rung orthopir fuzzy set on X nd B = (µ B ν B ) is q-rung orthopir fuzzy reltion on X. Construt digrph D = (P Q ) s follows: Let u v X e ny two verties of G suh tht µp (uv) > 0 or ν P (uv) > 0. For eh edge uv G dd verties δ i uv (1 i m) remove the edge uv nd drw direted pths from u nd v to δ m uv of length m. The digrph D = (P Q ) whose q-rof verties onsists of verties of G plus m q-rof isolted verties δ i uv(1 i m) for eh edge uv in G n e defined s P = A I r suh tht for 2 i m µ P (δuv) 1 = µ P (u) µ P (v) µ P (δuv) i = µ P (u) µ P (v) µ P (δuv) 1... µ P (δ i 1 ν P (δuv) 1 = ν P (u) ν P (v) ν P (δuv) i = ν P (u) ν P (v) µ P (δuv) 1... ν P (δ i 1 uv ) uv ) nd Q = uv [ uδuv 1 vδuv} 1 m 1 ] δuvδ i uv i+1 } i=1 suh tht power of ompetition of q-rof verties u nd v towrds the vertex δ 1 uv is (P uδ 1 uv ) µ = (P vδ 1 uv ) µ = µ B(uv) µ P (δ 1 uv) (P uδ 1 uv ) ν = (P vδ 1 uv ) ν = ν B(uv) ν P (δ 1 uv) nd power of ompetition of q-rof vertex δ i uv towrds δ i+1 uv (P δuvδ i uv i+1 (P δuvδ i uv i+1 ) µ = µ B(uv) µ P (δ i+1 re uv ) ) ν = ν B(uv) ν P (δuv i+1 ) for ll 2 i m. Continuing the proess we get n yli digrph D whih gives n m-step q-rung orthopir fuzzy ompetition grph C m ( D ) = G I r where I r is q-rung orthopir fuzzy set of r isolted q-rof verties in G. This ompletes the proof. The method of onstruting orresponding m-step digrph D of q-rofg is demonstrted in Algorithm 2. The omplexity of Algorithm 2 is O(n 2 m).

24 Mthemtis of 33 Algorithm 2. q-rung ORTHOPAIR FUZZY DIGRAPH INPUT: A q-rofg G = (A B). OUTPUT: A q-rof direted grph D = (P Q ). proedure Digrph D = (P Q ) for i := 1 to n do for j := i + 1 to n do if µ B (u i u j ) > 0 or ν B (u i u j ) > 0 then dd δ t u i v j (1 t m) suh tht µ P (δ 1 u i v j ) = µ P (u i ) µ P (u j ) ν P (δu 1 i v j ) = ν P (u i ) ν P (u j ) t := 1 while t m do µ P (δu t i u j ) = µ P (u i ) µ P (u j ) µ P (δu 1 i u j )... µ P (δu t 1 i u j ) ν P (δu t i u j ) = ν P (u i ) ν P (u j ) µ P (δu 1 i u j )... ν P (δu t 1 i u j ) t := t + 1 end while end if end for end for for i := 1 to n do for j := i + 1 to n do if µ B (u i u j ) > 0 or ν B (u i u j ) > 0 then remove u i u j drw direted pths Pu m i δu m nd P m i v j u j δu m suh tht i u j (P ) uδ 1 µ = (P ) uv vδ 1 µ = µ B(uv) uv µ P (δuv) 1 (P ) uδ ν = (P ) uv vδ 1 ν = ν B(uv) uv ν P (δuv) t := 1 1 while t m do (P ) δu t 1 i u δ t µ = µ B(u i u j ) µ j u i u P (δu t j i u j ) (P δ t 1 t := t + 1 end while end if end for end for end proedure ) u i u δ t ν = ν B(u i u j ) ν j u i u P (δu t j i u j ) Remrk 2. In Algorithm 2 the only informtion out q-rof rs of direted pths of D re the power of ompetition of q-rof verties u nd v towrds m-step ommon q-rof prey. Thus q-rung orthopir fuzzy m-step out-neighorhoods of verties n e defined in similr mnner y tking into ount the power of ompetition of rs of direted pths insted of their memership grdes. Theorem 15 nturlly guides us to define m-step ompetition numer k m (G) of q-rung orthopir fuzzy grph G. Definition 31. Let G e ny q-rung orthopir fuzzy grph. The smllest possile numer of isolted q-rof verties whih when dd in G leds n m-step q-rung orthopir fuzzy ompetition grph of some yli digrph is lled m-step ompetition numer of G denoted y k m (G ). Cho et l. [8] proved reltion etween m-step ompetition numer nd ompetition numer of risp grphs. We now generlize this reltion for q-rung orthopir fuzzy grphs. Theorem 16. If G = (A B) is ny q-rung orthopir fuzzy grph then k(g ) k m (G ) where m is positive integer.

25 Mthemtis of 33 Proof. Let G = (A B) is ny q-rung orthopir fuzzy grph nd k m (G ) e the m-step ompetition numer of G. Then there exists n yli q-rof digrph D suh tht C m ( D ) = C( D m ). The q-rof digrph D m is lerly yli nd y the definition of ompetition numer of q-rofg G it follows tht This ompletes the proof. k(g ) k m (G ). Theorem 17. If G = (A B) is q-rung orthopir fuzzy grph defined on G = (A B) then k m (G ) m B where m is positive integer. Proof. Let G = (A B) e ny q-rung orthopir fuzzy grph defined on G = (A B). By definition of m-step ompetition numer of q-rofg G we hve to dd m verties for t most eh q-rof edge. Sine there re B edges in G therefore t most m B verties must dd in G to mke it q-rof ompetition grph of yli digrph onstruted ording to Algorithm 2. Hene This ompletes the proof. k m (G ) m B. Theorem 18. If G = (A B) is q-rung orthopir fuzzy grph without isolted q-rof verties then mxm θ e (G ) + A m + 1} k m (G ) m θ e (G ) where m is positive integer. Proof. Let G = (A B) e ny q-rung orthopir fuzzy grph defined on G = (A B) where A = n. Let C = C 1 C 2... C l } e smllest q-rof edge lique over of G. Then θ e (G ) = l. Construt n yli direted grph D s ording to Algorithm 2. Then it n esily e heked tht D is yli nd C m ( D ) = G I lm. Thus k m (G ) m θ e (G ). Now onsider n yli q-rof digrph D so tht G long with k m (G ) isolted q-rof verties defines C m ( D ). We n ssign n yli leling to D s v 1 v 2... v n+k suh tht v n+1 v n+2... v n+k re the k dded isolted q-rof verties. Then v 1 v 2... v m+1 nnot e exerted s m-step q-rof prey. Despite this sine two distint q-rof liques in C should prey on different m-step ommon q-rof prey there should e t lest θ e (G ) distint q-rof verties operted s m-step ommon q-rof prey. Therefore k m (G ) θ e (G ) + A m + 1. To onlude the proof of first inequlity we notie tht v n is djent to t lest one q-rof vertex of G s G hs no isolted q-rof verties. Consequently v n should hve n m-step ommon q-rof prey in D. Sine ny q-rof vertex possessing lel less thn n nnot e n q-rof out-neighor of v n in D it follows tht k m (G ) m. Hene k m (G ) mxm θ e (G ) + A m + 1}. The speil se of the ove theorem (i.e. for m = 1) is proved y Opsut [40] for ny risp grph G. The next orollry is n instnt onsequene of the ove theorem. Corollry 1. For ny omplete q-rung orthopir fuzzy grph K n with n 2 k m (K n ) = m.

26 Mthemtis of Applition Competition grphs re eoming inresingly signifint s they n pply to mny res in whih there ours ompetition etween entities. A generl outlook of this ft s soure of n interesting grph theoretil ide tht n e seen in the soil eosystem. The q-rung orthopir fuzzy sets provide system modelers with more freedom nd is less restritive in permissile memership grdes. To fully understnd the onept of q-rung orthopir fuzzy ompetition grphs we now disply n importnt pplition of the ompetition grph under the Pythgoren fuzzy environment (tking q = 2) to oserve the strength of ompetition etween plnt-ssoited teri in the rhizosphere or the soil eosystem with n lgorithm Plnt-Bteril Intertions in Soil Eosystem The rhizosphere represents nutrient-rih hitt for miroorgnisms. Soil is hu of ountless living orgnisms tht ount of the proper mintenne of lned nutrients in the soil eosystem s well s for etter yield nd growth of plnts. These orgnisms inlude teri fungi soil lge or tinomyete. Among them teri re of gret importne with respet to soil fertility nd plnt helth. They my e enefiil or hzrdous for plnts nd the soil eosystem. There re some teri ommonly known s plnt growth-promoting rhizoteri (PGPR) whih hve growth-stimulting potentil nd enhne ntioxidnt enzymti tivity in plnts promote disese suppression nd redue stress suseptiility [44]. Contrry to this pthogeni miroorgnisms ffeting plnt helth re mjor nd hroni thret to food prodution nd eosystem stility worldwide. Thus there re some teri whih re reported s pthogeni for soil environment plnt growth nd development. These soil-orne teri inhiit plnt growth due to the relese of some toxi ompounds in the rhizosphere. The inhiition in plnt growth ultimtely results in the lower yield of plnts [45]. The Figure 17 explores ompetition mong plnt-ssoited teri in soil eosystem. The meliorting effets of PGPR nd deleterious effets of phytopthogeni teri not only result in the promotion nd retrdtion of growth prmeters of plnts respetively ut lso iohemil prmeters suh s hlorophyll ontent proline ontent rohydrtes lipids protein ontents nd phenoli ompounds. Hene these fts employ the ritiques mde y severl reserhers tht there is ompetition etween PGPR nd soil-orne phytopthogeni teri whih re present in soil eosystem. These oth types of teri ompete to hve their domintion effet on plnts. Side y side there is lso severe ompetition etween ll PGPR nd in etween ll soil-orne phytopthogeni teri with eh other to influene the growth of plnts. Consider n exmple of 12 teri in soil eosystem. The set of PGPR Billus pumulis Billus tropheus Stphyloous lentus Billus ereus Ahromoter piehudii Azospirillum rsilense Pseudomons fluoresens} nd phytopthogeni teri Agroterium tumefiens Xnthomons oryze Xylell fstidios Pseudomons syringe Rlstoni solnerum} re ompeting for following set of growth nd iohemil prmeters of plnt Shoot length Root length Plnt iomss Numer of leves Chlorophyll ontent Protein ontent Proline ontent Auxin ontent}. The estimted vlues for these growth nd iohemil prmeters with respet to seleted teril speies re given in Tle 2. A Pythgoren fuzzy digrph presenting suh soil eosystem shown in Figure 18 displys teril ompetition for prtiulr prmeters resulting in plnt s growth. The memership degree of eh soil-orne phytopthogeni teri represents the extent of inhiition nd non-memership degree represents the extent of non-inhiition in plnt growth. The phytopthogeni teri n e represented y orthopir s: (Inhiition in plnt growth Not inhiition in plnt growth). Moreover the prtiiption of plnt prmeters in the growth of plnt is shown y their memership vlues nd non-prtiiption is represented y its non-memership vlues. For exmple the orthopir orresponding to shoot length is ( ). The support for memership 0.25 shows

27 growth of plnts. These orgnisms inlude teri fungi soil lge or tinomyete. Among them teri hve gret importne with respet to soil fertility nd plnt helth. They my e enefiil or hzrdous for plnts nd soil eosystem. There re some teri ommonly known s plnt growth-promoting rhizoteri(pgpr) whih hve growth stimulting potentil nd enhne ntioxidnt enzymti tivity in plnts promote disese suppression nd redue stress suseptiility. [14]. Contrry to this pthogeni miroorgnisms Mthemtis of 33 ffeting plnt helth re mjor nd hroni thret to food prodution nd eosystem stility worldwide. Thus there re some teri whih re reported s pthogeni for soil environment plnt s growth nd development. These soil-orne teri inhiit plnt growth euse of relesing some toxi ompounds in rhizosphere. tht the role of shoot length is 6.25% in growth of plnt 49% of it does not tke prt in growth nd The inhiition in plnt growth ultimtely results in lower yield of plnts [24]. there is hesittion of out 44.75%. Plnt-ssoited teri in soil eosystem Plnt growth-promoting rhizoteri (PGPR) Competition Soil-orne pthogeni teri Enhne plnt growth Plnts Inhiit plnt growth Figure Figure 5.1: 17. Potentil role of plnt-ssoited teri in soil in soil eosystem. eosystem Tle 2. Impt of Bteril Speies on Plnt Prmeters. The meliorting effets of PGPR nd deletrious effets of phytopthogeni teri not only results in the promotion nd retrdtion Plnt of growth prmeters of plnts Bteril respetively Effetiveness ut lso iohemil of Bterilprmeters Speies like hlorophyll ontent proline Prmeters ontent rohydrtes lipids Speies protein ontents nd on Plnt phenoli Prmeters ompounds. Hene these fts employ the ritiques mde y severl reserhers tht there is ompetition etween PGPR nd B. pumulis 13.3 inhes soil-orne phytopthogeni teri whih re present in A. soil rsilense eosystem. These oth 11.7 types inhes of teri ompete with eh other to hve their domintion Shoot effet length on plnts. P. Side fluoresens y side there is lso14.1 severe inhes ompetition etween ll PGPR nd lso in etween ll soil-orne phytopthogeni R. solnerum teri with eh3.2 other inhes to influene over the growth of plnts. B. ereus 8.9 inhes Root length Plnt growth prmeters A. piehudii 11.6 inhes Consider n exmple of 12 teri in soil eosystem. The set of PGPR Billus pumulis Billus tropheus Stphyloous lentus Billus ereus Ahromoter piehudii B. tropheus Azospirillum rsilense 7.2 g Pseudomons fluoresens} nd phytopthogeni teri Plnt Agroterium iomss tumefiens S. lentus Xnthomons oryze 9.6 g Xylell fstidios Pseudomons syringe Rlstoni solnerum} re ompeting P. syringe for following set of 1.7 growth g nd iohemil S. lentus 12 No. of leves A. rsilense 10 25P. fluoresens 15 A. tumefiens 35 µg/g Proline ontent X. oryze 40 µg/g X. fstidios 51 µ/g Biohemil prmeters Protein ontent Auxin ontent Chlorophyll ontent X. fstidios 43 µg/g P. syringe 59 µg/g B. pumulis 266 µg/g R. solnerum 68 µg/g A. rsilense 6.7 µg/g R. solnerum 2.9 µg/g The rs in the Pythgoren fuzzy digrph indite the influene of teri on plnt growth nd iohemil prmeters. For exmple the teri Billus pumulis hs the influene on two prmeters nmely shoot length nd uxin ontent. It n influene oth eqully s the degree of memership (µ) is 0.7 while hving different degrees of non-memership (ν) nd hesittion (π = 1 µ ν). Thus the influentil pproh of teri towrds plnt prmeters n e displyed in the form of n orthopir: (Effetiveness of teri on plnt prmeters Ineffetiveness of teri on plnt prmeters). The Pythgoren fuzzy ompetition grph n e onstruted to investigte the strength of ompetition etween teri for growth of plnt. The Pythgoren fuzzy out-neighorhoods of teri re displyed in Tle 3.

28 Mthemtis of 33 ( ) ( ) B.pumulis ( ) ( ) ( ) P.fluoresens ( ) B.tropheus ( ) ( ) Root length Shoot length ( ) Plnt iomss ( ) ( ) ( ) A.tumefiens ( ) S.lentus ( ) ( ) ( ) X.oryze ( ) ( ) No. of leves ( ) Proline ontent ( ) ( ) ( ) ( ) B.ereus ( ) A.piehudii ( ) ( ) ( ) ( ) A.rsilense ( ) Protein ontent ( ) ( ) ( ) Chlorophyll ontent ( ) Auxin ontent ( ) ( ) ( ) R.solnerum ( ) P.syringe ( ) X.fstidios ( ) ( ) ( ) Figure 5.2: Figure 18. Pythgoren fuzzy soil soil eosystem eosystem digrph. digrph This is n yli Pythgoren fuzzy digrph of the soil eosystem/rhizosphere in whih the This is n yli orthopirpythgoren ssigned to ehfuzzy vertex nd digrph r indites of soil its support eosystem/rhizosphere for memership (memership in whih degree) the orthopir ned to eh vertex nd support ndginst r indites memership its(non-memership support for memership(memership degree) under Pythgoren fuzzy degree) environment. nd support gi mership(non-memership The memership degree) of eh under PGPR Pythgoren represents thefuzzy extent of environment. meliortion ndthe non-memership degree of e degree represents extent of non-meliortion in the growth of plnt. The PGPR n e represented PR represents extent of meliortion nd non-memership degree represents extent of not meliortion y pir of disjoint sets lled orthopir: growth of plnt. The PGPR n e represented y pir of disjoint sets lled orthopir s: (Ameliortion in plnt growth Not meliortion in plnt growth). (Ameliortion in plnt growth Not meliortion in plnt growth). The orresponding ompetition grph of Pythgoren fuzzy soil eosystem Figure 18 is displyed e memershipindegree Figure 19. of eh soil-orne phytopthogeni teri represents extent of inhiition nd n mership degree represents extent of not inhiition in plnt growth. The phytopthogeni teri n Tle 3. Pythgoren fuzzy out-neighorhoods of teri. resented y orthopir s: Bteri N + (Bteri) (Inhiition in plnt growth Not inhiition in plnt growth). Billus pumulis (Shoot length ) (Auxin ontent )} Billus tropheus (Plnt iomss )} reover the prtiiption Stphyloous lentus of plnt (Plnt prmeters iomss 0.6 in0.4) growth (No. of leves of plnt )} is shown y their memership vlues Billus ereus (Root length )} -prtiiptionahromoter is represented piehudii y its (Root non-memership length )} vlues. For exmple the orthopir orresponding ot length is (0.25 Azospirillum 0.7). rsilense The support (Shoot for length memership ) (No. of 0.25 leves shows ) (Chlorophyll tht the role ontent of0.7 shoot 0.4)} length is 6.25% Pseudomons fluoresens (Shoot length ) (No. of leves )} wth of plnt 49% of it does not tke prt in growth nd there is hesittion of out 44.75%. Agroterium tumefiens (Proline ontent )} Xnthomons oryze (Proline ontent )} e rs in Pythgoren Xylell fstidios fuzzy digrph (Proline indite ontent 0.6 the 0.7) influene (Protein ontent of 0.2 teri 0.5)} on plnt s growth nd iohe Pseudomons syringe (Plnt iomss ) (Protein ontent )} prmeters. For Rlstoni exmple solnerum the teri (Shoot length Billus ) (Auxin pumulis ontent hs ) influene (Chlorophyll on ontent two prmeters )} nmely sh gth nd uxin ontent. It n influene oth of them eqully s degree of memership (µ) is 0.7 while hv erent degree of non-memership (ν) nd hesittion (π = 1 µ ν). Thus the influentil pproh of te rds plnt prmeters n e displyed in the form of orthopir s: (Effetiveness of teri on plnt prmeters Ineffetiveness of teri on plnt prmeters).

29 Agroterium tumefiens (Proline ontent )} Xnthomons oryze (Proline ontent )} Xylell fstidios (Proline ontent ) (Protein ontent )} Pseudomons syringe (Plnt iomss ) (Protein ontent )} Rlstoni solnerum (Shoot length ) (Auxin ontent ) (Chlorophyll ontent Mthemtis of 33 ( ) B.tropheus ( ) ( ) S.lentus ( ) B.pumulis ( ) ( ) Root length No. of leves Shoot length Proline ontent Plnt iomss ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P.fluoresens ( ) ( ) A.tumefiens ( ) ( ) ( ) X.oryze ( ) ( ) B.ereus ( ) ( ) ( ) Protein ontent ( ) Auxin ontent ( ) X.fstidios ( ) ( ) A.piehudii ( ) Chlorophyll ontent ( ) P.syringe ( ) A.rsilense ( ) ( ) R.solnerum ( ) Figure Figure 5.3: 19. Pythgoren Fuzzy Fuzzy Competition Competition Grph. Grph The edge onneting two teri in Pythgoren fuzzy ompetition grph highlights tht oth teri re ompeting for prtiulr iohemil or growth prmeter of plnt. Tle 4 gives the strength of ompetition of eh terium for prmeter p with respet to plnt growth promotion. For instne the teri B. tropheus S. lentus nd P. syringe re ompeting for the growth prmeter plnt iomss with strengths nd 1.75 respetively. We see tht P. syringe hs mximum strength of ompetition mong these teri. Consequently the terium P. syringe hs more influentil power on the growth prmeter plnt iomss s ompred to teri B. tropheus nd S. lentus nd finlly it inhiits the iomss of plnt eing destrutive phytopthogeni teri. In other words the toxiity of phytopthogeni teri P. syringe is more effetive/dominting thn growth-promoting potentil of PGPR B. tropheus nd S. lentus. In Tle 4 some teri hve equl mximum strength of ompetition for plnt prmeters. To ssess whih one hs dominnt influene on plnt prmeters their support for meliortion or inhiition n e tken into ount. For exmple teri A. rsilense nd R. solnerum re ompeting for iohemil prmeter hlorophyll ontent with sme strength The support for meliortion of A. rsilense is 49% whih is more thn its support for not meliortion i.e. 25% while the terium R. solnerum hs 9% support for inhiition nd 64% support for non-inhiition. Thus we onlude tht the terium A. rsilense is more effetive to enhne the hlorophyll ontent in the plnt s ompred to R. solnerum whih leds to inhiition of hlorophyll ontent. The method for onstruting q-rung orthopir fuzzy ompetition grph of digrph depiting plnt-teril intertions in the soil eosystem nd28 to find strength of ompetition etween teri is illustrted in Algorithm 3. The omplexity of Algorithm 3 is O(n 2 + lr k ).

30 Mthemtis of 33 Tle 4. Strength of ompetition of teri for plnt prmeters. Plnt Prmeters Bteri Bteri in Competition R (Bteri Prmeter) S (Bteri Prmeter) Shoot length Root length Plnt iomss Numer of leves Proline ontent Protein ontent Auxin ontent Chlorophyll ontent B. pumulis A. rsilense P. fluoresens R. solnerum ( ) 1.74 A. rsilense B. pumulis P. fluoresens R. solnerum ( ) P. fluoresens B. pumulis A. rsilense R. solnerum ( ) 1.74 R. solnerum B. pumulis A. rsilense P. fluoresens ( ) 1.74 B. ereus A. piehudii ( ) 1.64 A. piehudii B. ereus ( ) 1.64 B. tropheus S. lentus P. syringe ( ) S. lentus B. tropheus P. syringe ( ) P. syringe B. tropheus S. lentus ( ) 1.75 S. lentus A. rsilense P. fluoresens ( ) 1.54 A. rsilense S. lentus P. fluoresens ( ) P. fluoresens S. lentus A. rsilense ( ) A. tumefiens X. oryze X. fstidios ( ) X. oryze A. tumefiens X. fstidios ( ) 1.73 X. fstidios A. tumefiens X. oryze ( ) X. fstidios P. syringe ( ) P. syringe X. fstidios ( ) B. pumulis R. solnerum ( ) 1.74 R. solnerum B. pumulis ( ) 1.74 A. rsilense R. solnerum ( ) 1.74 R. solnerum A. rsilense ( ) 1.74

31 Mthemtis of 33 Algorithm 3. STRENGTH OF COMPETITION OF BACTERIA INPUT: A q-rof digrph D = (P Q ) where P = (u i µ P (u i ) ν P (u i )) : 1 i n(= l + m)} u i s involve m teri nd l plnt prmeters. OUTPUT: Strength of ompetition of teri j 1 j m for plnt prmeters p k 1 k l. proedure Strength of ompetition S( p) for i := 1 to n do for j := i + 1 to n do if µ ( u Q i u j ) > 0 or ν ( u Q i u j > 0 then (u j µ ( u Q i u j ) ν ( u Q i u j )) N 2 + (u i) else (u j µ ( u Q i u j ) ν ( u Q i u j )) N 2 + (u i) end if end for end for for i := 1 to n do for j := i + 1 to n do if N 2 + (u i) N 2 + (u j) = then µ B (u i u j ) := (µ P (u i ) µ P (u j )) h µ (N 2 + (u i) N 2 + (u j)) ν B (u i u j ) := (ν P (u i ) ν P (u j )) h ν (N 2 + (u i) N 2 + (u j)) else (µ B (u i u j ) ν B (u i u j )) = (0 0) end if end for end for Numer of teri jk ompeting for p k := r k for k := 1 to l do for j := 1 to r k do R µ ( jk p k ) := µ B ( jk 1k ) + µ B ( jk 2k ) µ B ( jk j 1k ) + µ B ( jk j+1k ) + µ B ( jk rk k ) R ν ( jk p k ) := ν B ( jk 1k ) + ν B ( 2k ) ν B ( jk j 1k ) + ν B ( jk j+1k ) + ν B ( jk rk k ) R( jk p k ) := (R µ( jk p k ) R ν ( jk p k )) r k 1 S( jk p k ) := R µ ( jk p k ) R ν ( jk p k ) end for end for end proedure 6. Conlusions One of the most importnt reserh diretions is how to express unertin informtion in humn nlysis. IFSs nd PFSs re oth good wy to del with fuzzy informtion s pirs of disjoint sets lled orthopirs. The q-rofss superior to IFSs nd PFSs roden the spe of vgue informtion. Our urrent work hs delt with ompetition grphs under the q-rung orthopir fuzzy environment. The present investigtion hs provided more preision pliility nd onsisteny for speies nd prey in food we. Thus lot of ompetition in the rel world n e designed y q-rofcgs due to its notle hrteristis µ(x) q + ν(x) q 1 (q 1) whih provides sustntil enefit in modeling humn knowledge. The proposed onept of the q-rung orthopir fuzzy lique n well express severl rel-world prolems. This pper hs explored one exmple in this ontext. Moreover the study hs lerly nlyzed the onept of ompetition numer of q-rofgs whih governs the ltering of ny q-rofg into orresponding q-rofcg of some digrph. In onlusion we hve explored the neessity of q-rofcgs with n pplition in soil eosystem nd designed n lgorithm to evlute the strength of ompetition mong teri. In future reserh we will extend this work to (1) intervl-vlued q-rung orthopir fuzzy ompetition grphs; (2) q-rung orthopir fuzzy ompetition hypergrphs; nd (3) q-rung piture fuzzy ompetition grphs.

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