itlin journl of pure nd pplied mthemtis n. 31 2013 (97 110) 97 ANTIPODAL BIPOLAR FUZZY GRAPHS Muhmmd Akrm Punj University College of Informtion Tehnology University of the Punj Old Cmpus, Lhore-54000 Pkistn e-mil: mkrmmth@yhoo.om m.krm@puit.edu.pk Sheng-Gng Li College of Mthemtis nd Informtion Siene Shnxi Norml University 710062, Xi n Chin e-mil: shenggngli@yhoo.om.n K.P. Shum Institute of Mthemtis Yunnn University Kunming, 650091 Chin e-mil: kpshum@ynu.edu.n Astrt. The onept of n ntipodl ipolr fuzzy grph of given ipolr fuzzy grph is introdued. Chrteriztions of ntipodl ipolr fuzzy grphs re presented when the ipolr fuzzy grph is omplete or strong. Some isomorphi properties of ntipodl ipolr fuzzy grph re disussed. The notion of self medin ipolr fuzzy grphs of given ipolr fuzzy grph is lso introdued. Keywords nd phrses: ntipodl ipolr fuzzy grphs, medin ipolr fuzzy grphs, self medin ipolr fuzzy grphs. 2000 Mthemtis Sujet Clssifition: 05C99. 1. Introdution Conepts of grph theory hve pplitions in mny res of omputer siene (suh s dt mining, imge segmenttion, lustering, imge pturing, networking et.). For exmples, dt struture n e designed in the form of trees, modeling of network topologies n e done using grph onepts. The most importnt
98 muhmmd krm, sheng-gng li, k.p. shum onept of grph oloring is utilized in resoure llotion nd sheduling. The onepts of pths, wlks nd iruits in grph theory re used in trveling slesmn prolem, dtse design onepts, nd resoure networking. This leds to the development of new lgorithms nd new theorems tht n e used in tremendous pplitions. A notion hving ertin influene on grph theory is fuzzy set, whih is introdued y Zdeh [17] in 1965; tully, the theory of fuzzy sets hs lredy eome vigorous reserh re whih intersets with mny reserh res, suh s medil nd life sienes, mngement sienes, soil sienes, engineering, sttistis, grph theory, rtifiil intelligene, signl proessing, multi-gent systems, pttern reognition, rootis, omputer networks, expert systems, deision mking nd utomt theory, et. A fuzzy set on given set X is just mpping A : X [0,1]. Bsed on the sme ide, Zhng [20] defined the notion of ipolr fuzzy set on given set X in 1994, whih is just mpping A : X [ 1,1], where the memership degree 0 of n element x mens tht the element x is irrelevnt to the orresponding property, the memership degree in (0, 1] of n element x indites tht the element somewht stisfies the property, nd the memership degree in [ 1,0) of n element x indites tht the element somewht stisfies the impliit ounter-property. In 1975, Rosenfeld [13] disussed the onept of fuzzy grph whose si ide ws introdued y Kuffmnn [11] in 1973. By onsidering fuzzy reltions etween fuzzy sets nd developing struture of fuzzy grphs, Rosenfeld otined nlogs of severl grph theoretil onepts. Bhtthry [9] gve some remrks on fuzzy grphs. The omplement of fuzzy grph ws defined y Mordeson [12] nd further studied y Sunith nd Vijykumr [15]. Ahmed nd Gni disussed the onepts of perfet fuzzy grph nd self medin fuzzy grph in [7]. Bsed on the notion of intuitionisti fuzzy set [5], Atnssov [5] introdued the onepts of intuitionisti fuzzy reltion nd intuitionisti fuzzy grphs. Reently, the ipolr fuzzy grphs hve een defined nd disussed in [1-3] sed on the notion of ipolr fuzzy set. The present pper ontinues to study ipolr fuzzy grphs. We introdue the onepts of ntipodl ipolr fuzzy grph nd self medin ipolr fuzzy grph of given ipolr fuzzy grph, nd prove severl hrteriztions theorems of ntipodl ipolr fuzzy grphs whose ipolr fuzzy grph re omplete or strong. We lso disuss isomorphi properties of ntipodl ipolr fuzzy grph. 2. Preliminries In this setion, we review some elementry onepts whose understnding is neessry fully enefit from this pper. By grph G = (V,E), we men non-trivil, finite, onneted nd undireted grph without loops or multiple edges. Formlly, given grph G = (V,E), two verties x, y V re sid to e neighors, or djent nodes, if {x,y} E. The ntipodl grph of grph G, denoted y A(G ), hs the sme vertex set s G with n edge joining verties u nd v if d(u,v) is equl to the
ntipodl ipolr fuzzy grphs 99 dimeter of G. For grph G of order p, the ntipodl grph A(G ) = G if nd only if G = K p. If G is non-omplete grph of order p, then A(G ) G. For grph G, the ntipodl grph A(G ) = G if nd only if () G is of dimeter 2 or () G is disonneted nd the omponents of G re omplete grphs. A grph G is n ntipodl grph if ut only if it is the ntipodl grph of its omplement. A grph G is n ntipodl grph if nd only if (i) dim(g ) = 2 or (ii) G is disonneted nd the omponents of G re omplete grphs. The self medin fuzzy grphs were introdued y Ahmed nd Gni in [7] The medin of grph G is the set of ll verties v of G for whih the vlue d G (v) is minimized. A grph is distne-lned (lso lled self-medin) if its medin is the whole vertex set. Thus, grph G is self-medin if nd only if the vlue d G (v) is onstnt over ll verties v of G. The sttus, or distne sum, of given vertex v in grph is defined y s(v) = u vd(u,v), where d(u,v) is the distne from vertex u to v. In other words, self medin grph G is one in whih ll the nodes hve the sme sttus s(v). The grphs C n, K n,n nd K n re self medin. The sttus of vertex v i is denoted y S(v i ) nd is defined s S(v i ) = δ(v i,v j ). The totl sttus of v j V fuzzy grph G is denoted y t[s(g )] nd is defined s t[s(g )] = S(v i ). v i V The medin of fuzzy grph G, denoted, is the set of nodes with minimum sttus. A fuzzy grph G is sid to e self-medin if ll the verties hve the sme sttus. Every self-medin fuzzy grph is self entered fuzzy grph. Every ue Q n is self-medin fuzzy grph. Definition 2.1 [17, 18] A fuzzy suset µ on set X is mp µ : X [0,1]. A fuzzy inry reltion on X is fuzzy suset µ on X X. By fuzzy reltion we men fuzzy inry reltion given y µ : X X [0,1]. Definition 2.2 [20] Let X e nonempty set. A ipolr fuzzy set B in X is n ojet hving the form B = {(x, µ P B (x), µn B (x)) x X} where µ P B : X [0, 1] nd µn B : X [ 1, 0] re mppings. We use the positive memership degree µ P B (x) to denote the stisftion degree of n element x to the property orresponding to ipolr fuzzy set B, nd the negtive memership degree µ N B (x) to denote the stisftion degree of n element x to some impliit ounter-property orresponding to ipolr fuzzy set B. If µ P B (x) 0 nd µn B (x) = 0, it is the sitution tht x is regrded s hving only positive stisftion for B. If µ P B (x) = 0 nd µn B (x) 0, it is the sitution tht x does not stisfy the property of B ut somewht stisfies the ounter property of B. It is possile for n element x to e suh tht µ P B (x) 0 nd µn B (x) 0 when the memership funtion of the property overlps tht of its ounter property over some portion of X.
100 muhmmd krm, sheng-gng li, k.p. shum For the ske of simpliity, we shll use the symol B = (µ P B, µn B ) for the ipolr fuzzy set B = {(x, µ P B(x), µ N B(x)) x X}. A nie pplition of ipolr fuzzy onept is politil epttion (mp to[0, 1]) nd non-epttion (mp to [-1, 0]). Definition 2.3 [20] Let X e nonempty set. Then, we ll mpping A = (µ P A,µN A ) : X X [0,1] [ 1,0] ipolr fuzzy reltion on X suh tht µ P A (x,y) [0,1] nd µn A (x,y) [ 1,0]. Definition 2.4 [1] Let A = (µ P A,µN A ) nd B = (µp B,µN B ) e ipolr fuzzy sets on set X. If A = (µ P A,µN A ) is ipolr fuzzy reltion on set X, then A = (µp A,µN A ) is lled ipolr fuzzy reltion on B = (µ P B,µN B ) if µp A (x,y) min(µp B (x),µp B (y)) nd µ N A (x,y) mx(µn B (x),µn B (y)) for ll x, y X. A ipolr fuzzy reltion A on X is lled symmetri if µ P A (x,y) = µp A (y,x) nd µn A (x,y) = µn A (y,x) for ll x, y X. Definition 2.5 [3] Let G e onneted ipolr fuzzy grph. The µ P -distne, δ µ P(v i,v j ), is the smllest µ P -length of ny v i v j pth P in G, where v i,v j V. Tht is, δ µ P(v i,v j ) = min(l µ P(P)). The µ N -distne, δ µ N(v i,v j ), is the lrgest µ N -length of ny v i v j pth P in G, where v i,v j V. Tht is, δ µ N(v i,v j ) = mx(l µ N(P)). Thedistne, δ(v i,v j ),isdefinedsδ(v i,v j )=(δ µ P(v i,v j ),δ µ N(v i,v j )). For eh v i V, the µ P -eentriity of v i, denoted y e µ P(v i ) nd is defined s e µ P(v i )=mx{δ µ P(v i,v j ) : v i V,v i v j }. Forehv i V, theµ N -eentriity of v i,denotedye µ N(v i )ndisdefinedse µ N(v i )=min{δ µ N(v i,v j ) : v i V,v i v j }. For eh v i V, the eentriity of v i, denoted y e(v i ) nd is defined s e(v i ) = (e µ P(v i ),e µ N(v i )). The µ P -rdius of G is denoted y r µ P(G) nd is defined s r µ P(G) = min{e µ P(v i ) : v i V}. The µ N -rdius of G is denoted y r µ N(G) nd is defined s r µ N(G) = mx{e µ N(v i ) : v i V}. The rdius of G is denoted y r(g) nd is defined s r(g) = (r µ P(G),r µ N(G)). The µ P -dimeter of G is denoted y d µ P(G) nd is defined s d µ P(G) =mx{e µ P(v i ) : v i V}. The µ N -dimeter of G is denoted y d µ N(G) nd is defined s d µ N(G) =min{e µ N(v i ) : v i V}. The dimeter of G is denoted y d(g) nd is defined s d(g) = (d µ P(G),d µ N(G)). A onneted ipolr fuzzy grph G is self entered grph, if every vertex of G is entrl vertex, tht is r µ P(G) = e µ P(v i ) nd r µ N(G) = e µ N(v i ), v i V. 3. Antipodl ipolr fuzzy grphs Definition 3.1 Let G = (A,B) e ipolr fuzzy grph. An ntipodl ipolr fuzzy grph A(G) = (E,F) is ipolr fuzzy grph G = (A,B) in whih: () An ipolr fuzzy vertex set of G is tken s ipolr fuzzy vertex set of A(G), tht is, µ P E (x) = µp A (x) nd µn E (x) = µn A (x) for ll x V,
ntipodl ipolr fuzzy grphs 101 () If δ(x,y) = d(g), then { µ P µ P F (xy) = B (xy) if x nd y re neighors in G, min(µ P A (x),µp A (y)) if x nd y re not neighors in G, { µ N µ N F (xy) = B (xy) if x nd y re neighors in G, mx(µ N A (x),µn A (y)) if x nd y re not neighors in G. Tht is, two verties in A(G) re mde s neighorhood if the µ P µ N distne etween them is dimeter of G. Exmple 3.2 Consider ipolr fuzzy grph G suh tht A = {v 1,v 2,v 3 }, B = {v 1 v 2,v 1 v 3,v 2 v 3 }. v 1 v 1 ( 1 3, 1 6 ) ( 1 3, 1 6 ) ( 1 3, 1 4 ) ( 1 6, 1 5 ) ( 1 6, 1 5 ) (1 2, 1 5 ) ( 1 3, 1 2 ) ( 1 3, 1 7 ) (1 2, 1 5 ) ( 1 3, 1 7 ) v 3 v 2 v 3 v 2 Bipolr Fuzzy Grph By routine lultions, we hve, Antipodl Bipolr Fuzzy Grph Figure 1 δ µ P(v 1,v 2 ) = 6, δ µ P(v 1,v 3 ) = 3, δ µ P(v 2,v 3 ) = 3, δ µ N(v 1,v 2 ) = 6, δ µ N(v 1,v 3 ) = 7, δ µ N(v 2,v 3 ) = 9, e µ P(v 1 ) = 6, e µ P(v 2 ) = 6, e µ P(v 3 ) = 3, e µ N(v 1 ) = 6, e µ N(v 2 ) = 6, e µ N(v 3 ) = 7, d(g) = (6, 6), δ(v 1,v 2 ) = (6, 6) = d(g). Hene A(G) = (E,F), suh tht E = {v 1,v 2,v 3 } nd F = {v 1 v 2 }. Theorem 3.3 Let G = (A,B) e omplete ipolr fuzzy grph where (µ P A,µN A ) is onstnt funtion then G is isomorphi to A(G). Proof. Given thtg = (A,B)eomplete ipolr fuzzygrphwith(µ P 1,µN 1 ) = (k 1,k 2 ), where k 1 nd k 2 re onstnts, whih implies tht δ(v i,v j ) = (l 1,l 2 ), v i,v j V. Therefore, eentriity e(v i ) = (l 1,l 2 ), v i V, whih implies tht d(g) = (l 1,l 2 ). Hene δ(v i,v j ) = (l 1,l 2 ) = d(g), v i,v j V. Hene every pir of verties re mde s neighors in A(G) suh tht
102 muhmmd krm, sheng-gng li, k.p. shum () An ipolr fuzzy vertex set of G is tken s ipolr fuzzy vertex set of A(G), tht is, µ P E (v i) = µ P A (v i) nd µ N E (v i) = µ N A (v i) for ll v i V, () µ P F (v iv j ) = µ P B (v iv j ), sine v i nd v j re neighors in G µ N F (v iv j ) = µ N B (v iv j ), sine v i nd v j re neighors in G. It hs sme numer of verties, edges nd it preserves degrees of the verties. Hene G = A(G). Theorem 3.4 Let G : (A,B) is onneted ipolr fuzzy grph. Every ntipodl ipolr fuzzy grph is spnning sugrph of G. Proof. By the definition of n ntipodl ipolr fuzzy grph, A(G) ontins ll the verties of G. Tht is, () µ P E (x) = µp A (x) nd µn E (x) = µn A (x) for ll x V, nd () If δ(x,y) = d(g), then { µ P µ P F (xy) = B (xy) if x nd y re neighors in G, min(µ P A (x),µp A (y)) if x nd y re not neighors in G, { µ N µ N F (xy) = B (xy) if x nd y re neighors in G, mx(µ N A (x),µn A (y)) if x nd y re not neighors in G. Hene A(G) is spnning sugrph of G. Theorem 3.5 Let G e ipolr fuzzy grph, where risp grph G is n even or odd yle. If lternte edges hve sme memership vlues nd non-memership vlues, then G is self entered ipolr fuzzy grph. Theorem 3.6 Let G e ipolr fuzzy grph, where risp grph G is n even or odd yle. If lternte edges hve sme positive nd negtive vlues, then A µ P(G) nd A µ N(G) is the edge indued ipolr fuzzy sugrph of Ḡ, whose end verties of A µ P(G) nd A µ N(G) re with mximum µ P - eentriity nd minimum µ N - eentriity in G. Proof. If lternte edges hve sme positive nd negtive vlues, then µ P - distne etween non-djent verties is greter thn the djent verties nd µ N -distne etween non-djent verties is lesser thn the djent verties. Let µ P B (v i,v j ) e the lest mong ll other edges, then δ µ P(v i,v j ) = 1 µ P B (v i,v j ). Clim (i): Neighors in G re not neighors in A(G). Consider n ritrry pth onneting v k,v t suh tht (1) (v k,v t ) µ P B
ntipodl ipolr fuzzy grphs 103 If P is pth of length 2 etween v k,v t, then (2) l µ P(P) Hene δ(v k,v t ) 1 µ P B (v i,v j ) 1 µ P B (v i,v j ), sine y eqution (1) nd (2), whih implies tht δ(v i,v j ) < δ(v k,v t ) d(g), where δ(v k,v t ) µ P B nd(v i,v j ) µ P B. Tht is, δ(v i,v j ) < d(g), if (v i,v j ) µ P B. Therefore, if (v i,v j ) µ P B, then v i nd v j re not neighors in A(G). Clim (ii): Edges in A(G) re edges in Ḡ. If (v m,v n ) µ P F, then y Clim (i), (v i,v j ) µ P B. So, µ P F (v m,v n ) = min(µ P A (v m),µ P A (v n)), sine y the definition of A(G), whih implies tht edges in A(G) re edges in Ḡ. Hene A(G) is ipolr fuzzy sugrph of Ḡ, indued y the edges of Ḡ, whose end verties re with mximum µ P eentriity in G. Let (v i,v j ) E, then δ µ N(v i,v j ) = k. Clim (i): Neighors in G re not neighors in A(G). Consider n ritrry pth onneting v k,v t suh tht (3) (v k,v t ) µ N B If Q is pth of length 2 etween v k,v t, then (4) δ µ P(P) k Hene δ µ N(v k,v t ) δ µ N(v i,v j ), sine y eqution (3) nd (4), whih implies tht δ µ N(v k,v t ) d(g), where (v k,v t ) µ N B nd (v i,v j ) µ N B. Tht is, δ µ N(v i,v j ) d(g), if (v i,v j ) µ N B. Therefore, if (v i,v j ) µ N B, then v i nd v j re not neighors in A(G). Clim (ii): Edges in A(G) re edges in Ḡ. If (v m,v n ) µ N F, then y Clim (i), (v i,v j ) µ P B. So, µ N F(v m,v n ) = mx(µ N A(v m ),µ N A(v n )), sine y the definition of A(G), whih implies tht edges in A µ N(G) re edges in Ḡ. Hene A µ N(G) is ipolr fuzzy sugrph of Ḡ, indued y the edges of Ḡ, whose end verties re with minimum µ N eentriity in G.
104 muhmmd krm, sheng-gng li, k.p. shum Theorem 3.7 Let G e ipolr fuzzy grph, where risp grph G is n even or odd yle. If lternte edges hve sme positive nd negtive vlues, then A(G) is iprtite ipolr fuzzy grph. Theorem 3.8 Let G : (A,B) e onneted strong ipolr fuzzy grph, where risp grph G is n even or odd yle, suh tht (µ P A,µN A )(v i) = (k 1,k 2 ), v i µ P A nd v i µ N A. Then A(G) is the spnning ipolr fuzzy grph sugrph of Ḡ, indued y the edges of Ḡ, whose end verties re mximum µp - eentriity nd minimum µ N - eentriity in G. Proof. Let (v i,v j ) µ P B, δ µ P(v i,v j ) = 1 k 1. But for ny (v i,v j ) µ P B, δ(v k,v m ) 2 k 1. Tht isδ µ P(v i,v j ) = 1 k 1 < 2 k 1 δ µ P(v k,v m ), where (v i,v j ) µ P B, whih implies tht v i,v j re verties in A(G), ut re not neighors in A(G). Now, let (v i,v j ) µ N B, δ µ N(v i,v j ) > 1 k 1. But for ny (v i,v j ) µ N B, δ µ N(v i,v j ) δ µ N(v k,v m ), where (v i,v j ) µ N B, whih implies tht v i,v j re verties in A(G), ut re not neighors in A(G). The remining proof is similr to lim (ii) of ove Theorem nd hene we omit it. Theorem 3.9 If G 1 nd G 2 re isomorphi to eh other, then A(G 1 ) nd A(G 2 ) re lso isomorphi. Proof. As G 1 nd G 2 re isomorphi, the isomorphism h, etween them preserves the edge weights, so the µ P µ N -length nd µ P µ N -distne will lso e preserved. Hene, if the vertex v i hs the mximum µ P -eentriity nd minimum µ N -eentriity, in G 1, then h(v i ) hs the mximum µ P -eentriity nd minimum µ N -eentriity, in G 2. So, G 1 nd G 2 will hve the sme dimeter. If the µ P µ N - distne etween v i nd v j is (k 1,k 2 ) in G 1, then h(v i ) nd h(v j ) will lso hve their µ P µ N -distne s (k 1,k 2 ). The sme mpping h itself is ijetion etween A(G 1 ) nd A(G 2 ) stisfying the isomorphism ondition. (i) µ P E 1 (v i ) = µ P A 1 (v i ) = µ P A 2 (h(v i )) = µ P E 2 (h(v i )), v i G 1 (ii) µ N E 1 (v i ) = µ N A 1 (v i ) = µ N A 2 (h(v i )) = µ N E 2 (h(v i )), v i G 1 (iii) µ P F 1 (v i,v j ) = µ P B 1 (v i,v j ), if v i nd v j re neighors in G 1 µ P F 1 (v i,v j ) = min(µ P E 1 (v i ),µ P E 1 (v j )), if v i nd v j re not neighors in G 1 nd (iv) µ N F 1 (v i,v j ) = µ N B 1 (v i,v j ), if v i nd v j re neighors in G 1 µ N F 1 (v i,v j ) = mx(µ N E 1 (v i ),µ N E 1 (v j )), if v i nd v j re not neighors in G 1 As h : G 1 G 2 is n isomorphism, µ N F 1 (v i,v j ) = µ N B 2 (h(v i ),h(v j )),if v i nd v j re neighors in G 1 µ N F 1 (v i,v j ) = mx(µ N B 2 (v i ),µ N B 2 (v j )),if v i nd v j re not neighors in G 1 µ N F 1 (v i,v j ) = µ N B 2 (h(v i ),h(v j )),if v i nd v j re neighors in G 1 µ N F 1 (v i,v j ) = mx(µ N B 2 (v i ),µ N B 2 (v j )),if v i nd v j re not neighors in G 1
ntipodl ipolr fuzzy grphs 105 Hene µ P F 1 (v i,v j ) = µ P F 2 (h(v i ),h(v j )) nd µ N F 1 (v i,v j ) = µ N F 2 (h(v i ),h(v j )). So, the sme h is n isomorphism etween A(G 1 ) nd A(G 2 ). Theorem 3.10 If G 1 nd G 2 re omplete ipolr fuzzy grph suh tht G 1 is o-wek isomorphi to G 2 then A(G 1 ) is o-wek isomorphi to A(G 2 ). Proof. As G 1 is o-wek isomorphi to G 2, there exists ijetion h : G 1 G 2 stisfying, µ P A (v i) µ P A (h(v i)), µ P B (v i,v j ) = µ P B (h(v i),h(v j )), v i,v j V 1. If G 1 hs n verties, rrnge the verties of G 1 in suh wy tht µ P A (v 1) µ P A (v 2) µ P A (v 3)...µ P A (v n). As G 1 nd G 2 re omplete, o-wek isomorphi ipolr fuzzy grph, µ P B (v i,v j ) = µ P B (h(v i),h(v j )), v i,v j V 1. By Theorem 3.9 nd the definition of ntipodl ipolr fuzzy grph, we hve A(G i ) ontins ll the verties of G, where i = 1,2. Tht is, µ P E (x) = µp A (x) nd µn E (x) = µn A (x) for ll x V nd µ P F (v i,v j ) = µ P F (h(v i),h(v j )), v i,v j V 1. So, the sme ijetion h is o-wek isomorphism etween A(G 1 ) nd A(G 2 ). We stte the following Theorem without its proof. Theorem 3.11 If G 1 nd G 2 re omplete ipolr fuzzy grph suh tht G 1 is o-wek isomorphi to G 2 then A(G 1 ) is homomorphi to A(G 2 ). Remrk 1 If G is self omplementry ipolr fuzzy grph, then its ntipodl ipolr fuzzy grph my not e self omplementry. Exmple 3.12 Consider ipolr fuzzy grph G ( 1 8, 1 5 ) (1 5, 1 6 ) ( 1 8, 1 5 ) (1 5, 1 6 ) ( 1 10, 1 2 ) ( 1 10, 1 2 ) (1 5, 1 2 ) ( 1 8, 1 7 ) (1 3, 1 8 ) ( 1 8, 1 7 ) (1 3, 1 8 ) d G By routine lultions, we hve Figure 2 d A(G) δ(,) = (15, 4), δ(,) = (10, 2), δ(,d) = (10, 2), δ(,) = (5, 2), δ(,d) = (25, 6), δ(,d) = (20, 4), e() = (15, 2), e() = (20, 2), e(d) = (25, 2), d(g) = (25, 2). e() = (15, 2), Sine d(g) δ(x,y) for ll x, y V. Hene A(G) is n ntipodl ipolr fuzzy grph of G hving sme verties s in G only, nd no two verties in A(G) re mde s neighorhood sine their µ P µ N distne etween them is not equl to the dimeter of G.
106 muhmmd krm, sheng-gng li, k.p. shum Consider ipolr fuzzy grph G ( 8 1, 1 5 ) ( 1 8, 1 5 ) (1 5, 1 6 ) ( 1 8, 1 5 ) (1 5, 1 6 ) ( 1 8, 1 6 ) ( 1 8, 1 7 ) (1 ( 1 3, 1 8 ) ( 8 1, 1 7 ) (1 3, 1 8 ) 8, 1 7 ) d d G A(G) Figure 3 By routine lultions, we hve δ(,) = (8, 5), δ(,) = (24, 18), δ(,d) = (16, 11), δ(,) = (16, 13), δ(,d) = (8, 6), δ(,d) = (8, 7), e() = (24, 5), e() = (16, 5), e() = (24, 7), e(d) = (16, 6), d(g) = (24, 5). Sine d(g) δ(x,y) for ll x, y V. Hene A(G) is n ntipodl ipolr fuzzy grph of G hving sme verties s in G only, nd no two verties in A(G) re mde s neighorhood sine their µ P µ N distne etween them is not equl to the dimeter of G. ( 1 8, 1 5 ) ( 1 8, 1 5 ) (1 5, 1 6 ) ( 1 8, 1 5 ) ( 1 8, 1 6 ) ( 1 8, 1 5 ) ( 1 5, 1 6 ) ( 1 8, 1 7 ) (1 3, 1 8 ) ( 1 8, 1 7 ) d A(G) Figure 4 Clerly, A(G) is not isomorphi to A(G) Hene G is self omplementry, ut its ntipodl ipolr fuzzy grph A(G) is not self omplementry ipolr fuzzy grph.
ntipodl ipolr fuzzy grphs 107 Exmple 3.13 Consider ipolr fuzzy grph G ( 6 1, 1 3 ) ( 1 5, 1 4 ) (1 5, 1 5 ) ( 1 5, 1 4 ) (1 5, 1 5 ) ( 1 5, 1 4 ) ( 1 6, 1 3 ) ( 1 5, 1 4 ) (1 6, 1 3 ) ( 1 5, 1 4 ) (1 6, 1 3 ) G d A(G) d Figure 5 By routine lultions, we hve δ(,) = (5, 4), δ(,) = (11, 7), δ(,d) = (17, 10), δ(,) = (6, 3), δ(,d) = (12, 6), δ(,d) = (6, 3), e() = (17, 4), e() = (11, 3), e(d) = (17, 3), d(g) = (17, 3). e() = (12, 3), Sine d(g) δ(x,y) for ll x, y V. Hene A(G) is n ntipodl ipolr fuzzy grph of G hving sme verties s in G only, nd no two verties in A(G) re mde s neighorhood sine their µ P µ N distne etween them is not equl to the dimeter of G. ( 1 5, 1 4 ) (1 5, 1 5 ) ( 1 5, 1 4 ) (1 5, 1 5 ) ( 1 6, 1 3 ) ( 1 5, 1 4 ) ( 1 5, 1 4 ) (1 ( 6 1 6, 1 3 ), 1 3 ) d Ḡ ( 1 5, 1 4 ) (1 6, 1 3 ) d A(Ḡ) Figure 6 By routine lultions, we hve δ(,) = (12, 6), δ(,) = (5, 4), δ(,d) = (6, 3), δ(,) = (17, 10), δ(,d) = (6, 3), δ(,d) = (11, 7), e() = (12, 3), e() = (17, 3), e() = (17, 4), e(d) = (11, 3), d(g) = (17, 3).
108 muhmmd krm, sheng-gng li, k.p. shum Sine d(g) δ(x,y) for ll x, y V. Hene A(G) is n ntipodl ipolr fuzzy grph of G hving sme verties s in G only, nd no two verties in A(G) re mde s neighorhood sine their µ P µ N distne etween them is not equl to the dimeter of G. ( 1 5, 1 4 ) ( 1 5, 1 4 ) (1 5, 1 5 ) ( 1 5, 1 4 ) ( 1 5, 1 4 ) ( 1 6, 1 3 ) ( 1 6, 1 3 ) ( 1 5, 1 4 ) (1 6, 1 3 ) ( 1 6, 1 3 ) A(G) Figure 7 Clerly, A(G) is not isomorphi to A(G), though A(G) is isomorphi to A(G). Hene G is self omplementry ipolr fuzzy grph ut A(G) is not self omplementry ipolr fuzzy grph. We now present the onept of medin ipolr fuzzy grphs. Definition 3.14 Let G e onneted ipolr fuzzy grph. The µ P -sttus of vertex v i is denoted y S µ P(v i ) nd is defined s S µ P(v i ) = δ µ P(v i,v j ). d v j V The µ N -sttus of vertex v i is denoted y S µ N(v i ) nd is defined s S µ N(v i ) = δ µ N(v i,v j ). The minimum µ P -sttus of G is denoted y m[s µ P(G)] nd v j V is defined s m[s µ P(G)] = min(s µ P(v i ), v i V). The minimum µ N -sttus of G is denoted y m[s µ N(G)] nd is defined s m[s µ N(G)] = min(s µ N(v i ), v i V). The minimum µ P µ N sttus of G is denoted y m[s µ P µ N(G)] nd is defined s m[s µ P µ N(G)] = (m[s µ P(G)],m[S µ N(G)]). The mximum µp -sttus of G is denoted y M[S µ P(G)] nd is defined s M[S µ P(G)] = mx(s µ P(v i ), v i V). Themximumµ N -sttusofgisdenotedym[s µ N(G)]ndisdefineds M[S µ N(G)] = mx(s µ N(v i ), v i V). The mximum µ P µ N sttusof G isdenoted y M[S µ P µ N(G)] nd is defined s M[S µ P µ N(G)] = (M[S µ P(G)],M[S µ N(G)]). The totl µ P -sttus of ipolr fuzzy grph G is denoted y t[s µ P(G)] nd is defined s t[s µ P(G)] = S µ P(v i ). The totl µ N -sttus of ipolr fuzzy grph v i V G is denoted y t[s µ N(G)] nd is defined s t[s µ N(G)] = S µ N(v i ). The v i V totl µ N -sttus of ipolr fuzzy grph G is denoted y t[s µ P µ N(G)] nd is defined s t[s µ P µ N(G)] = (t[s µ P(G)],t[S µ N(G)]). The medin of ipolr fuzzy
ntipodl ipolr fuzzy grphs 109 grph G is denoted y M(G) nd is defined s the set of nodes with minimum µ P µ N sttus. An ipolr fuzzy grph G is sid to e self-medin if ll the verties hve the sme sttus. In other words, G is self-medin if nd only if m[s µ P µ N(G)] = M[S µ P µ N(G)]. Exmple 3.15 v 1 v 2 ( 1 3, 1 3 ) ( 1 2, 1 3 ) (1 3, 1 4 ) ( 1 3, 1 4 ) (1 3, 1 4 ) ( 1 2, 1 5 ) (1 2, 1 3 ) ( 1 3, 1 3 ) By routine lultions, we hve, v 4 v 3 Figure 8. Selfmedinipolrfuzzygrph. δ µ P(v 1,v 2 ) = 3, δ µ P(v 1,v 3 ) = 6, δ µ P(v 1,v 4 ) = 3, δ µ P(v 2,v 3 ) = 3, δ µ P(v 2,v 4 ) = 6, δ µ P(v 3,v 4 ) = 3, δ µ N(v 1,v 2 ) = 11, δ µ N(v 1,v 3 ) = 7, δ µ N(v 1,v 4 ) = 10, δ µ N(v 2,v 3 ) = 10, δ µ N(v 2,v 4 ) = 7, δ µ N(v 3,v 4 ) = 11, S µ P(v 1 ) = 12, S µ P(v 2 ) = 12, S µ P(v 3 ) = 12, S µ P(v 4 ) = 12, S µ N(v 1 ) = 28, S µ N(v 2 ) = 28, S µ N(v 3 ) = 28, S µ N(v 4 ) = 28. Therefore, S µ P µ N(v 1) = (12, 28), S µ P µ N(v 2) = (12, 28), S µ P µ N(v 3) = (12, 28), S µ P µ N(v 4) = (12, 28)ndt[S µ P µ N(G)] = (48, 112). Here,S µ P µ N(v i) = (12, 28), v i V. Hene G is self medin ipolr fuzzy grph. Theorem 3.15 Let G e ipolr fuzzy grph, where risp grph G is n even yle. If lternte edges hve sme positive vlues nd negtive vlues, then G is self medin ipolr fuzzy grph. Proof. Given tht G is ipolr fuzzy grph. Sine risp grph G is n even yle. Also, lternte edges of G hve sme positive vlues nd negtive vlues, we hve, δ(v 1,v 2 ) = δ(v 3,v 4 ) = δ(v 1,v 2 ) =... = δ(v n 1,v n ) nd, similrly, δ(v 2,v 3 ) = δ(v 4,v 5 ) =... = δ(v n,v 1 ), δ(v 1,v 3 ) = δ(v 2,v 4 ) = δ(v 3,v 5 ) =... = l, so on. Hene S µ P(v i ) = k nd S µ N(v i ) = m, v i V. Hene G is self medin ipolr fuzzy grph. Remrk 2 Let G e ipolr fuzzy grph, where risp grph G is n odd yle. If lternte edges hve sme positive nd negtive vlues, then G my not e self medin ipolr fuzzy grph.
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