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Annals of Fuzzy Mathematics and Informatics FMI @ Volume14, No. 1, (July 2017), pp. 99–120 ISSN:2093–9310 (print version) (cid:13)c KyungMoon Sa Co. ISSN:2287–6235 (electronic version) http://www.kyungmoon.com http://www.afmi.or.kr Interval-valued neutrosophic competition graphs Muhammad Akram and Maryam Nasir Received 19 April2017; Revised 2 May 2017; Accepted 10 May 2017 Abstract. We first introduce the concept of interval-valued neutrosophic competition graphs. We then discuss certain types, including k- competition interval-valued neutrosophic graphs, p-competition interval- valued neutrosophic graphs and m-step interval-valued neutrosophiccom- petition graphs. Moreover, we present the concept of m-step interval- valued neutrosophic neighbourhood graphs. 2010 AMS Classification: 03E72, 05C72,05C78, 05C99 Keywords: Interval-valued neutrosophic digraphs, Interval-valued neutrosophic competition graphs. Corresponding Author: Muhammad Akram ([email protected]) 1. Introduction I n 1975, Zadeh [35] introduced the notion of interval-valued fuzzy sets as an extensionoffuzzy sets [34] in whichthe values of the membershipdegreesare inter- vals of numbers instead of the numbers. Interval-valued fuzzy sets provide a more adequate description of uncertainty than traditional fuzzy sets. It is therefore im- portant to use interval-valued fuzzy sets in applications, such as fuzzy control. One of the computationally most intensive part of fuzzy control is defuzzification [19]. Atanassov[12]proposedtheextendedformoffuzzysettheorybyaddinganewcom- ponent,called,intuitionisticfuzzysets. Smarandache[26,27]introducedtheconcept of neutrosophic sets by combining the non-standard analysis. In neutrosophic set, the membership value is associated with three components: truth-membership (t), indeterminacy-membership (i) and falsity-membership (f), in which each member- shipvalueisarealstandardornon-standardsubsetofthenon-standardunitinterval ]0−,1+[ and there is no restrictionon their sum. Smarandache [28] and Wang et al. [29] presented the notion of single-valued neutrosophic sets to apply neutrosophic setsinreallifeproblemsmoreconveniently. Insingle-valuedneutrosophicsets,three components are independent and their values are taken from the standard unit interval [0,1]. Wang et al. [30] presented the concept of interval-valued neutrosophic MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 sets,whichismorepreciseandmoreflexiblethanthesingle-valuedneutrosophicset. Aninterval-valuedneutrosophicsetisageneralizationoftheconceptofsingle-valued neutrosophicset,inwhichthree membership(t,i,f)functions areindependent, and their values belong to the unit interval [0,1]. Kauffman [18] gave the definition of a fuzzy graph. Fuzzy graphs were narrated by Rosenfeld [22]. After that, some remarks on fuzzy graphs were represented by Bhattacharya [13]. He showed that all the concepts on crisp graph theory do not have similarities in fuzzy graphs. Wu [32] discussed fuzzy digraphs. The concept of fuzzy k-competition graphs and p-competition fuzzy graphs was first developed by Samanta and Pal in [23], it was further studied in [11, 21, 25]. Samanta et al. [24] introducedthe generalizationoffuzzycompetitiongraphs,calledm-stepfuzzy competition graphs. Samanta et al. [24] also introduced the concepts of fuzzy m-step neighbourhood graphs, fuzzy economic competition graphs, and m-step economic competition graphs. The concepts of bipolar fuzzy competition graphs and intu- itionistic fuzzy competition graphs are discussed in [21, 25]. Hongmei and Lianhua [16], gave definition of interval-valued fuzzy graphs. Akram et al. [1, 2, 3, 4] have introducedseveralconceptsoninterval-valuedfuzzygraphsandinterval-valuedneutrosophic graphs. Akram and Shahzadi [6] introduced the notion of neutrosophic softgraphswithapplications. Akram[7]introducedthenotionofsingle-valuedneu- trosophicplanargraphs. AkramandShahzadi[8]studiedpropertiesofsingle-valued neutrosophic graphs by level graphs. Recently, Akram and Nasir [5] have discussed some concepts of interval-valued neutrosophic graphs. In this paper, we first introduce the concept of interval-valued neutrosophic competition graphs. We then discuss certain types, including k-competition interval-valued neutrosophic graphs, p-competition interval-valued neutrosophic graphs and m-step interval-valued neutrosophic competition graphs. Moreover,we present the concept of m-step interval- valued neutrosophic neighbourhood graphs. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [6, 9, 10, 13, 14, 15, 17, 20, 26, 33, 36]. 2. Interval-Valued Neutrosophic Competition Graphs Definition 2.1 ([35]). The interval-valued fuzzy set A in X is defined by A={(s,[tl (s),tu(s)]):s∈X}, A A where,tl (s)andtu(s)arefuzzysubsets ofX suchthattl (s)≤tu(s)forallx∈X. A A A A An interval-valued fuzzy relation on X is an interval-valued fuzzy set B in X ×X. Definition 2.2 ([30, 31]). The interval-valued neutrosophic set (IVN-set) A in X is defined by A={(s,[tl (s),tu(s)],[il (s),iu(s)],[fl (s),fu(s)]):s∈X}, A A A A A A where, tl (s), tu(s), il (s), iu(s), fl(s), and fu(s) are neutrosophic subsets of X A A A A A A suchthattl (s)≤tu(s), il (s)≤iu(s)andfl(s)≤fu(s)foralls∈X. Aninterval- A A A A A A valuedneutrosophicrelation(IVN-relation)onX is aninterval-valuedneutrosophic set B in X ×X. 100 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 Definition 2.3 ([5]). An interval-valued neutrosophic digraph (IVN-digraph) on a −→ non-emptysetX isapairG=(A, B),(inshort,G),whereA=([tl ,tu],[il ,iu],[fl, A A A A A fu])is anIVN-setonX andB =([tl ,tu],[il ,iu],[fl ,fu])is anIVN-relationonX, A B B B B B B such that: −−−→ −−−→ (i) tl (s,w)≤tl (s)∧tl (w), tu(s,w)≤tu(s)∧tu(w), B −−−→ A A B−−−→ A A (ii) il (s,w)≤il (s)∧il (w), iu(s,w)≤iu(s)∧iu(w), B −−−→ A A B −−−→ A A (iii) fl (s,w)≤fl(s)∧fl(w), fu(s,w)≤fu(s)∧fu(w), for all s,w∈X. B A A B A A −→ Example2.4. WeconstructanIVN-digraphG=(A, B)onX ={a,b,c}asshown in Fig. 1. a([0.2,0.4],[0.3,0(.[5b0],[.10.6,,00..72])],([[00.1.,10,.30].,2[0],.2[,00..62],)0.3],[0.1bb,([00..([660.1,],0.2)],0[0..2,80.3],][,0.2,[0.05]).b3,0.8],[0.2,0.9]) c([0.1,0.2],[0.2,0.4],[0.3,0.7]) - Figure 1. IVN-digraph −→ Definition 2.5. Let G be an IVN-digraph then interval-valued neutrosophic out- neighbourhoods (IVN-out-neighbourhoods) of a vertex s is an IVN-set N+(s)=(X+, [t(l)+, t(u)+], [i(l)+, i(u)+], [f(l)+, t(u)+]), s s s s s s s where −−−→ −−−→ −−−→ −−−→ −−−→ X+ ={w|[tl (s,w)>0, tu(s,w)>0], [il (s,w)>0,iu(s,w)>0], [fl (s,w)>0, s B B −−−→B B B fu(s,w)>0]}, B such that t(l)+ : X+ → [0,1], defined by t(l)+(w) = tl −(s−,−w→), t(u)+ : X+ → [0,1], s s s B s s defined by t(u)+(w) = tu−(s−,−w→), i(l)+ : X+ → [0,1], defined by i(l)+(w) = il −(s−,−w→), s B s s s B i(u)+ : X+ → [0,1], defined by i(u)+(w) = iu−(s−,−w→), f(l)+ : X+ → [0,1], defined by s s s B s s f(l)+(w)=fl −(s−,−w→), f(u)+ :X+ →[0,1], defined by f(u)+(w)=fu−(s−,−w→). s B s s s B −→ Definition 2.6. Let G be an IVN-digraph then interval-valued neutrosophic in- neighbourhoods (IVN-in-neighbourhoods) of a vertex s is an IVN-set N−(s)=(X−, [t(l)−, t(u)−], [i(l)−, i(u)−], [f(l)−, t(u)−]), s s s s s s s where −−−→ −−−→ −−−→ −−−→ −−−→ X− ={w|[tl (w,s)>0, tu(w,s)>0], [il (w,s)>0,iu(w,s)>0], [fl (w,s)>0, s B B −−−→B B B fu(w,s)>0]}, B 101 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 such that t(l)− : X− → [0,1], defined by t(l)−(w) = tl −(w−−,→s), t(u)− : X− → [0,1], s s s B s s defined by t(u)−(w) = tu−(w−−,→s), i(l)− : X− → [0,1], defined by i(l)−(w) = il −(w−−,→s), s B s s s B i(u)− : X− → [0,1], defined by i(u)−(w) = iu−(w−−,→s), f(l)− : X− → [0,1], defined by s s s B s s f(l)−(w)=fl −(w−−,→s), f(u)− :X− →[0,1], defined by f(u)−(w)=fu−(w−−,→s). s B s s s B −→ Example 2.7. Consider an IVN-digraph G=(A, B) on X ={a,b,c} as shown in Fig. 2. ([0c10.2,].[0,20.4,].[0,30.7,]).b([0.1,0.2],[0.1,(0.[3]0,[.01.2,,00.6.])2]a,([[00b..22([,0,0.01.,4.03].,2][],0,[[.003.2,.,200.,.530]],,.[0[50.1].),60,.60].)7]) b )]902.0,[].8,03.0,[].8,06.0,[(.b Figure 2. IVN-digraph We have Table 1 and Table 2 representing interval-valued neutrosophic out and in-neighbourhoods, respectively. Table 1. IVN-out-neighbourhoods s N+(s) a {(b, [0.1,0.2],[0.2,0.3],[0.1,0.6]),(c, [0.1,0.2],[0.1,0.3],[0.2,0.6])} b ∅ c {(b, [0.1,0.2],[0.2,0.3],[0.2,0.5])} Table 2. IVN-in-neighbourhoods s N−(s) a ∅ b {(a, [0.1,0.2],[0.2,0.3],[0.1,0.6]),(c, [0.1,0.2],[0.2,0.3],[0.2,0.5])} c {(a, [0.1,0.2],[0.1,0.3],[0.2,0.6])} Definition 2.8. The heightofIVN-set A=(s,[tl ,tu],[il ,iu],[fl,fu]) inuniverse A A A A A A of discourse X is defined as: for all s∈X, h(A)=([hl(A),hu(A)],[hl(A),hu(A)],[hl(A),hu(A)]), 1 1 2 2 3 3 =([suptl (s),suptu(s)],[supil (s),supiu(s)],[inf fl(s), inf fu(s)]). A A A A A A s∈X s∈X s∈X s∈X s∈X s∈X 102 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 Definition 2.9. An interval-valued neutrosophic competition graph (IVNC-graph) −→ −→ of an interval-valued neutrosophic graph (IVN-graph) G =(A,B) is an undirected −−→ −→ IVN-graph C(G) = (A,W) which has the same vertex set as in G and there is an edge between two vertices s and w if and only if N+(s)∩N+(w) 6= ∅. The truth- membership, indeterminacy-membership and falsity-membership values of the edge (s,w) are defined as: for all s,w∈X, (i) tl (s,w)=(tl (s)∧tl (w))hl(N+(s)∩N+(w), W A A 1 tu (s,w)=(tu(s)∧tu(w))hu(N+(s)∩N+(w), W A A 1 (ii) il (s,w)=(il (s)∧il (w))hl(N+(s)∩N+(w), W A A 2 iu (s,w)=(iu(s)∧iu(w))hu(N+(s)∩N+(w), W A A 2 (iii) fl (s,w)=(fl(s)∧fl(w))hl(N+(s)∩N+(w), W A A 3 fu (s,w)=(fu(s)∧fu(w))hu(N+(s)∩N+(w). W A A 3 −→ Example 2.10. Consider an IVN-digraph G = (A, B) on X = {a,b,c} as shown in Fig. 3. ([0.1,0.2],[0.1,0.3],[0.2,0.6]a)(([0[0.1.b2,,00..24]],,[[(00[0..32.1,,,000...523]]],,,[[0[00.2.6.,2,0,0.30.]7.,5][0)].)1,0.6])b 90.2,0[.]8,0.3,0[.]8,0.6,0[.(b b )] c([0.1,0.2],[0.2,0.4],[0.3,0.7]) Figure 3. IVN-digraph We have Table 3 and Table 4 representing interval-valued neutrosophic out and in-neighbourhoods, respectively. Table 3. IVN-out-neighbourhoods s N+(s) a {(b, [0.1,0.2],[0.2,0.3],[0.1,0.6]),(c, [0.1,0.2],[0.1,0.3],[0.2,0.6])} b ∅ c {(b, [0.1,0.2],[0.2,0.3],[0.2,0.5])} Table 4. IVN-in-neighbourhoods s N−(s) a ∅ b {(a, [0.1,0.2],[0.2,0.3],[0.1,0.6]),(c, [0.1,0.2],[0.2,0.3],[0.2,0.5])} c {(a, [0.1,0.2],[0.1,0.3],[0.2,0.6])} 103 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 Then IVNC-graph of Fig. 3 is shown in Fig. 4. b([0.6,0.8],[0.3,0.8],[0.2,0.9]) b a([0.2,0.4],[0.3,0.b5],[0.(6,[00..70])1,0.04],[0.04,0.12],[0.06c,([00..14,20.]2)],[0.2,b0.4],[0.3,0.7]) Figure 4. IVNC-graph Definition 2.11. Consider an IVN-graph G = (A, B), where A = ([Al, Au], [Al, 1 1 2 Au], [Al, Au)] and B = ([Bl, Bu], [Bl, Bu], [Bl, Bu)]. then, an edge (s, w), s, w 2 3 3 1 1 2 2 3 3 ∈X is called independent strong, if 1 1 [Al(s)∧Al(w)] <Bl(s,w), [Au(s)∧Au(w)]<Bu(s,w), 2 1 1 1 2 1 1 1 1 1 [Al(s)∧Al(w)] <Bl(s,w), [Au(s)∧Au(w)]<Bu(s,w), 2 2 2 2 2 2 2 2 1 1 [Al(s)∧Al(w)] >Bl(s,w), [Au(s)∧Au(w)]>Bu(s,w). 2 3 3 3 2 3 3 3 Otherwise, it is called weak. We state the following theorems without their proofs. −→ Theorem 2.12. Suppose G is an IVN-digraph. If N+(s)∩N+(w) contains only −→ −→ one element of G, then the edge (s, w) of C(G) is independent strong if and only if |[N+(s)∩N+(w)]|tl >0.5, |[N+(s)∩N+(w)]|tu >0.5, |[N+(s)∩N+(w)]|il >0.5, |[N+(s)∩N+(w)]|iu >0.5, |[N+(s)∩N+(w)]|fl <0.5, |[N+(s)∩N+(w)]|fu <0.5. −→ Theorem 2.13. If all the edges of an IVN-digraph G are independent strong, then Bl(s,w) Bu(s,w) 1 >0.5, 1 >0.5, (Al(s)∧Al(w))2 (Au(s)∧Au(w))2 1 1 1 1 Bl(s,w) Bu(s,w) 2 >0.5, 2 >0.5, (Al(s)∧Al(w))2 (Au(s)∧Au(w))2 2 2 2 2 Bl(s,w) Bu(s,w) 3 <0.5, 3 <0.5, (Al(s)∧Al(w))2 (Au(s)∧Au(w))2 3 3 3 3 −→ for all edges (s, w) in C(G). Definition2.14. Theinterval-valuedneutrosophicopen-neighbourhood(IVN-open- neighbourhood) of a vertex s of an IVN-graph G = (A,B) is IVN-set N(s) = (X , s [tl, tu], [il, iu], [fl, fu]), where s s s s s s 104 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 X ={w|[Bl(s,w)>0, Bu(s,w)>0], [Bl(s,w)>0, Bu(s,w)>0], [Bl(s,w)>0, s 1 1 2 2 3 Bu(s,w)>0]}, 3 and tl : X → [0,1] defined by tl(w) = Bl(s, w), tu : X → [0,1] defined by s s s 1 s s tu(w) = Bu(s, w), il : X → [0,1] defined by il(w) = Bl(s, w), iu : X → [0,1] s 1 s s s 2 s s defined by iu(w) = Bu(s, w), fl : X → [0,1] defined by fl(w) = Bl(s, w), fu : s 2 s s s 3 s X → [0,1] defined by fu(w) = Bu(s, w). For every vertex s ∈ X, the interval- s s 3 valuedneutrosophicsingletonset, A˘ =(s,[Al′,Au′],[Al′, Au′],[Al′,Au′)suchthat: s 1 1 2 2 3 3 Al′ :{s}→[0,1], Au′ :{s}→[0,1], Al′ :{s}→[0,1], Au′ :{s}→[0,1], Al′ :{s}→ 1 1 2 2 3 [0,1], Au′ : {s}→[0,1], defined by Al′(s)= Al(s), Au′(s) =Au(s), Al′(s)=Al(s), 3 1 1 1 1 2 2 Au′(s) = Au(s), Al′(s) = Al(s) and Au′(s) = Au(s), respectively. The interval- 2 2 3 3 3 3 valued neutrosophic closed-neighbourhood (IVN-closed-neighbourhood) of a vertex s is N[s]=N(s)∪A . s Definition 2.15. Suppose G = (A, B) is an IVN-graph. Interval-valued neutrosophic open-neighbourhood graph (IVN-open-neighbourhood-graph) of G is an IVN-graphN(G)=(A, B′) which has the same IVN-set of vertices in G and has an interval-valuedneutrosophicedgebetweentwoverticess,w ∈X inN(G)ifandonly ifN(s)∩N(w) isanon-emptyIVN-setinG. Thetruth-membership,indeterminacy- membership, falsity-membership values of the edge (s, w) are given by: Bl′(s,w)=[Al(s)∧Al(w)]hl(N(s)∩N(w)), 1 1 1 1 Bl′(s,w)=[Al(s)∧Al(w)]hl(N(s)∩N(w)), 2 2 2 2 Bl′(s,w)=[Al(s)∧Al(w)]hl(N(s)∩N(w)), 3 3 3 3 Bu′(s,w)=[Au(s)∧Au(w)]hu(N(s)∩N(w)), 1 1 1 1 Bu′(s,w)=[Au(s)∧Au(w)]hu(N(s)∩N(w)), 2 2 2 2 Bu′(s,w)=[Au(s)∧Au(w)]hu(N(s)∩N(w)), respectively. 3 3 3 3 Definition 2.16. Suppose G = (A, B) is an IVN-graph. Interval-valued neutrosophic closed-neighbourhood graph (IVN-closed-neighbourhood-graph) of G is an IVN-graphN(G)=(A, B′) which has the same IVN-set of vertices in G and has an interval-valuedneutrosophicedgebetweentwoverticess, w ∈X inN[G]ifandonly if N[s]∩N[w] is a non-empty IVN-set in G. The truth-membership, indeterminacy- membership, falsity-membership values of the edge (s, w) are given by: Bl′(s,w)=[Al(s)∧Al(w)]hl(N[s]∩N[w]), 1 1 1 1 Bl′(s,w)=[Al(s)∧Al(w)]hl(N[s]∩N[w]), 2 2 2 2 Bl′(s,w)=[Al(s)∧Al(w)]hl(N[s]∩N[w]), 3 3 3 3 Bu′(s,w)=[Au(s)∧Au(w)]hu(N[s]∩N[w]), 1 1 1 1 Bu′(s,w)=[Au(s)∧Au(w)]hu(N[s]∩N[w]), 2 2 2 2 Bu′(s,w)=[Au(s)∧Au(w)]hu(N[s]∩N[w]), respectively. 3 3 3 3 We now discuss the method of construction of interval-valued neutrospohic competition graph of the Cartesianproduct of IVN-digraph in following theorem which can be proof using similar method as used in [21], hence we omit its proof. 105 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 −→ −→ Theorem 2.17. Let C(G )=(A , B ) and C(G )=(A , B ) be two IVNC-graphs 1 1 1 2 2 2 −→ −→ −→ −→ −→ −→ ofIVN-digraphsG =(A , L )andG =(A ,L ),respectively. ThenC(G ✷G )= 1 1 1 2 2 2 1 2 G −→ −→ ∪ G✷, where G −→ −→ is an IVN-graph on the crisp graph C(G1)∗✷C(G2)∗ C−(→G1)∗✷C(G2)∗ −→ (X ×X ,E −→ ✷E −→ ), C(G )∗ and C(G )∗ are the crisp competition graphs −1→ 2 −→C(G1)∗ C(G2)∗ 1 2 of G and G , respectively. G✷ is an IVN-graph on (X ×X ,E✷) such that: 1 2 1 2 (1) E✷ ={(s ,s )(w ,w ):w ∈N−(s )∗,w ∈N+(s )∗} 1 2 1 2 1 1 2 2 E −→ ✷E −→ ={(s ,s )(s ,w ):s ∈X ,s w ∈E −→ } C(G1)∗ C(G2)∗ 1 2 1 2 1 1 2 2 C(G2)∗ ∪{(s ,s )(w ,s ):s ∈X ,s w ∈E −→ }. 1 2 1 2 2 2 1 1 C(G1)∗ (2) tl = tl (s )∧tl (s ), il = il (s )∧il (s ), fl = A1✷A2 A1 1 A2 2 A1✷A2 A1 1 A2 2 A1✷A2 fl (s )∧fl (s ), A1 1 A2 2 tu = tu (s )∧tu (s ), iu = iu (s )∧iu (s ), fu = A1✷A2 A1 1 A2 2 A1✷A2 A1 1 A2 2 A1✷A2 fu (s )∧fu (s ). A1 1 A2 2 (3) tlB((s1,s2)(s1,w2))=[tlA1(s1)∧tlA2(s2)∧tlA2(w2)]×∨a2{tlA1(s1)∧tl−L→2(s2a2)∧ tl−→(w2a2)}, L2 (s ,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (4) ilB((s1,s2)(s1,w2))=[ilA1(s1)∧ilA2(s2)∧ilA2(w2)]×∨a2{ilA1(s1)∧il−L→2(s2a2)∧ il−→(w2a2)}, (Ls2,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (5) fBl ((s1,s2)(s1,w2))=[fAl1(s1)∧fAl2(s2)∧fAl2(w2)]×∨a2{fAl1(s1)∧f−Ll→2(s2a2)∧ f−l→(w2a2)}, (sL2,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (6) tuB((s1,s2)(s1,w2))=[tuA1(s1)∧tuA2(s2)∧tuA2(w2)]×∨a2{tuA1(s1)∧tu−L→2(s2a2)∧ tu−→(w2a2)}, L2 (s ,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (7) iuB((s1,s2)(s1,w2))=[iuA1(s1)∧iuA2(s2)∧iuA2(w2)]×∨a2{iuA1(s1)∧iu−L→2(s2a2)∧ iu−→(w2a2)}, (Ls2,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (8) fBu((s1,s2)(s1,w2))=[fAu1(s1)∧fAu2(s2)∧fAu2(w2)]×∨a2{fAu1(s1)∧f−Lu→2(s2a2)∧ f−u→(w2a2)}, L2 (s ,s )(s ,w )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 2 2 2 (9) tlB((s1,s2)(w1,s2))=[tlA1(s1)∧tlA1(w1)∧tlA2(s2)]×∨a1{tlA2(s2)∧tl−L→1(s1a1)∧ tl−→(w1a1)}, L1 (s ,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 (10) ilB((s1,s2)(w1,s2))=[ilA1(s1)∧ilA1(w1)∧ilA2(s2)]×∨a1{ilA2(s2)∧il−L→1(s1a1)∧ il−→(w1a1)}, L1 (s ,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 106 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 (11) fBl ((s1,s2)(w1,s2))=[fAl1(s1)∧fAl1(w1)∧fAl2(s2)]×∨a1{tlA2(s2)∧f−Ll→1(s1a1)∧ f−l→(w1a1)}, (sL1,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 (12) tuB((s1,s2)(w1,s2))=[tuA1(s1)∧tuA1(w1)∧tuA2(s2)]×∨a1{tuA2(s2)∧tu−L→1(s1a1)∧ tu−→(w1a1)}, L1 (s ,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 (13) iuB((s1,s2)(w1,s2))=[iuA1(s1)∧iuA1(w1)∧iuA2(s2)]×∨a1{iuA2(s2)∧iu−L→1(s1a1)∧ iu−→(w1a1)}, (Ls1,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 (14) fBu((s1,s2)(w1,s2))=[fAu1(s1)∧fAu1(w1)∧fAu2(s2)]×∨a1{tuA2(s2)∧f−Lu→1(s1a1)∧ f−u→(w1a1)}, L1 (s ,s )(w ,s )∈E −→ ✷E −→ , a ∈(N+(s )∩N+(w ))∗. 1 2 1 2 C(G1)∗ C(G2)∗ 1 1 1 (15) tl ((s ,s )(w ,w )) = [tl (s )∧tl (w )∧tl (s )∧tl (w )]×[tl (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 t(l−Ls→1,(ww1s)(1s)∧,wtlA)2(∈w2E)✷∧.tl−L→2(s2w2)], 1 1 2 2 (16) il ((s ,s )(w ,w )) = [il (s )∧il (w )∧il (s )∧il (w )] ×[il (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 i(l−Ls→1,(ww1s)(1s)∧,wilA)2(∈w2E)✷∧.il−L→2(s2w2)], 1 1 2 2 (17) fl ((s ,s )(w ,w ))= [fl (s )∧fl (w )∧fl (s )∧fl (w )]×[fl (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 f(s−Ll→1,(ww1)s(1s)∧,wfAl)2∈(wE2)✷∧. f−Ll→2(s2w2)], 1 1 2 2 (18) tu((s ,s )(w ,w )) = [tu (s )∧tu (w )∧tu (s )∧tu (w )]×[tu (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 t(u−Ls→1,(ww1s)(1s)∧,wtuA)2(∈w2E)✷∧.tu−L→2(s2w2)], 1 1 2 2 (19) iu((s ,s )(w ,w )) = [iu (s )∧iu (w )∧iu (s )∧iu (w )] ×[iu (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 i(u−Ls→1,(ww1s)(1s)∧,wiuA)2(∈w2E)✷∧.iu−L→2(s2w2)], 1 1 2 2 (20) fu((s ,s )(w ,w ))= [fu (s )∧fu (w )∧fu (s )∧fu (w )]×[fu (s )∧ B 1 2 1 2 A1 1 A1 1 A2 2 A2 2 A1 1 f(s−Lu→1,(ww1)s(1s)∧,wfAu)2∈(wE2)✷∧. f−Lu→2(s2w2)], 1 1 2 2 A. k-competition interval-valued neutrosophic graphs We now discuss an extension of IVNC-graphs, called k-competition IVN-graphs. Definition 2.18. The cardinality of an IVN-set A is denoted by |A|= |A|tl, |A|tu , |A|il, |A|iu , |A|fl, |A|fu . Where |A|tl, |A|tu , (cid:0)|(cid:2)A|il, |A|iu (cid:3)an(cid:2)d |A|fl, |A(cid:3)|f(cid:2)u represent(cid:3)(cid:1)the sum of truth- membership values, indeterminacy-membership values and falsity-membership val- (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) ues, respectively, of all the elements of A. 107 MuhammadAkrametal./Ann. FuzzyMath. Inform. 14(2017),No. 1,99–120 Example 2.19. The cardinality of an IVN-set A = {(a, [0.5, 0.7], [0.2, 0.8],[0.1, 0.3]),(b,[0.1,0.2],[0.1,0.5],[0.7,0.9]),(c,[0.3,0.5],[0.3,0.8],[0.6,0.9])}inX ={a, b, c} is |A|= |A|tl,|A|tu , |A|il,|A|iu , |A|fl,|A|fu =([0.9,1.4],[0.6,2.1],[1.4,2.1]). (cid:0)(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)(cid:1) We now discuss k-competition IVN-graphs. Definition 2.20. Letk be anon-negativenumber. Thenk-competitionIVN-graph −→ −→ −→ C (G) of an IVN-digraph G = (A, B) is an undirected IVN-graph G = (A, B) k −→ which has same IVN-set of vertices as in G and has an interval-valued neutro- −→ sophic edge between two vertices s, w ∈ X in C (G) if and only if |(N+(s) ∩ k N+(w))|tl > k, |(N+(s) ∩ N+(w))|tu > k, |(N+(s) ∩ N+(w))|il > k, |(N+(s) ∩ N+(w))|iu > k, |(N+(s) ∩ N+(w))|fl > k and |(N+(s) ∩ N+(w))|fu > k. The −→ interval-valued truth-membership value of edge (s, w) in C (G) is tl (s, w) = k B k1lk−lk[tlA(s)∧tlA(w)]hl1(N+(s)∩N+(w)), where k1l = |(N+(s)∩N+(w))|tl and tuB(s, w)1= k1uk−uk[tuA(s)∧tuA(w)]hu1(N+(s)∩N+(w)), where k1u =|(N+(s)∩N+(w))|tu, the 1 −→ interval-valued indeterminacy-membership value of edge (s, w) in C (G) is il (s, k B w) = k2lk−lk[ilA(s)∧ilA(w)]hl2(N+(s)∩N+(w)), where k2l = |(N+(s)∩N+(w))|il, and iuB(s, w)2= k2uk−uk[iuA(s)∧iuA(w)]hu2(N+(s)∩N+(w)), wherek2u =|(N+(s)∩N+(w))|iu, 2 −→ the interval-valued falsity-membership value of edge (s, w) in C (G) is fl (s, w)= k B k3lk−lk[fAl(s)∧fAl(w)]hl3(N+(s)∩N+(w)), where k3l =|(N+(s)∩N+(w))|fl, andfBu(s, w)3= k3uk−uk[fAu(s)∧fAu(w)]hu3(N+(s)∩N+(w)), where k3u =|(N+(s)∩N+(w))|fu. 3 −→ Example 2.21. Consider an IVN-digraph G = (A, B) on X = {s,w,a,b,c}, such that A = {(s, [0.4,0.5], [0.5,0.7], [0.8,0.9]), (w, [0.6,0.7], [0.4,0.6], [0.2,0.3]), (a, [0.2,0.6],[0.3,0.6],[0.2,0.6]),(b,[0.2,0.6],[0.1,0.6],[0.2,0.6]),(c, [0.2,0.7],[0.3,0.5], −−−→ −−−→ [0.2,0.6])},andB ={((s,a),[0.1,0.4],[0.3,0.6],[0.2,0.6]),((s,b),[0.2,0.4],[0.1,0.5], −−−→ −−−→ [0.2,0.6]),((s,c), [0.2,0.5],[0.3,0.5],[0.2,0.6]),((w,a), [0.2,0.5],[0.2,0.5],[0.2,0.3]), −−−→ −−−→ ((w,b),[0.2,0.6],[0.1,0.6],[0.2,0.3]),((w,c),[0.2,0.7],[0.3,0.5],[0.2,0.3])},asshown in Fig. 5. We calculate N+(s) = {(a, [0.1,0.4], [0.3,0.6], [0.2,0.6]), (b, [0.2,0.4], [0.1,0.5], [0.2,0.6]), (c, [0.2,0.5], [0.3,0.5], [0.2,0.6])} and N+(w) = {(a, [0.2,0.5], [0.2,0.5], [0.2,0.3]),(b,[0.2,0.6],[0.1,0.6],[0.2,0.3]),(c,[0.2,0.7],[0.3,0.5],[0.2,0.3])}. There- fore, N+(s) ∩ N+(w) = {(a, [0.1,0.4], [0.2,0.5], [0.2,0.3]), (b, [0.2,0.4], [0.1,0.5], [0.2,0.3]), (c, [0.2,0.5], [0.3,0.5], [0.2,0.3)}. So, kl = 0.5, ku = 1.3, kl = 0.6, 1 1 2 ku = 1.5, kl = 0.6 and ku = 0.9. Let k = 0.4, then, tl (s, w) = 0.02, tu(s, 2 3 3 B B w) = 0.56, il (s, w) = 0.06, iu(s, w) = 0.82, fl (s, w) = 0.02 and fu(s, w) = 0.11. B B B B This graph is depicted in Fig. 6. 108