A Study on Operations of Bipolar Neutrosophic Cubic Fuzzy Graphs

In 1975 Rosenfeld [7] introduced fuzzy graphs based on fuzzy set.Fuzzy graph theory plays essential roles in various discipines including information theory, neural networks,clustering problems and control theory,etc.Fuzzy models is more compatiable to the systemin compare with classical mode.Bhattacharya [5] gave some remarks on fuzzy graphs.Some operations on fuzzy graphs were introduced by mordeson and peng[6].Akram et al. has introduced several new concepts including bipolar fuzzy graphs,regular bipolar fuzzy graphs, irregular bipolar fuzzy graphs etc.In this paper, we present certain operations on bipolar fuzzy graphs structures and investigate some of their properties.

1.    Basic Definitions

Definition 1.1 Let X be a space of points with generic elements in X denoted by % . A neutrosophic fuzzy set A is characterized by truth-membership function f_Tt(x)> an indeterminacy-membership function A_ai(x) a falsity - membership function y_af(x).

For each point x in X h _Aт(х), Т_А ^x) ,y_{a F}(x) e [0,1] A neutrosophic fuzzy set A can be written as

A = {< x: ^at(x)-A_ai(x), YafW >,x eX}

Definition 1.2 Let X be a space of points with generic elements in X denoted by x . A neutrosophic cubic fuzzy set in X is a pair G = (M,N where M = {< x: h_Mt&\Ami&)>Ymf(x) >,x e X} is an interval neutrosophic fuzzy set in X and N = {< x: h_nt(x),A_ni(x),y_nf(x) >,x e X} is a neutrosophic fuzzy set in X.

Definition 1.3 Let G* = (V,E) be a fuzzy graph. By neutrosophic cubic fuzzy graph of G * , we mean a pair G = (M,N where M = (A ,B ) = (Hf_Tt> F_Tt,>(F_ai,F_bi,,(t_ff,y_ff) ) is the neutrosophic cubic fuzzy set representation of vertex set V and N = ( C, D ) = ((h_tt, d_tf), 0 ci>D i)> (Yc_F,Ydf) ) is the neutrosophic cubic fuzzy set representation of edge set E such that

• (i)    (Цтс^Уд < r min^^x^ ^A_AT^yy^}] , р_От^ух^ < max^BT^i), Рвт^уУ i^)})

• (ii)    ^ic^iyd < гтт&_А1(х1\ X_AI(y_t}}, ^     < max^B^x^^B^yd})

• (iii)    (Yfc^yxiyd< mmin{y_AF^yx)), y_Ap^i)}, d р_F^yxy) < max{y b_F⁽x)), у _bp^yyi^)})

2.    Bipolar Neutrosophic Cubic Fuzzy Graphs(BNCFG)

Definition 2.1 Let X be a space of points with generic elements in X denoted by x. A Bipolar neutrosophic cubic fuzzy set in X is a pair G = (^yM^p, N^p), (Mⁿ , ^pNps is defined as

M^p = {p: /ЛмтУхХАм/УхХУмрУх) >/ x 6 X}

MN = {< xw: Ммт(х),Л^1(х),у^рУх) >/x 6 X} is an interval neutrosophic fuzzy set in X and

Np = {< xp : 11ртСх\ЛшСх'),урр(х) >/x 6 X}

Nn = {< xN ■. ^ntCxUniCxUnpCx) >/x 6 X} is a neutrosophic fuzzy set in X, where рртУх),Вр!1Ух),урРУх) ^ [0,1] and рртУх),Вр1(х),урРУх) ^ [-1,0].

Definition 2.2

Let G* = (V,E) be a fuzzy graph. By a Bipolar neutrosophic cubic fuzzy graph of G * . We mean a pair G = (^yM^p, N^p),^yM^N,N^N)) where

M^p = (A ^,B ) = ^ууртт > ^ртт\ ^ypAi > B^P_Bj\ ^yy_AF, ур)^ ^

M^N = ^{(a ,b} ) = ^уур_Ат, ^р_Вт\ ^yNAi>B_B)\^yy^p_F, y_B)S^

is the neutrosophic cubic fuzzy set representation of vertex set V and

^^P = ⁽C> ) ) = ^уур Tt> ^т)' ^yc^Pi ' Dp)), ^уУср,Урр~) )

Nⁿ = ^yC, D) = ^Суаст, ^Npp), ^yNcj, ^Np)), ^уурр, урр)^

is the neutrosophic cubic fuzzy set representation of edge set E such that

• (i)     ^Tc^iyd < mmm^T&iX Елт^у^^        < тах^т^О,^^)})

• (ii)     V^c&iyd < m max^i^x^ X^^           <  min ^{ N Bi ^y% i ⁾ ,N Bi ^yy i ^)})

^y^^pc^yx 1У) >  rmax^{A ^pi ^y% i\ D^y )}, В.^р1^ухtyd > min^{ABi^yx' i), NB i^yy i^)})

• (iii)     ^yy^pc^yXiyi'⁾< гтах^р&^ур^у.УУ т^рр^уху^ < min^{y^pp^yXi⁾,y^pp^yyi^)})

• 3.    Operations of Two Bipolar Neutrosophic Cubic Fuzzy Graphs

^yy^pc ^(x yi) > r max^{y^pp ^yx)), у^рр ^yyi^)}, у ^рр ^yxy) > min^{y ^рр ^yx)), p^p_F^yy i)})

Definition 3.1 Let G^ =       , N-/) , (м^ , ^1^W)) be a bipolar neutrosophic cubic fuzzy graph of

G∗= ( Vf , Ei),(V? , Ei) and G2 =       , n2p), (m2n, n2n)^ be a bipolar neutrosophic cubic fuzzy graph of G∗= ( vi , e2),(vi , Ei) . The Cartesian product of G^ and G2 is denoted by

Gi × G₂ =       × M₂^P) , (Mi^N × M₂^N)) , ((Nl^p × N₂^P) , (N-i' × ^2^{^})))

=   ( Ai , Bi ),( Ai , Bi ) × ( A^p₂ , Bi ),( Aⁿ₂ , Bi ) , ( Ci , Di ),( ci , Di ) × ( c^p , Di ),( Ci , Di )

^i_A1 × ta₂ , Bta₁ × ta₂) ^, (Btb₁ × TB₂ , Btb₁ × tbS) ^, {^lAi × ia₂ , ^Ai × ia₂) ^, ^В₁ × I $2 , ^Bi × I $2 ((.Yfa, × fa₂ , Yfa₁ × fa₂) ^, (Yfb₁ × fb₂ , Yfb₁ × FB^

^ T Cj × TC₂ , Btc₁ × TcJ ^, (Btd₁ × td₂ , Btd₁ × tdS) ^, {i^ici × IC₂ , ^ici × ke) ^, ^ior × ID2 , ^о_± × idS} ((vic, × FC₂ , YFCi × fc₂) ^, (y^pd₁ × fd₂ , Yfd₁ × fb^

and is defined as follows

(bta₁ × ta₂ ⁽ и , V )= _ГР min (b^Pa₁ ⁽ и ), Bta₂ ⁽ V ⁾), Btb₁ × TB₂ ⁽ и , V )=max (^Втв₁ ⁽ и ), Втв₂ ⁽ v ⁾)) , (^a, × ta₂ ⁽ и , V )= _rN max (bta₁ ⁽ и ), Bta₂ ⁽ v ⁾), EtB₁ × TB₂ ⁽ и , V )=min (вт_В1 ⁽ и ), Втв₂ ⁽ V ⁾ )) (^^1× IA2 ( и , V )= _ГР min ^( и ), ^_A2 ( V )), ^/И, × I в-2 ( и , V )=max ^IB, ( и ), ^■^РВ₂ ( V ) )) , (^^1 × 1^2 ( и , V )= _rN max ^( и ), ^а₂ ( V )), ^в_± × 1В₂ ( и , V )=min(X ( и ), ^В₂ ( V ))) (?^Р_А1× fa₂ ( и , V )= _ГР max (?F_A1 ( и ), Yfa₂ ( V )), Yfb_± × fb₂ ( и , V )=min (yfb, ( и ), Yfb₂ ( V ))) , (?F_A1 × FA₂ ( и , V )= _rN min (yfa, ( и ), Yfa₂ ( V )), Yfb, × fb₂ ( и , V )=max (yfb₁ ( и ), Yfb₂ ( V )))

(iv)      i^BTCr × TC₂( ( и , Vl )( и , V₂ ))=              ( и ), Втс₂ (^1^2) )) , ^i_C1 × тс₂( ( и , V1 )( и , ^2 )=

iuTL 1× ГА 2 Vu , !■ 1i,I 2= min/.iTb 1 Vu , цТА 2 Vi 1 ’, 2

∀ v ∈ ^2 and u^ ∈ Ei

• (v)      ((fc× ic₂ ( ⁽ и , V1 )( и , ^2 ⁾)=             ⁽ u ), ^ic₂ ⁽^1^2⁾ )) , (2(⁽ и , V1 )( и , ^2 ⁾=

rNmaxA/A 1 Vu ,       Л/С 2 Vi 11 2,              Л/А 1× D 2Vu ,11^ ,1 2= min Л/А 1 Vu ,       Л/А 2 Vi 11 2,

Л/А 1× (A 2 Pu , ', 1 ’Л , ^ 2= 'пахЛ/Ь 1 Pu , Л/А 2 Pi 1^ 2, Л/А 1× (A 2 Vu ,11^ ,1 2= типЛ/А 1 Vu , Л/А 2 Vi 1^2

∀ v ∈ ^2 and щи₂ ∈ Ei

(vi)     ((у £q × fc₂( ( и , Vl )( и , ^2 ))=   max (yfa, ( и ), Yfc₂ ( ViV₂ ) )) , (yf_C1 × fc₂( ( и , Vl )( и , ^2 )=

Г. min yFA 1 Vi ,      yFL 2 Vi 1^ 2,

yFL 1× PL 2 Pl , ’, 1 ’л , ^ 2=minyFb 1 Pu ,      yFL 2 Pl 1^ 2, yFL 1× PL 2 Vi ,;1 'л , ^ 2=maxyFL 1 Vi , yFL 2 Vi 1^2

∀ v ∈ ^2 and u^ ∈ Ei

(vii)    [^Ercr ×tc2(( Ui ,V)(u2 ,V))= min ^TC1 ( uyu2), Ёта2 (V))) , (etC1 ×TC2(( Ui ,V)(u2 ,V)= r. max//FL 1 Vi 1 ’л 2,       у Fa 2 Vi ,              //FL 1× FL 2 Pi 1, '/i 2, ^ =max//FL 1 Pl 1 'л 2,       //FBPi ,

//FL 1× FL 2 Vi 1, vi 2, ^ = min //FL 1 Vi 1 ’л 2, //FL 2 Vp

(viii)     fe ×ic2 ( ( Ui ,V)(u2 ,V))=   min (Xq (U1U2), Va2 (V) )) , (^ ×/C2((Ui ,V)(u2 ,V)= r. maxALL 1 Vi 1 ’л 2,       ALA 2 Vi ,

ALL 1× ¥2 Pi 1, vi 2, ’, = max ALL 1 Pi 1 'л 2,       ALL 2 Pi ,

ALL 1× ¥2 Vi 1, vi 2, ! = min ALL 1 Vi 1 •л 2, ALL 2 Vn

(ix) [(fe ×fc2(( Ui ,V)(u2 ,V))=   max ^FCX (U1U2), Yfa2 (V))) , ^FC, ×fc2(( Ui ,V)(u2 ,V)= r. minyFL 1 Vi 1 ’л 2,      yFA 2 Vi ,             yFL 1× PL 2 Pl 1, vi 2, i =minyFL 1 Pl 1 'л 2,      yFL 2 Pl , yFL 1× PL 2 Vi 1, vi 2, ^ =maxyFL 1 Vi 1 'л 2, yFL 2 Vis

∀ ( и , V ) ∈ ( Vi , V₂ )

Example 3.2

Let G^ =      , n/), (M1N, N^)^ be a bipolar neutrosophic cubic fuzzy graph of G∗=(vi , Ei) where vi ={и,V,w}, E={uv, vw, uw }

{ и , ([0.1,0.1],0.4),([0.3, 0.4],0.2),([0.5,0.6], 0.1)}

М/ = 〈 { v , ([0.1,0.3],0.1), ([0.4,0.5],0.3),([0.1, 0.1],0.2)} 〉

{ w ,([0.2, 0.3],0.1),([0.1,0.2], 0.6),([0.3,0.5],0.2)}

{uv, ([0.1,0.1], 0.4), ([0.3,0.4], 0.2), ([0.5,0.6], 0.1)}

N ^P = ({ vw, ([0.1,0.3], 0.1), ([0.1,0.2], 0.6), ([0.3,0.5], 0.2)})

{uw, ([0.2,0.3], 0.1), ([0.1,0.2], 0.6), ([0.5,0.6], 0.1)}

{u, ([—0.1, — 0.1], — 0.4), ([—0.3, —0.4], —0.2), ([—0.5, —0.6], —0.1)}

M^ = ( { v, ([—0.1, —0.3], —0.1), ([—0.4, —0.5], —0.3), ([—0.1, —0.1], —0.2)} )

{w, ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], —0.2)}

{uv, ([—0.1, —0.1], — 0.4), ([—0.3, —0.4], —0.2), ([—0.5, —0.6], —0.1)}

N^ = ({ vw, ([—0.1, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], — 0.2)})

{uw, ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.5, — 0.6], —0.1)} and G2 = ((M2p,N2p), (M2n, N2w) ) be a bipolar neutrosophic cubic fuzzy graph of G} = (V2,E2) where Vi = { a, v, c} and E2 = { ab, be, ac}

{ a, ([0.6,0.7], 0.5), ([0.1,0.3], 0.4), ([0.2,0.3], 0.6)}

M 2 ^p = ({ b , ([0.1,0.2], 0.3), ([0.5,0.6], 0.2), ([0.8,0.9], 0.4)})

{ c, ([0.3,0.4], 0.1), ([0.2,0.3], 0.1), ([0.5,0.6], 0.3)}

{ ab , ([0.1,0.2], 0.5), ([0.1,0.3], 0.4), ([0.8,0.9], 0.4)}

N 2 ^p = ({ bc, ([0.1,0.2], 0.3), ([0.2,0.3], 0.2), ([0.8,0.9], 0.3)})

{ ac, ([0.3,0.4], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6], 0.3)}

{ a, ([—0.6, —0.7], —0.5), ([—0.1, —0.3], —0.4), ([—0.2, — 0.3], —0.6)}

M₂ "  = ({ b , ([—0.1, —0.2], —0.3), ([—0.5, —0.6], —0.2), ([—0.8, —0.9], —0.4)})

{ c, ([—0.3, —0.4], —0.1), ([—0.2, — 0.3], —0.1), ([—0.5, —0.6], — 0.3)}

{ ab , ([—0.1, —0.2], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.4)}

N₂ⁿ = ( { b c,([—0.1,—0.2],—0.3),([—0.2,—0.3],—0.2),([—0.8,—0.9],—0.3)} )

{ ac, ([—0.3, —0.4], — 0.5), ([—0.1, —0.3], — 0.4), ([—0.5, — 0.6], —0.3)} then Gj x G2 is a bipolar neutrosophic cubic fuzzy graph of G} x G}, where 1^x1^ = {(u, a), (u, b), (u, c), (v, a), (v, b), (v, c), (w, a), (w, b), (w, c)} and

{(и, а), ([0.1,0.1], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6], 0.1)}

{(и, й), ([0.1,0.1], 0.4), ([0.3,0.4], 0.2), ([0.8,0.9], 0.1)}

{(и, с), ([0.1,0.1], 0.6), ([0.2,0.3], 0.2), ([0.5,0.6], 0.1)}

{(v, а), ([0.1,0.3], 0.5), ([0.1,0.3], 0.4), ([0.2,0.3], 0.2)}

M f xM f = <{(v, й), ([0.1,0.2], 0.3), ([0.4,0.5], 0.3), ([0.8,0.9], 0.2)}>

{(v, с), ([0.1,0.3], 0.1), ([0.2,0.3], 0.3), ([0.5,0.6], 0.2)}

{(w, а), ([0.2,0.3], 0.5), ([0.1,0.2], 0.6), ([0.3,0.5], 0.2)}

{(w, й), ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{(w, с), ([0.2,0.3], 0.1), ([0.1,0.2], 0.6), ([0.5,0.6], 0.2)}

{(и, а), ([—0.1, —0.1], —0.5), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], — 0.1)}

{(и, й ), ([—0.1, — 0.1], —0.4), ([—0.3, —0.4], —0.2), ([—0.8, —0.9], — 0.1)}

{(и, с), ([—0.1, —0.1], —0.6), ([—0.2, —0.3], —0.2), ([—0.5, —0.6], —0.1)}

{(v, а), ([—0.1, —0.3], —0.5), ([—0.1, —0.3], —0.4), ([—0.2, —0.3], —0.2)}

М^ хМ^ = <  {(v, й), ([—0.1, —0.2], —0.3), ([—0.4, —0.5], —0.3), ([—0.8, —0.9], —0.2)} >

{(v, с), ([—0.1, —0.3], —0.1), ([—0.2, —0.3], —0.3), ([—0.5, —0.6], —0.2)}

{(w, а), ([—0.2, —0.3], —0.5), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], —0.2)}

{(w, b ), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{(w, с), ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.5, — 0.6], —0.2)}

{((и, а), (и, b)) , ([0.1,0.1], 0.5), ([0.1,0.3], 0.4), ([0.8,0.9],0.1)}

{((u, b), (и, с)), ([0.1,0.1], 0.4), ([0.2,0.3], 0.2), ([0.8,0.9],0.1)}

{((и, а), (v, с)), ([0.1,0.1], 0.4), ([0.1,0.3], 0.4), ([0.5,0.6],0.2)}

{((v, а), (v, с)), ([0.1,0.3], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6],0.2)}

V ^P x Vf = < {((v,a),(v,b)),([0.1,0.2],0.5),([0.1,0.3],0.4),([0.8,0.9],0.2)} )

{ ((v, b ), (w, b )) , ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{ ((w, b ), (w, с)) , ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{ ((w, a), (w, с)) , ([0.2,0.3], 0.5), ([0.1,0.2], 0.6), ([0.5,0.6], 0.2)}

{ ((u, ab ), (w, a)) , ([0.1,0.1], 0.5), ([0.1,0.2], 0.6), ([0.5,0.6], 0.1)}

{ ((u, a), (u, b )) , ([—0.1, —0.1], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.1)}

{ ((u, b ), (и, с)) , ([—0.1, —0.1], —0.4), ([—0.2, —0.3], —0.2), ([—0.8, —0.9], —0.1)}

{((и, a), (v, с)), ([—0.1, —0.1], —0.4), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], —0.2)}

{((v, a), (v, с)), ([—0.1, —0.3], —0.5), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], —0.2)}

V ^p x V₂ " = ( {((v, a), (v, b)), ([—0.1, —0.2], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.2)} )

{((v, b), (w, b)), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{((w, b), (w, с)), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{((w, a), (w, с)), ([—0.2, —0.3], —0.5), ([—0.1, —0.2], —0.6), ([—0.5, —0.6], —0.2)}

{ ((и, ab), (w, a)) , ([—0.1, — 0.1], —0.5), ([—0.1, —0.2], —0.6), ([—0.5, —0.6], —0.1)}

Definition 3.3 Let G ^ = ( (M-jPv ^P ) ^M/jV i" )) be a bipolar neutrosophic cubic fuzzy graph of G * = (^,1;^ and G₂ - ( (M₂ ^p ,V₂^p), (M₂ ⁿ, N₂ " ) ) be a Bipolar neutrosophic cubic fuzzy graph of G * = (^₂ M₂) . Then composition of G₁ and G₂ is denoted by G.]_[G₂ ] and defined as follows

G i [G 2 ]= ((M i^P ,V i ^P) ,(M/,V i " )) [(M 2^P ,V 2 ^p),(M 2 ^W ,V 2 " )]

= {(M P , M " )[M₂ ^P , M " ], (V^p, V " ) [V₂ ^P , V " ]}

((л pm p ), oof)) [ (m pm 2 ), (b₂^pm P )) ]

(( C ^p , D " ), ( C ^p , D " )) [ (( C ^p , D^ ), ( C ^p , D " ) )]

( Al , Al ),[ A^p₂ , Al ],( Bl , Bl ),[ Bl , Bl ]

( Cl , Cl ),[ Cl , cl ],( Bl , Dl ),[ Dl , Dl ]

< (SS , Вт A J ^∘ (bIa₂ , Вт aS) , (SS , BtbJ ^∘ (Btb₂ , BtbS) , ⎫ (й, , ^mJ ^∘ SlA₂ , ^s , (&, ^IbD ^∘ Sb₂ , <₂)) ,    ⎪^⎪

⎪ {S^a^ , YfaJ ∘ (Yfa2 , YfaS) , SIb, , YfbJ ∘ (.Yfb2 , YfbS) > ⎪ < ((Btc, , B-tS ∘ (Btc2 , BtcS) , ((bId,, bS) ∘ (bId2 , BtdS) , ⎪⎪ ((^ , ^ ∘ (Sc2 , o) , (Ж , ^J ∘ (Sd2 , ^S , ⎪⎪ ⎩⎪ (Me., US ∘ Wc2 , YfcS) , ((SS, yIdJ ∘ (.YPFd2 , YfdS) > ⎭⎪ where (i) ∀((up , UN)(vp, VN))∈(V1 , У2 )

⎧ SS ^∘ Bta₂) ⁽ u^p , v^p )= _rp min (bta₂ ⁽ u^p ), Bta₂ ⁽ v^{p )} , ⎫

⎪ (bS ∘ Btb2)( up , vP)=max[btb1 ( up), Btb2 (vp )

SS ^∘ Bta₂) ⁽ U^N , V^N )= _rN max (bIa. ⁽ u^N ), Bta₂ ⁽ v^N ) , ⎬

⎩⎪ (SS ∘ Втв2)( uN , vN)=min(bIB1 ( uN), Btb2 (vN)

⎧ SS ∘ ^IaD ( UP , VP)=rP min ^A1 ( up), %a2 (vp) ,⎫ ⎪ Як ∘ *Ib2)( up , vp)=maxSB1 ( up), ^Ib2 (vp)

(Л1А1 ∘ ^Ia2)( UN , VN)=rN max (X ( UN), ^Ia2 ( VN) , ⎩⎪ SiB1 ∘ ^B2) ( UN , VN)=min^Br (UN), ^Ib2 (VN)

⎧ ^Yfa₁ ^∘ ~'fa₂ ) ⁽ u^p , v^p )= _rp max IYfa, ⁽ u^p ), Yfa₂ ⁽ v^{p )} , ⎫

⎪ (yPb2 ∘ ' 'fb2 ) ( up , vp)=min'?FB1( up ), Yfb2 (vp )

⎨ ^FA_X ^∘ ~'fa₂ ) ⁽ U^N , V^N )= _rN min IYfa, ⁽ U^N ), Yfa₂ ⁽ V^N ) , ⎬

⎩⎪ (Yfb! ∘ 'fbD ( UN , VN)=max[Yfb, ( UN ), Yfb2 (VN )

• (ii)    ∀ ( u^p , U^N ) ∈ ^1 and ( vlvl )( vlvl ) ∈ E

⎧ {b^Pc₁ ^∘ BtcS ( u^p , ^V1^P )( u^p , V₂^P ))= min {btc₁ ( u^p ), Btc₂ ( vlvl ) , ⎫ ^⎪ (Btd₁ ^∘ BtdS ( u^p , v^ )( u^p , V₂^P )) =max (ss ( u^p ), Btd₂ ( vlvl ) ^⎪ (.Втс₂ ^∘ bIcS ⁽ u^N , v_±^N )( U^N , V₂^{N )})= max (bS ⁽ u^N ), Btc₂ ⁽ vlvl ) , ⎩ ^⎪ (fiS ^∘ BtdS ( u^N , v_±^N )( u^N , v₂ⁿ )) =min (bI_D1 ( u^N ), Btd₂ ( vlvl ) ⎭ ^⎪

⎧       ∘       (   ,    )(   ,    )  = min      (   ),     () ,

⎪          ∘        (   ,    )(   ,    ) =max       (   ),     (      )

⎨       ∘       (   ,    )(   ,    )  = max      (   ),     () ,

⎩        ∘        (   ,     )(   ,     )  =min       (   ),     (      )

⎧       ∘        (   ,    )(   ,    )  = max      (   ),     () ,

⎪           ∘         (   ,     )(   ,     )  =min        (   ),      (      )

,ч z

∘        (   ,     )(   ,     )  = min       (   ),     () ,

⎩       ∘       (   ,    )(   ,    ) =max      (   ),     (      )

• (iii)    ^(v^p,v^N) e ^and (ulU^UiU) £ E_r

⎧        ∘        (    ,   )(    ,   )  = min       (      ),     () ,

⎪           ∘         (    ,   )(    ,   ) =max       (      ),      (   )

∘        (    ,    )(    ,    )  = max       (      ),      () ,

⎩        ∘        (    ,   )(    ,   )  =min       (      ),     (   )

⎧       ∘       (    ,   )(    ,   )  = min      (      ),     () ,

⎪          ∘        (    ,   )(    ,   ) =max      (      ),     (   )

⎨       ∘       (    ,   )(    ,   )  = max      (      ),     () ,

⎩        ∘       (    ,   )(    ,   )  =min      (      ),     (   )

⎧       ∘        (    ,   )(    ,   )  = max      (      ),     () ,

⎪          ∘        (    ,   )(    ,   ) =min       (      ),      (   )

к4 ?

∘        (    ,    )(    ,    )  = min       (      ),      () ,

⎩       ∘       (    ,   )(    ,   ) =max      (      ),     (   )

⎧       ∘       (    ,   )(    ,   )  = min      (      ),     () ,

⎪          ∘        (    ,   )(    ,   ) =max      (      ),     (   )

⎨       ∘       (    ,   )(    ,   )  = max      (      ),     () ,

⎩        ∘       (    ,   )(    ,   )  =min      (      ),     (   )

⎧ ^fc₂ ^∘ Yfc^ ( и^ , v^p )( U₂^P , v^p ))= max (yF_C1 ( u^pu^p ), Yfa₂ ( v^p ) , ⎫ ^⎪ (y'fd^ ^∘ Yfd^( ( u^ , v^p )( U₂^P , v^p )) =min (yfd, ( u^pu₂ ), Yfb₂ ( v^p ) ^⎪ ⎨ [Yfc! ^∘ Yfc^^ ( u_±^N , V^N )( U₂^N , V^N ))= min (yfc! ( u^u₂ ), Yfa₂ ( V^N ) , ⎬ ⎩ ^⎪ (.Y^ ^∘ Yfd^ ( u^ , v^N )( u₂ⁿ , V^N )) =max ^FD_X ( u^u₂ ), Yfb₂ ( V^N ) ⎭ ^⎪

• (iv)    ∀ (( u^p , v^p )( U^p , V₂ )),((Ui , V? )( U^ , v₂ ) ∈ E °- E

⎧ (pT^ ∘Etc2)((up , vp)(Up , V2 )= min (Pta2 ( V1 ), Pta2 (v2p), Pt^ (upup)),⎫ ⎪ (pTDt ∘ Ptd.^ (up , ^1 )(^2 , ^2 ) =max(ртв2 ( Vi ), Ptb2 (v2p), Ptd1 (upup)

⎨ (Ptc₁ ^∘ PtC₂)( ( u^ , V? )( u₂ , V₂ )= max (ртл₂ ( Vi ), Pta₂ ( v₂ⁿ ), Ртс_г ( u^u₂ ) , ⎬ ⎩ ^⎪ (flrD! ^∘ PtD₂)( ( Ui , V? )( U^ , V2 ) =min (р‘тв₂ ( v_±^N ), Ptb₂ ( v₂ⁿ ), P1D_a ( u^u₂ ) ⎭ ^⎪

⎧ ^C1 ∘ ^C2)(( uf , Vp)(Up2 , V2 )= min (%A2 (V1P), ^A-; (v2p), *PC1 (upu2 ) ,⎫ ⎪ (%, ∘ ^D2)((uf , ^1 )(Up , V? ) =maxi^B2 (V1P), Wb2 (v2p), %D1 (upup)

⎨ 0/q ∘ <2)(( u^ , V?)(U% , V2 )= max {^A2 (V±N), ^a2 (v2n ), WC1 (U1U2) ,⎬ ⎩⎪ 0^Dr ∘ ^2)(( u? , V?)(U% , V2 ) =min(^B2 ( v^ ), ^B2 (V2N), ^Bi (u^u2 )

⎧ (.Yfc1 ∘YfcJ{( u% , Vp)(Up , ^2 )= max ^FA2 ( V1P), Yfa2 (v2p ), Yfc! (upup) ,⎫ ⎪ (Yfd! ∘ Yfd^ ( u^ , Vp)(U^ , ^2 ) =min(yPb2 (V1P), Yfb2 (v2p), Yfd^ (upup)

⎨ [Yfc! ^∘ Yfc^ ( u^ , V? )( U^ , V₂ )= min (yfa₂ ( v^ ), Yfa₂ ( v₂ⁿ ), Yfc^ ( u^u₂ ) , ⎬ ⎩ ^⎪ (yFDi ^∘ Yfd₂)( ( u^ , V? )( U? , ^2 ) =max (yfb₂ ( V_±^N ), Y^b₂ ( v₂ⁿ ), Y^ ( U1U₂ ) ⎭ ^⎪

Example 3.4 Let G ^∗ =( ^1 , Ei ) and G ^∗ =( ^2 , E2 ) be two fuzzy graphs, where ^1 =( и , V ) and = ( X , У ) . Suppose M_± and M₂ be the bipolar neutrosophic fuzzy cubic set representations of ^1 and ^2 . Also ^N1 and ^2 be the bipolar neutrosophic fuzzy cubic set representations of Ei and E2 and defined as

{ и ,([0.4,0.5],0.1),([0.1,0.1],0.4),([0.7,0.8],0.2)}

M_r^p = 〈                                         〉

{ v ,([0.3,0.4],0.2),([0.1,0.2],0.1),([0.4,0.5],0.5)}

{ и , ([-0.4, -0.5], -0.1), ([-0.1, -0.1], - 0.4), ([-0.7, -0.8], -0.2)}

M_±^N = 〈                                                    〉

{ V , ([-0.3, -0.4], -0.2), ([-0.1, - 0.2], -0.1), ([-0.4, -0.5], -0.5)}

Mi = 〈 { uv ,([0.3,0.4],0.2),([0.1,0.1],0.4),([0.7,0.8],0.2)} 〉

N^ =〈{uv, ([-0.3, -0.4], -0.2), ([-0.1, -0.1], - 0.4), ([-0.7, -0.8], -0.2)}〉 and

{%, ([0.5,0.6], 0.3), ([0.7,0.8], 0.7), ([0.1,0.1], 0.5)} м2 р — (                                              )

{у, ([0.2,0.3], 0.6), ([0.5,0.6], 0.4), ([0.8,0.9], 0.8)}

{%, ([-0.5, -0.6], -0.3), ([-0.7, -0.8], - 0.7), ([-0.7, -0.8], -0.2)}

М₂^N — (                                                             )

{у, ([—0.2, —0.3], —0.6), ([—0.5, — 0.6], —0.4), ([—0.8, —0.9], —0.8)}

N₂^р = <{%у, ([0.2,0.3], 0.6), ([0.5,0.6], 0.7), ([0.8,0.9], 0.5)})

N₂ⁿ = <{%у, ([-0.2, -0.3], -0.6), ([-0.5, -0.6], -0.7), ([-0.8, -0.9], -0.5)})

Clearly G_r— ( (М₁^Р, N^) ,(M j ^p ,N₁ ^p ) ) and G₂ — ( (M^pP, N₂ ^p ) , (M₂ ^p ,N₂ ^p ) ) are bipolar neutrosophic cubic fuzzy graphs. So, the composition of two bipolar neutrosophic cubic fuzzy graphs G₁ and G₂ is again a bipolar neutrosophic cubic fuzzy graph, where

{( , ),([0.4,0.5],0.3),([0.1,0.1],0.7),([0.7,0.8],0.2)}

{( , ),([0.2,0.3],0.6),([0.1,0.1],0.4),([0.8,0.9],0.2)}

M P [M P ] — (                                                  )

{( , ),([0.3,0.4],0.3),([0.1,0.2],0.7),([0.4,0.5],0.5)}

{( ,  ),([0.2,0.3],0.6),([0.1,0.2],0.4),([0.8,0.9],0.5)}

{(u, %), ([—0.4, —0.5], —0.3), ([—0.1, —0.1], —0.7), ([—0.7, —0.8], —0.2)}

{(u, у), ([—0.2, —0.3], —0.6), ([—0.1, —0.1], —0.4), ([—0.8, —0.9], —0.2)}

M P [M P ] — (                                                             )

{(v, %), ([—0.3, —0.4], —0.3), ([—0.1, —0.2], —0.7), ([—0.4, —0.5], —0.5)}

{( , ),([—0.2,—0.3],—0.6),([—0.1,—0.2],—0.4),([—0.8,—0.9],—0.5)}

• {    ((u, %), (u, у)) , ([0.2,0.3], 0.6), ([0.1,0.1], 0.7), ([0.8,0.9], 0.2)}

• {    ((u,y), (v,y)) , ([0.2,0.3], 0.6), ([0.1,0.1], 0.4), ([0.8,0.9], 0.2)}

• {    ((v, у), (v, %)) , ([0.2,0.3], 0.6), ([0.1,0.2], 0.7), ([0.8,0.9], 0.5)}

N i^P И — (                                           )

• {    ((v, %), (u, %)) , ([0.3,0.4], 0.3), ([0.1,0.1], 0.7), ([0.7,0.8], 0.2)}

• {    ((u, %), (v, у)) , ([0.2,0.3], 0.6), ([0.1,0.1], 0.7), ([0.8,0.9], 0.2)}

• { ((u, у), (v, %)) , ([0.2,0.3], 0.6), ([0.1,0.1], 0.7), ([0.8,0.9], 0.2)}

{(( и , X ),( и , У )), ([-0.2, -0.3], -0.6), ([-0.1, -0.1], -0.7), ([-0.8, -0.9], -0.2)}

{(( и , У ),( v , У )), ([-0.2, -0.3], -0.6), ([-0.1, -0.1], -0.4), ([-0.8, -0.9], -0.2)}

{(( V , У ),( v , х )), ([-0.2, -0.3], -0.6), ([-0.1, -0.2], -0.7), ([-0.8, -0.9], -0.5)}

<[<]= 〈                                               〉

{(( V , X ),( и , X )), ([-0.3, -0.4], -0.3), ([-0.1, -0.1], -0.7), ([-0.7, -0.8], -0.2)}

{(( и , X ),( V , У )), ([-0.2, -0.3], -0.6), ([-0.1, -0.1], -0.7), ([-0.8, -0.9], -0.2)}

{(( и , У ),( v , х )), ([-0.2, -0.3], -0.6), ([-0.1, -0.1], -0.7), ([-0.8, -0.9], -0.2)}

Proposition 3.5            Let Gi =      , ^р) , (М^, ^w)) and G2 =      , N2P), (M2N, ^2^)) be two bipolar neutrosophic cubic fuzzy graphs, then the Cartesian product of two bipolar neutrosophic cubic fuzzy graphs is again a bipolar neutrosophic cubic fuzzy graph.

Proof: Condition is oblivious for (M-^ , M^) × (m2p, M2N). Therefore we verify conditions only for (n^, ^lw) × (N2p, n2n), where w, N^) × (n2p, n2n)

{ ((pTCi × TC₂ ^, Ptd₁ × tdJ ^, (firCi × TC₂ ^, Ptd_± × tdS) ^, [(Jtfci × IC₂ , ID^ × ID^ ^, ^^Pc₁ × IC₂ , ID^ × idS) ^, (^Yfc^ × fc₂ , FD₁ × fdJ ^, {Yfc₁ × FC₂ , FD₁ × fdS)                      )

Let ( u^p , U^N ) ∈ ^1 and U₂v₂ ∈ E₂ .

Then

^^ T c^ × TC₂ ( ( u^p , U^p₂ )( u^p , V₂ ) , Ртс_г × TC₂ ( ( U^N , u₂ )( U^N , V₂ )

=    min {(pTA, ( u^p ), Ptc₂ ( U₂^P , V₂ ) )} , _rN max [(^4 ( u^p ), Ptc₂ ( U₂^P , V₂ ))}}

(r^p min {(pTA, ⁽ U^P ), _rP min ^ТА₂ ⁽ U₂^P ), Pta₂ ⁽ v₂^{P )} ))} ,

\r^N max {(pTAr ( U^N ),     _rN max {рТЛ₂ ( u₂ⁿ ), Pta₂ ( v₂ⁿ ) ))p

( r^p min [r^p min {(Pta₁ ( u^p ), Pta₂ ( U₂^P ) , _rp min (рТА! ( u^p ), Ата₂ ( V₂^P )))},

|r^wmax [r^N max ((^TA, ( U^N ), &та₂ ( u₂ⁿ ) , _rN max ^TA, ( U^N ), &та₂ ( v₂ⁿ ) w

(r^p min {(firAt × Pta^ ⁽ u^p , U^p ), (^TA₁ × Pta^ ⁽ u^p , ^2 ⁾},

|r^w max {(pTA! × P-TA^ ( U^N , U% ), (&TA! × Pta^ ( U^N , V₂ )})

^Ptd₁ × TD₂ ( ( u^p , U^ )( u^p , ^2 ) , Ptd₁ × TD₂ ( ( U^N , U% )( U^N , V₂ )

max max ^TBi ( u^p ), Atb₂ ( U₂^P ) ,max ^TBi ( u^p ), Atb₂ ( V₂ ) »1

min min ^TBi ( UN), Ртв2 (u2n ) ,min ^TBi ( UN), Atb2 (v2n)ж max {^TB1 × AtB2)( up , u2 ), ^TBi × РтВ2)( up , V2 ) )} , min {^TB± × РтВ2)( UN , u2 ), ((fir Bi × АтВ^ ( UN , V2 )ж

^Ci × IC₂ ( ( u^p , U^p₂ )( u^p , V₂ ) ,< × IC₂ (( U^N , u₂ )( U^N , V₂ )

=    min {^ ( u^p ), Vc₂ ( U₂^P , ^2 ) )} , {(X ( U^N ), <₂ ( U₂^N , v₂ ))}}

(r^p min {(.^Ai ( u^p ),     _rp min $a₂ ( U₂^P ), %a₂ ( v₂^p )))} ,

Ж max {(X ( U^N ),    _rN max ^A₂ ( u₂ⁿ ), ^a₂ ( v₂ⁿ ) Ш

(r^p min {r^P min ^^Ai ( u^p ), л^₂ ( U₂^P ) , _rp min ^_A1 ( u^p ), ^( v₂^p )))},

]r^N min [r^N max {^IAi ( U^N ), ^( u₂ⁿ ) , _rN min {^Ai ( U^N ), ^ia₂ ( v₂ⁿ ) )) }j

(r^p min {(Л?Ai × ^A^ ( U^P , U₂ ), ($Ai × ^A^ ( U^P , V₂ )},

Ж max {^Ai × ^A^ ( U^N , U% ), ^lAi × ^A^ ( U^N , V₂ )})

^Di × ^2 ( ( u^p , U^p )( u^p , V₂ ) , ^Di × ^2 ( ( U^N , U? )( U^N , V? )

= {max {(%Bi ( up ), ^■PD2 (U2P, ^2 ) )},min{(X ( UN), ^id2 (u2n, V2 ) )} max {(■%! (up), max (^ib2 (U2P)), ~^PB2 (V2P)},

^≤ min       (   ),     (    (    ) ,     (    )

max max (^Bi (up), ^PB2 (U2P) ,maxfe ( up), ^■m2 (V2 )ж min min (X ( UN), ^B2 (U2N) ,min ^IBi ( UN), ^B2 (V2N)ж max {^IBi × ^IB^) ( up , U^), (f^Bi × ^IB,}( up , ^2 ))} , min {«! × ^2)( UN , U%), («X × ^IB2)( UN , V2 )ж

(^FQ × FC₂ ( ( u^p , U^p )( u^p , ^2 ) , YFCi × FC₂ ( ( U^N , U% )( U^N , V₂ )

=    max K^Ml ( u^p ), Yfc₂ ( U₂^P , V^p ) )},rN min {(^1 ( U^N ), Yfc₂ ( u₂ⁿ , V₂ ))}}

(r^P max [(^1 ⁽ u^p ),     _ГР max (?fa₂ ⁽ U₂^P ), Yfa₂ ⁽ v₂^{p )} ))} ,

[rN min [(^1 ( UN),    rN max (yfa2 (u2n), Yfa2 (v2n) ))} rp max \rp max ((^i ( up ), Yfa2 (u2p) , rp max (y^ ( up), Yfa2 (V2P)))}, rN min \rN min ((^1 ( UN), Yfa2 (u2n ) , rN min (yfa, (UN), Yfa2 (v2n)))i rp max {(Yfa1 × Yfa2)( up , Up), (.Yfa1 × Yfa2)( up , ^2 )}, rN min{('Yfa1 × Yfa2)( UN , U^ ), (.Yfa! × Yfa2)( UN , V2 )}

(yfd₁ × fd₂ ( ( u^p , U^p )( u^p , V₂ ) , Yfd_± × fd₂ ( ( U^N , U₂ )( U^N , V? )

= {min K^Bl ( up ), Yfd2 (U2P, V2 ) )},maxK^Bl ( UN), Yfd2 (u2n, V2 ) )} min {min (yfb1 ( up ), Yfb2 (U2P) ,min (yfb1 ( up), Yfb2 (v2P ) )} , max max (yfb1 ( UN), Yfb2 (u2n ) ,max(yfb, ( UN), Yfb2 (u2n) )p min min (yfb1 ( up), Yfb2 (U2P) ,min (yfb1 ( up), Yfb2 ( ^2 )^ 1

max max ^FBX ( UN), Yfb2 (u2n) ,max (yfb! ( UN), Yfb2 (v2n))}J min {tel × Yfb^ ( up , U^), ((Yfb, × Yfb^ ( up , vp ) )} , max {tei × Yfb^ ( UN , U%), ((Yfb, × Yfb^ ( UN , V2 ) )} J

Similarly we can prove it for w ∈ ^2 and щу_г ∈ ^1 .

Proposition 3.6 Let Gi =      , NiP), tew, w/)) and G2 =      , n2p), {M2n, ^")) be two bipolar neutrosophic cubic fuzzy graphs, then the composition of two bipolar neutrosophic cubic fuzzy graphs is again a bipolar neutrosophic cubic fuzzy graph.

Example 3.7 Let ^1 =       , ^N1^P) , (^M1^W, ^1^W)) be a bipolar neutrosophic cubic fuzzy graph of

G ^∗ =( ^i , £i) where ^ri ={ и , V , w }, E ={ uv , VW , uw }

{и, ([0.1,0.1],0.4),([0.3, 0.4],0.2),([0.5,0.6], 0.1)} м/ =〈{V, ([0.1,0.3],0.1), ([0.4,0.5],0.3),([0.1, 0.1],0.2)}〉

{ w ,([0.2, 0.3],0.1),([0.1,0.2], 0.6),([0.3,0.5],0.2)}

{uv, ([0.1,0.1],0.4), ([0.3,0.4],0.2),([0.5, 0.6],0.1)} n/ =〈{VW, ([0.1,0.3],0.1),([0.1, 0.2],0.6),([0.3,0.5], 0.2)}〉

{ uw ,([0.2, 0.3],0.1),([0.1,0.2], 0.6),([0.5,0.6],0.1)}

{и, ([—0.1, — 0.1], — 0.4), ([—0.3, —0.4], —0.2), ([—0.5, —0.6], —0.1)}

М^ = ( { v, ([—0.1, —0.3], —0.1), ([—0.4, —0.5], —0.3), ([—0.1, —0.1], —0.2)} )

{w, ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], —0.2)}

{uv, ([—0.1, —0.1], — 0.4), ([—0.3, —0.4], —0.2), ([—0.5, —0.6], —0.1)}

N^ = ({ vw, ([—0.1, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], — 0.2)})

{uw, ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.5, — 0.6], —0.1)} and G2 = ((M2P, N2p), (M2w,N2w) ) be a bipolar neutrosophic cubic fuzzy graph of G2 = (V2,E2) where Vi = { a, v, c} and E2 = { ab, be, ac}

{ a, ([0.6,0.7], 0.5), ([0.1,0.3], 0.4), ([0.2,0.3], 0.6)}

М 2 ^p = ({ b , ([0.1,0.2], 0.3), ([0.5,0.6], 0.2), ([0.8,0.9], 0.4)})

{ c, ([0.3,0.4], 0.1), ([0.2,0.3], 0.1), ([0.5,0.6], 0.3)}

{ ab , ([0.1,0.2], 0.5), ([0.1,0.3], 0.4), ([0.8,0.9], 0.4)}

N 2 ^p = ({ bC( ([0.1,0.2], 0.3), ([0.2,0.3], 0.2), ([0.8,0.9], 0.3)})

{ ac, ([0.3,0.4], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6], 0.3)}

{ a, ([—0.6, —0.7], —0.5), ([—0.1, —0.3], —0.4), ([—0.2, — 0.3], —0.6)}

M₂^N = ({ b , ([—0.1, —0.2], —0.3), ([—0.5, —0.6], —0.2), ([—0.8, —0.9], —0.4)})

{ c, ([—0.3, —0.4], —0.1), ([—0.2, — 0.3], —0.1), ([—0.5, —0.6], — 0.3)}

{ ab , ([—0.1, —0.2], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.4)}

N₂ⁿ = ( { b c,([—0.1,—0.2],—0.3),([—0.2,—0.3],—0.2),([—0.8,—0.9],—0.3)} )

{ cC( ([—0.3, —0.4], — 0.5), ([—0.1, —0.3], — 0.4), ([—0.5, — 0.6], —0.3)} then Gr x G2 is a bipolar neutrosophic cubic fuzzy graph of G^ x G2, where Vi x V2 = {(u, a), (u, b), (u, c), (v, a), (v, b), (v, c), (w, a), (w, b), (w, c)} and

{(и, а), ([0.1,0.1], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6], 0.1)}

{(и, й), ([0.1,0.1], 0.4), ([0.3,0.4], 0.2), ([0.8,0.9], 0.1)}

{(и, с), ([0.1,0.1], 0.6), ([0.2,0.3], 0.2), ([0.5,0.6], 0.1)}

{(v, а), ([0.1,0.3], 0.5), ([0.1,0.3], 0.4), ([0.2,0.3], 0.2)}

M f xM f = <{(v, й), ([0.1,0.2], 0.3), ([0.4,0.5], 0.3), ([0.8,0.9], 0.2)}>

{(v, с), ([0.1,0.3], 0.1), ([0.2,0.3], 0.3), ([0.5,0.6], 0.2)}

{(w, а), ([0.2,0.3], 0.5), ([0.1,0.2], 0.6), ([0.3,0.5], 0.2)}

{(w, й), ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{(w, с), ([0.2,0.3], 0.1), ([0.1,0.2], 0.6), ([0.5,0.6], 0.2)}

{(и, а), ([—0.1, —0.1], —0.5), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], — 0.1)}

{(и, й ), ([—0.1, — 0.1], —0.4), ([—0.3, —0.4], —0.2), ([—0.8, —0.9], — 0.1)}

{(и, с), ([—0.1, —0.1], —0.6), ([—0.2, —0.3], —0.2), ([—0.5, —0.6], —0.1)}

{(v, а), ([—0.1, —0.3], —0.5), ([—0.1, —0.3], —0.4), ([—0.2, —0.3], —0.2)}

М^ хМ^ = <  {(v, й), ([—0.1, —0.2], —0.3), ([—0.4, —0.5], —0.3), ([—0.8, —0.9], —0.2)} >

{(v, с), ([—0.1, —0.3], —0.1), ([—0.2, —0.3], —0.3), ([—0.5, —0.6], —0.2)}

{(w, а), ([—0.2, —0.3], —0.5), ([—0.1, —0.2], —0.6), ([—0.3, —0.5], —0.2)}

{(w, b ), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{(w, с), ([—0.2, —0.3], —0.1), ([—0.1, —0.2], —0.6), ([—0.5, — 0.6], —0.2)}

{((и, а), (и, b)) , ([0.1,0.1], 0.5), ([0.1,0.3], 0.4), ([0.8,0.9],0.1)}

{((u, b), (и, с)), ([0.1,0.1], 0.4), ([0.2,0.3], 0.2), ([0.8,0.9],0.1)}

{((и, а), (v, с)), ([0.1,0.1], 0.4), ([0.1,0.3], 0.4), ([0.5,0.6],0.2)}

{((v, а), (v, с)), ([0.1,0.3], 0.5), ([0.1,0.3], 0.4), ([0.5,0.6],0.2)}

Nf x^f = ( {((v,a),(v,b)),([0.1,0.2],0.5),([0.1,0.3],0.4),([0.8,0.9],0.2)} )

{ ((v, b ), (w, b )) , ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{ ((w, b ), (w, с)) , ([0.1,0.2], 0.3), ([0.1,0.2], 0.6), ([0.8,0.9], 0.2)}

{ ((w, a), (w, с)) , ([0.2,0.3], 0.5), ([0.1,0.2], 0.6), ([0.5,0.6], 0.2)}

{ ((u, ab ), (w, a)) , ([0.1,0.1], 0.5), ([0.1,0.2], 0.6), ([0.5,0.6], 0.1)}

{ ((u, a), (u, b )) , ([—0.1, —0.1], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.1)}

{ ((u, b ), (и, с)) , ([—0.1, —0.1], —0.4), ([—0.2, —0.3], —0.2), ([—0.8, —0.9], —0.1)}

{((и, a), (v, с)), ([—0.1, —0.1], —0.4), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], —0.2)}

{((v, a), (v, с)), ([—0.1, —0.3], —0.5), ([—0.1, —0.3], —0.4), ([—0.5, —0.6], —0.2)}

< x Л^ = ( {((v, a), (v, b)), ([—0.1, —0.2], —0.5), ([—0.1, —0.3], —0.4), ([—0.8, —0.9], —0.2)} )

{((v, b), (w, b)), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{((w, b), (w, с)), ([—0.1, —0.2], —0.3), ([—0.1, —0.2], —0.6), ([—0.8, —0.9], —0.2)}

{((w, a), (w, с)), ([—0.2, —0.3], —0.5), ([—0.1, —0.2], —0.6), ([—0.5, —0.6], —0.2)}

{ ((и, ab), (w, a)) , ([—0.1, — 0.1], —0.5), ([—0.1, —0.2], —0.6), ([—0.5, —0.6], —0.1)}