INTUITIONISTIC FUZZY GRAPH STRUCTURES

Transcription

1 Kragujevac Journal of Mathematcs Volume 41(2) (2017), Pages INTUITIONISTIC FUZZY GRAPH STRUCTURES MUHAMMAD AKRAM 1 AND RABIA AKMAL 2 Abstract. In ths paper, we ntroduce the concept of an ntutonstc fuzzy graph structure (IFGS). We dscuss certan notons, ncludng ntutonstc fuzzy B -cycles, ntutonstc fuzzy B -trees and φ-complement of an ntutonstc fuzzy graph structure wth several examples. We also present φ-complement of an ntutonstc fuzzy graph structure along wth self-complementary and strong self-complementary ntutonstc fuzzy graph structures. 1. Introducton Fuzzy set was ntroduced by Zadeh n A fuzzy set gves the degree of membershp of an object n a gven set. Kaufmann s ntal defnton of a fuzzy graph [10] was based on Zadeh s fuzzy relatons [22]. The fuzzy relatons between fuzzy sets were consdered by Rosenfeld and he developed the structure of fuzzy graphs, obtanng analogs of several graph theoretcal concepts. Later on, Bhattacharya [7] gave some remarks on fuzzy graphs and some operatons on fuzzy graphs were ntroduced by Mordeson and Peng [14]. In 1983, Atanassov [5] extended the dea of a fuzzy set and ntroduced the concept of an ntutonstc fuzzy set. He added a new component, degree of non-membershp, n the defnton of a fuzzy set wth the condton that sum of two degrees must be less or equal to one. Atanassov [6] also ntroduced the concept of ntutonstc fuzzy graphs and ntutonstc fuzzy relatons. Shannon and Atanassov nvestgated some propertes of ntutonstc fuzzy relatons and ntutonstc fuzzy graphs n [20]. Parvath et al. defned operatons on ntutonstc fuzzy graphs n [16]. Karunambga et al. used ntutonstc fuzzy graphs to fnd shortest paths n networks [11]. Akram et al. [1 4] ntroduced many Key words and phrases. Intutonstc fuzzy graph structure (IFGS), strong IFGS, Intutonstc fuzzy B -cycles, Intutonstc fuzzy B -trees, φ-complement of an IFGS Mathematcs Subject Classfcaton. Prmary: 05C72, 68R10. Secondary: 03E72, 05C78. Receved: September 5, Accepted: September 12,

2 220 M. AKRAM AND R. AKMAL new concepts, ncludng strong ntutonstc fuzzy graphs, ntutonstc fuzzy trees, ntutonstc fuzzy hypergraphs, and ntutonstc fuzzy dgraphs n decson support systems. Fuzzy graph theory s fndng an ncreasng number of applcatons n modelng real tme systems where the level of nformaton nherent n the system vares wth dfferent levels of precson. Fuzzy models are becomng useful because of ther am n reducng the dfferences between the tradtonal numercal models used n engneerng and scences and the symbolc models used n expert systems. Intutonstc fuzzy set has got an advantage over fuzzy set because of ts addtonal component whch explans the defcency of knowledge n assgnng the degree of membershp to an object because there s a far chance of the exstence of a non-zero hestaton part at each moment of evaluaton of anythng. The advantages of ntutonstc fuzzy sets and graphs are that they gve more accuracy nto the problems, reduce the cost of mplementaton and mprove effcency. Intutonstc fuzzy sets are very useful n provdng a flexble model to descrbe uncertanty and vagueness nvolved n decson makng, so ntutonstc fuzzy graphs are playng a substantal role n chemstry, economcs, computer scences, engneerng, medcne and decson makng problems, now a days. Graph structures or generalzed graph structures ntroduced by Sampathkumar n 2006 [19], are a generalzaton of graphs whch s qute useful n studyng sgned graphs and graphs n whch every edge s labeled or colored because they help to study varous relatons and correspondng edges smultaneously. Dnesh and Ramakrshnan [9] ntroduced fuzzy graph structures. In ths paper, we have worked on ntutonstc fuzzy graph structures, some of ther fundamental concepts and propertes due to the mproved nfluence of ntutonstc fuzzy sets and partcular use of graph structures. In ths paper, we ntroduce the concept of an ntutonstc fuzzy graph structure (IFGS). We dscuss certan notons, ncludng ntutonstc fuzzy B -cycles, ntutonstc fuzzy B -trees and φ-complement of an ntutonstc fuzzy graph structure wth several examples. We also present φ-complement of an ntutonstc fuzzy graph structure along wth self-complementary and strong self-complementary ntutonstc fuzzy graph structures. 2. Prelmnares We frst revew some defntons from [19] that are necessary for ths paper. A graph structure G = (U, E 1, E 2,..., E k ), conssts of a non-empty set U together wth relatons E 1, E 2,..., E k on U, whch are mutually dsjont such that each E s rreflexve and symmetrc. If (u, v) E for some, 1 k, we call t an E -edge and wrte t as uv. A graph structure G = (U, E 1, E 2,..., E k ) s complete, f () each edge E, 1 k appears at least once n G ; () between each par of vertces uv n U, uv s an E -edge for some, 1 k. A graph structure G = (U, E 1, E 2,..., E k ) s connected, f the underlyng graph s connected. In a graph structure, E -path between two vertces u and v, s the path

3 INTUITIONISTIC FUZZY GRAPH STRUCTURES 221 whch conssts of only E -edges for some, and smlarly, E -cycle s the cycle, whch conssts of only E -edges for some. A graph structure s a tree, f t s connected and contans no cycle or equvalently the underlyng graph of G s a tree. G s an E -tree, f the subgraph structure nduced by E-edges s a tree. Smlarly, G s a E 1 E 2... E k -tree, f G s a E j -tree for each j, 1 j k. A graph structure s an E -forest, f the subgraph structure nduced by E -edges s a forest,.e., f t has no E -cycles. Let S U, then the subgraph structure S nduced by S, has vertex set S, where two vertces u and v n S are joned by an E -edge f, and only f, they are joned by an E -edge n G for 1 k For some, 1 k, the E -subgraph nduced by S, s denoted by E - S and t has only E -edges jonng the vertces n S. If T s a subset of edge set n G, then subgraph structure T nduced by T has the vertex set, the end vertces n T, and whose edges are those n T. Let G = (U 1, E 1, E 2,..., E m ) and H = (U 2, E 1, E 2,..., E n) be graph structures then G and H are somorphc, f m = n and there exsts a bjecton f : U 1 U 2 and a permutaton φ : {E 1, E 2,..., E n } {E 1, E 2,..., E n}, say E E j, 1, j n, such that for all u, v U 1, uv E mples f(u)f(v) E j. Two graph structures G = (U, E 1, E 2,..., E k ) and H = (U, E 1, E 2,..., E k ), on the same vertex set U, are dentcal, f there exsts a bjecton f : U U, such that for all u and v n U and an E -edge uv n G, f(u)f(v) s an E -edge n H, where 1 k and E E for all. Let φ be a permutaton on {E 1, E 2,..., E k } then the φ-cyclc complement of G denoted by (G ) φc s obtaned by replacng E wth φ(e ) for 1 k. Let G = (U, E 1, E 2,..., E k ) be a graph structure and φ be a permutaton on {E 1, E 2,..., E k }, then G s φ-self complementary, f G s somorphc to (G ) φc, the φ-cyclc complement of G and G s self-complementary, f φ dentty permutaton; G s strong φ-self complementary, f G s dentcal to (G ) φc, the φ-complement of G and G s strong self-complementary, f φ dentty permutaton. Defnton 2.1 ([6]). An ntutonstc fuzzy set (IFS) on an unverse X s an object of the form A = { x, µ A (x), ν A (x) x X}, where µ A (x)( [0, 1]) s called degree of membershp of x A, ν A (x)( [0, 1]) s called degree of nonmembershp of x A, and µ A and ν A satsfy the followng condton: for all x X, µ A (x) + ν A (x) 1. Defnton 2.2 ([6]). An ntutonstc fuzzy relaton R = (µ R (x, y), ν R (x, y)) n an unverse X Y (R(X Y )) s an ntutonstc fuzzy set of the form R = { (x, y), µ R (x, y), ν R (x, y) (x, y) X Y }, where µ R : X Y [0, 1] and ν R : X Y [0, 1]. The ntutonstc fuzzy relaton R satsfes µ R (x, y) + ν R (x, y) 1 for all x, y X.

4 222 M. AKRAM AND R. AKMAL Defnton 2.3 ([9]). Let G = (U, E 1, E 2,..., E k ) be a graph structure and ν, ρ 1, ρ 2,..., ρ k be the fuzzy subsets of U, E 1, E 2,..., E k, respectvely such that 0 ρ (xy) ν(x) ν(y), for all x, y U and = 1, 2,..., k. Then G = (ν, ρ 1, ρ 2,..., ρ k ) s a fuzzy graph structure of G. Defnton 2.4 ([9]). Let G = (ν, ρ 1, ρ 2,..., ρ k ) be a fuzzy graph structure of a graph structure G = (U, E 1, E 2,..., E k ). Then F = (ν, τ 1, τ 2,..., τ k ) s a partal fuzzy spannng subgraph structure of G f, τ ρ for = 1, 2,..., k. Defnton 2.5 ([9]). Let G be a graph structure and G be a fuzzy graph structure of G. If xy supp(ρ ), then xy s sad to be a ρ -edge of G. Defnton 2.6 ([9]). The strength of a ρ -path x 0 x 1... x n of a fuzzy graph structure G s n j=1 ρ (x j 1 x j ) for = 1, 2,..., k. Defnton 2.7 ([9]). In any fuzzy graph structure G, ρ 2 (xy) = ρ ρ (xy) = z {ρ (xz) ρ (zy)}, ρ j (xy) = (ρj 1 ρ )(xy) = z { ρ j 1 (xz) ρ (zy) }, j = 2, 3,..., m, for any m 2. Also ρ (xy) = { ρ j (xy), j = 1, 2,... }. Defnton 2.8 ([9]). G = (ν, ρ 1, ρ 2,..., ρ k ) s a ρ -cycle ff (supp(ν), supp(ρ 1 ), supp(ρ 2 ),..., supp(ρ k )) s a E -cycle. Defnton 2.9 ([9]). G = (ν, ρ 1, ρ 2,..., ρ k ) s a fuzzy ρ -cycle ff (supp(ν), supp(ρ 1 ), supp(ρ 2 ),..., supp(ρ k )) s a E -cycle and there exsts no unque xy n supp(ρ ) such that ρ (xy) = {ρ )(uv) uv supp(ρ )}. Defnton 2.10 ([9]). G = (ν, ρ 1, ρ 2,..., ρ k ) s a fuzzy ρ -tree f t has a partal fuzzy spannng subgraph structure, F = (ν, τ 1, τ 2,..., τ k ) whch s a τ -tree where for all ρ -edges not n F, ρ (xy) < τ (xy). 3. Intutonstc Fuzzy Graph Structures Defnton 3.1. Let {E : = 1, 2,..., n} be a set of rreflexve, symmetrc and mutually dsjont relatons on a non-empty set U. An ntutonstc fuzzy graph structure (IFGS) wth underlyng vertex set U s denoted by Ğs = (A, B 1, B 2,..., B n ), where () A s an ntutonstc fuzzy set of U wth µ A : U [0, 1] and ν A : U [0, 1], namely the degree of membershp and the degree of nonmembershp of x U, respectvely, such that 0 µ A (x) + ν A (x) 1, for all x U.

5 INTUITIONISTIC FUZZY GRAPH STRUCTURES 223 () Each B s an ntutonstc fuzzy set of E such that the functons µ B : E [0, 1] and ν B : E [0, 1] are defned by and µ B (xy) µ A (x) µ A (y), ν B (xy) ν A (x) ν A (y) 0 µ B (xy) + ν B (xy) 1, for all xy U U, = 1, 2,..., n. Equvalently, an IFGS of a graph structure may be defned n the followng way. Let G = (U, E 1, E 2,..., E n ) be a graph structure and let A, B 1, B 2,..., B n 1 andb n be ntutonstc fuzzy subsets of U, E 1, E 2,..., E n 1 and E n, respectvely. Then Ğ s = (A, B 1, B 2,..., B n ) s called an IFGS of G, f µ B (xy) µ A (x) µ A (y), ν B (xy) ν A (x) ν A (y), for all xy E, = 1, 2,..., n, and µ B (xy) + ν B (xy) 1, for all xy U U. Example 3.1. Let G = (U, E 1, E 2 ) be a graph structure such that U = {a 1, a 2, a 3, a 4 }, E 1 = {a 1 a 2, a 2 a 3 } and E 2 = {a 3 a 4, a 1 a 4 }. Let A, B 1 andb 2 be ntutonstc fuzzy subsets of U, E 1 and E 2, respectvely, such that A = {(a 1, 0.5, 0.2), (a 2, 0.7, 0.3), (a 3, 0.4, 0.3), (a 4, 0.7, 0.3)}, B 1 = {(a 1 a 2, 0.5, 0.3), (a 2 a 3, 0.4, 0.3)}, and B 2 = {(a 3 a 4, 0.4, 0.3), (a 1 a 4, 0.1, 0.2)}. Then Ğs = (A, B 1, B 2 ) s an IFGS of G as shown n Fg. 1. Fgure 1. IFGS Ğs = (A, B 1, B 2 ) Defnton 3.2. An IFGS H s = (C, D 1, D 2,..., D n ) s sad to be an ntutonstc fuzzy subgraph structure of an IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, f C A and D C for all, that s µ C (x) µ A (x), ν C (x) ν A (x), for all x U,

6 224 M. AKRAM AND R. AKMAL and for = 1, 2,..., n µ D (xy) µ B (xy), ν D (xy) ν B (xy), for all xy U U. H s s called an ntutonstc fuzzy spannng subgraph structure of an IFGS Ğs, f C = A. H s s called an ntutonstc fuzzy partal spannng subgraph structure of an IFGS Ğ s, f t excludes some edges of Ğ s. s s P s Fgure 2. Intutonstc Fuzzy Subgraph Structures Example 3.2. Consder an IFGS Ğs = (A, B 1, B 2 ), as shown n Fg. 1. Let C = {(a 1, 0.4, 0.4), (a 2, 0.0, 0.4), (a 3, 0.4, 0.3), (a 4, 0.6, 0.4)}, D 1 = {(a 1 a 2, 0, 0.4), (a 2 a 3, 0, 0.4)}, D 2 = {(a 3 a 4, 0.3, 0.4), (a 1 a 4, 0.1, 0.4)},

7 INTUITIONISTIC FUZZY GRAPH STRUCTURES 225 C 1 = {(a 1 a 2, 0.3, 0.3), (a 2 a 3, 0.4, 0.3)}, C 2 = {(a 3 a 4, 0.3, 0.3), (a 1 a 4, 0.1, 0.3)}, F 1 = {(a 1 a 2, 0.5, 0.3), (a 2 a 3, 0.4, 0.3)}, and F 2 = {(a 1 a 4, 0.1, 0.3)}. By routne calculatons, t s easy to see that H s = (C, D 1, D 2 ), J s = (A, C 1, C 2 ) and K s = (A, F 1, F 2 ) are respectvely the ntutonstc fuzzy subgraph structure, ntutonstc fuzzy spannng subgraph structure and ntutonstc fuzzy partal spannng subgraph structure of Ğ s. Ther respectve drawngs are shown n Fg. 2. Defnton 3.3. Let Ğs = (A, B 1, B 2,..., B n ) be an IFGS wth underlyng vertex set U. Then there s a B -edge between two vertces x and y of U, f one of the followng s true: () µ B (xy) > 0 and ν B (xy) > 0, () µ B (xy) > 0 and ν B (xy) = 0, () µ B (xy) = 0 and ν B (xy) > 0, for some. Defnton 3.4. For an ntutonstc fuzzy graph structure Ğs = (A, B 1, B 2,..., B n ) wth vertex set U, support of B s gven by: supp(b ) = {xy U U : µ B (xy) 0 or ν B (xy) 0}, = 1, 2,..., n. Defnton 3.5. B -path of an IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, s a sequence of dstnct vertces v 1, v 2,..., v m U (except the choce v m = v 1 ), such that v j 1 v j s a B -edge for all j = 2, 3,..., m. Defnton 3.6. In an IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, two vertces x and y of U are sad to be B -connected, f they are joned by a B -Path, for some {1, 2, 3,..., n}. Defnton 3.7. An IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, s sad to be B -strong, f for all B -edges xy for some {1, 2, 3,..., n}. µ B (xy) = µ A (x) µ A (y), ν B (xy) = ν A (x) ν A (y), Example 3.3. Consder the IFGS Ğs = (A, B 1, B 2 ), as shown n Fg. 1. Then () a 1 a 2, a 2 a 3 are B 1 -edges and a 3 a 4, a 1 a 4 are B 2 -edges; () a 1 a 2 a 3 and a 3 a 4 a 1 are B 1 - and B 2 -paths, respectvely; () a 1 and a 3 are B 1 -connected vertces of U; (v) Ğs s B 1 -strong, snce supp(b 1 ) = {a 1 a 2, a 2 a 3 } and µ B1 (a 1 a 2 ) = 0.5 = (µ A (a 1 ) µ A (a 2 )), ν B1 (a 1 a 2 ) = 0.3 = (ν A (a 1 ) ν A (a 2 )),

8 226 M. AKRAM AND R. AKMAL µ B1 (a 2 a 3 ) = 0.4 = (µ A (a 2 ) µ A (a 3 )), and ν B1 (a 2 a 3 ) = 0.3 = (ν A (a 2 ) ν A (a 3 )). Defnton 3.8. An IFGS Ğs = (A, B 1, B 2,..., B n ) s sad to be strong, f t s B -strong for all {1, 2, 3,..., n}. Defnton 3.9. An IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, s called complete or B 1 B 2... B n -complete f () Ğs s a strong IFGS; () supp(b ) for all = 1, 2, 3,..., n; () For each par of vertces x, y U, xy s a B -edge for some. Example 3.4. Let Ğs = (A, B 1, B 2 ) shown n Fg. 3, be IFGS of the graph structure G = (U, E 1, E 2 ) where U = {a 1, a 2, a 3, a 4 }, E 1 = {a 1 a 3, a 3 a 4, a 1 a 4 } and E 2 = {a 1 a 2, a 2 a 3, a 2 a 4 }. Then Ğs s a strong IFGS snce t s both B 1 -strong and B 2 -strong. Fgure 3. IFGS Ğs = (A, B 1, B 2 ) Moreover supp(b 1 ), supp(b 2 ), every par of vertces belongng to U, s ether a B 1 -edge or a B 2 -edge, so Ğs s a complete or B 1 B 2 -complete IFGS as well. Defnton In an IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U, µ B - and ν B -strengths of a B -path B P = v 1 v 2... v m, are denoted by B δ.p and B.P, respectvely, such that m m B δ.p = [µ B (v j 1 v j )] and B.P = [ν B (v j 1 v j )]. j=2 Then we wrte, strength of the path P B = (δ.p B,.P B ). Example 3.5. In Ğs = (A, B 1, B 2 ) shown n Fg. 3, P 1 = a 1 a 3 a 4 a 1 s a B 1 -path and P 2 = a 3 a 2 a 4 s a B 2 -path and δ.p 1 = µ B1 (a 1 a 3 ) µ B1 (a 3 a 4 ) µ B1 (a 4 a 1 ) = = 0.2, j=2

9 INTUITIONISTIC FUZZY GRAPH STRUCTURES 227.P 1 = ν B1 (a 1 a 3 ) ν B1 (a 3 a 4 ) ν B1 (a 4 a 1 ) = = 0.6, δ.p 2 = µ B2 (a 3 a 2 ) µ B2 (a 2 a 4 ) = = 0.3,.P 2 = ν B2 (a 3 a 2 ) ν B2 (a 2 a 4 ) = = 0.6. Thus strength of B 1 -path P 1 = (δ.p 1,.P 1) = (0.2, 0.6), strength of B 2 -path P 2 = (δ.p 2,.P 2) = (0.3, 0.6). Defnton In an IFGS Ğs = (A, B 1, B 2,..., B n ) wth underlyng vertex set U: () µ B -strength of connectedness between x and y, s defned by µ B (xy) = j 1 {µj B (xy)}, where µ j B (xy) = (µ j 1 B o µ B )(xy) for j 2 and µ 2 B (xy) = (µ B o µ B )(xy) = z {µ B (xz) µ B (zy)}; () ν B -strength of connectedness between x and y, s defned by ν B (xy) = j 1 {νj B (xy)}, where ν j B (xy) = (ν j 1 B o ν B )(xy) for j 2 and ν 2 B (xy) = (ν B o ν B )(xy) = z {ν B (xz) ν B (zy)}. Example 3.6. Let Ğs = (A, B 1, B 2 ), as shown n Fg. 4, be IFGS of graph structure G = (U, E 1, E 2 ), such that U = {a 1, a 2, a 3 }, E 1 = {a 1 a 2, a 1 a 3 } and E 2 = {a 2 a 3 }. Snce µ B1 (a 1 a 2 ) = 0.3, µ B1 (a 1 a 3 ) = 0.3, µ B1 (a 2 a 3 ) = 0, therefore µ 2 B 1 (a 1 a 2 ) = (µ B1 oµ B1 )(a 1 a 2 ) = µ B1 (a 1 a 3 ) µ B1 (a 3 a 2 ) = = 0, µ 2 B 1 (a 2 a 3 ) = (µ B1 oµ B1 )(a 2 a 3 ) = µ B1 (a 2 a 1 ) µ B1 (a 1 a 3 ) = = 0.3, µ 2 B 1 (a 1 a 3 ) = (µ B1 oµ B1 )(a 1 a 3 ) = µ B1 (a 1 a 2 ) µ B1 (a 2 a 3 ) = = 0, µ 3 B 1 (a 1 a 2 ) = (µ 2 B 1 oµ B1 )(a 1 a 2 ) = µ 2 B 1 (a 1 a 3 ) µ B1 (a 3 a 2 ) = = 0, µ 3 B 1 (a 2 a 3 ) = (µ 2 B 1 oµ B1 )(a 2 a 3 ) = µ 2 B 1 (a 2 a 1 ) µ B1 (a 1 a 3 ) = = 0, µ 3 B 1 (a 1 a 3 ) = (µ 2 B 1 oµ B1 )(a 1 a 3 ) = µ 2 B 1 (a 1 a 2 ) µ B1 (a 2 a 3 ) = = 0. Fgure 4. IFGS Ğs = (A, B 1, B 2 ) Thus, we have µ B 1 (a 1 a 2 ) = {0.3, 0.0, 0.0} = 0.3, µ B 1 (a 2 a 3 ) = {0.0, 0.3, 0.0} = 0.3, µ B 1 (a 1 a 3 ) = {0.3, 0.0, 0.0} = 0.3.

10 228 M. AKRAM AND R. AKMAL Snce ν B1 (a 1 a 2 ) = 0.7, ν B1 (a 1 a 3 ) = 0.3, ν B1 (a 2 a 3 ) = 0, therefore and Thus, we have ν 2 B 1 (a 1 a 2 ) = (ν B1 oν B1 )(a 1 a 2 ) = ν B1 (a 1 a 3 ) ν B1 (a 3 a 2 ) = = 0.3, ν 2 B 1 (a 2 a 3 ) = (ν B1 oν B1 )(a 2 a 3 ) = ν B1 (a 2 a 1 ) ν B1 (a 1 a 3 ) = = 0.7, ν 2 B 1 (a 1 a 3 ) = (ν B1 oν B1 )(a 1 a 3 ) = ν B1 (a 1 a 2 ) ν B1 (a 2 a 3 ) = = 0.7, ν 3 B 1 (a 1 a 2 ) = (ν 2 B 1 oν B1 )(a 1 a 2 ) = ν 2 B 1 (a 1 a 3 ) ν B1 (a 3 a 2 ) = = 0.7, ν 3 B 1 (a 2 a 3 ) = (ν 2 B 1 oν B1 )(a 2 a 3 ) = ν 2 B 1 (a 2 a 1 ) ν B1 (a 1 a 3 ) = = 0.3, ν 3 B 1 (a 1 a 3 ) = (ν 2 B 1 oν B1 )(a 1 a 3 ) = ν 2 B 1 (a 1 a 2 ) ν B1 (a 2 a 3 ) = = 0.3, ν 4 B 1 (a 1 a 2 ) = (ν 3 B 1 oν B1 )(a 1 a 2 ) = ν 3 B 1 (a 1 a 3 ) ν B1 (a 3 a 2 ) = = 0.3, ν 4 B 1 (a 2 a 3 ) = (ν 3 B 1 oν B1 )(a 2 a 3 ) = ν 3 B 1 (a 2 a 1 ) ν B1 (a 1 a 3 ) = = 0.7, ν 4 B 1 (a 1 a 3 ) = (ν 3 B 1 oν B1 )(a 1 a 3 ) = ν 3 B 1 (a 1 a 2 ) ν B (a 2 a 3 ) = = 0.7. ν B 1 (a 1 a 2 ) = {0.7, 0.3, 0.7, 0.3} = 0.7, ν B 1 (a 2 a 3 ) = {0.0, 0.7, 0.3, 0.7} = 0.7, ν B 1 (a 1 a 3 ) = {0.3, 0.7, 0.3, 0.7} = 0.7. By smlar calculatons, t can be easly checked that µ B 2 (a 1 a 2 ) = 0, µ B 2 (a 2 a 3 ) = 0.5, µ B 2 (a 1 a 3 ) = 0, ν B 2 (a 1 a 2 ) = 0.3, ν B 2 (a 2 a 3 ) = 0.3, ν B 2 (a 1 a 3 ) = 0.3. Defnton An IFGS Ğs = (A, B 1, B 2,..., B n ) of a graph structure G = (U, E 1, E 2,..., E n ) s a B -cycle, f G s an E -cycle. Defnton An IFGS Ğs = (A, B 1, B 2,..., B n ) of a graph structure G = (U, E 1, E 2,..., E n ) s an ntutonstc fuzzy B -cycle for some, f followng condtons hold: () Ğs s a B -cycle; () There s no unque B -edge uv n Ğs, such that µ B (uv) = mn{µ B (xy) : xy E = supp(b )} or ν B (uv) = max{ν B (xy) : xy E = supp(b )}. Example 3.7. IFGS Ğs = (A, B 1, B 2 ) shown n Fg. 3, s a B 1 -cycle as well as ntutonstc fuzzy B 1 -cycle, snce (supp(a), supp(b 1 ), supp(b 2 )) s an E 1 -cycle and there are two B 1 -edges wth mnmum degree of membershp and two B 1 -edges wth maxmum degree of nonmembershp of all B 1 -edges. Defnton An IFGS Ğs = (A, B 1, B 2,..., B n ) of a graph structure G = (U, E 1, E 2,..., E n ) s a B -tree, f (supp(a), supp(b 1 ), supp(b 2 ),..., supp(b n )) s an E -tree. In other words, Ğ s s a B -tree f the subgraph of Ğ s, nduced by supp(b ), forms a tree.

11 INTUITIONISTIC FUZZY GRAPH STRUCTURES 229 Defnton An IFGS Ğs = (A, B 1, B 2,..., B n ) of a graph structure G = (U, E 1, E 2,..., E n ) s an ntutonstc fuzzy B -tree (ntutonstc fuzzy B -forest), f Ğ s has an ntutonstc fuzzy partal spannng subgraph structure H s = (A, C 1, C 2,..., C n ), such that H s s a C -tree (C -forest) and µ B (xy) < µ C (xy) and ν B (xy) < ν C (xy) for all B edges not n H s. Example 3.8. The IFGS, shown n Fg. 3, s a B 2 -tree but not an ntutonstc fuzzy B 2 -tree. Whle IFGS Ğs = (A, B 1, B 2 ), shown n Fg. 5, s not a B 1 -tree but an ntutonstc fuzzy B 1 -tree, snce t has an ntutonstc fuzzy partal spannng subgraph structure (A, B 1, B 2) as a B 1 -tree, whch s obtaned by deletng B 1 -edge a 1 a 4 from Ğ s, wth µ B1 (a 1 a 4 ) = 0.3 < 0.4 = µ B (a 1 1 a 4 ) and ν B1 (a 1 a 4 ) = 0.5 < 0.6 = ν B 1(a 1 a 4 ). Fgure 5. IFGS Ğs = (A, B 1, B 2 ) Defnton An IFGS G s1 = (A 1, B 11, B 12,..., B 1n ) of GS G 1 = (U 1, E 11, E 12,..., E 1n ) s somorphc to an IFGS G s2 = (A 2, B 21, B 22,..., B 2n ) of G 2 = (U 2, E 21, E 22,..., E 2n ), f there exst a bjecton f : U 1 U 2 and a permutaton φ on the set {1, 2,..., n}, such that: and for φ() = j µ A1 (u 1 ) = µ A2 (f(u 1 )), ν A1 (u 1 ) = ν A2 (f(u 1 )), for all u 1 U 1 µ B1 (u 1 u 2 ) = µ B2j (f(u 1 )f(u 2 )), ν B1 (u 1 u 2 ) = ν B2j (f(u 1 )f(u 2 )), for all u 1 u 2 E 1, = 1, 2,..., n. Defnton An IFGS G s1 = (A 1, B 11, B 12,..., B 1n ) of GS G 1 = (U, E 11, E 12,..., E 1n ) s dentcal to an IFGS G s2 = (A 2, B 21, B 22,..., B 2n ) of G 2 = (U, E 21, E 22,..., E 2n ), f there exst a bjecton f : U U, such that: µ A1 (u) = µ A2 (f(u)), ν A1 (u) = ν A2 (f(u)), for all u U and µ B1 (u 1 u 2 ) = µ B2 (f(u 1 )f(u 2 )), ν B1 (u 1 u 2 ) = ν B2 (f(u 1 )f(u 2 )), for all u 1 u 2 E 1, = 1, 2,..., n.

12 230 M. AKRAM AND R. AKMAL Example 3.9. Ğs1 and Ğs2, as shown n Fg. 6 and Fg. 7, are IFGSs of graph structures G 1 = (U 1, E 1, E 2, E 3, E 4 ) and G 2 = (U 2, E 1, E 2, E 3, E 4), respectvely, where U 1 = {a 1, a 2, a 3, a 4, a 5 }, E 1 = {a 1 a 2, a 2 a 5 }, E 2 = {a 2 a 3, a 2 a 4 }, E 3 = {a 1 a 3, a 4 a 5 }, E 4 = {a 1 a 5, a 3 a 4 }, U 2 = {b 1, b 2, b 3, b 4, b 5 }, E 1 = {b 2 b 4, b 3 b 4 }, E 2 = {b 1 b 4, b 4 b 5 }, E 3 = {b 1 b 2, b 3 b 5 }, E 4 = {b 1 b 5, b 2 b 3 }. Fgure 6. IFGS Ğs1 = (A 1, B 1, B 2, B 3, B 4 ) Then Ğs1 s somorphc to Ğs2 under the mappng f : U 1 U 2, gven by f(a 1 ) = b 5, f(a 2 ) = b 4, f(a 3 ) = b 3, f(a 4 ) = b 2, f(a 5 ) = b 1, and a permutaton φ gven by such that for all a U 1, and φ(1) = 2, φ(2) = 1, φ(3) = 3, φ(4) = 4, µ A1 (a ) = µ A2 (f(a )), ν A1 (a ) = ν A2 (f(a )) µ Bk (a a j ) = µ Bφ(k) (f(a )f(a j )), ν Bk (a a j ) = ν Bφ(k) (f(a )f(a j )), for all a a j E k, k = 1, 2, 3, 4. Also, Ğ s1 s dentcal wth Ğs2 under the mappng f : U 1 U 2, gven by such that f(a 1 ) = b 3, f(a 2 ) = b 4, f(a 3 ) = b 5, f(a 4 ) = b 1, f(a 5 ) = b 2, µ A1 (a ) = µ A2 (f(a )), ν A1 (a ) = ν A2 (f(a )),

13 INTUITIONISTIC FUZZY GRAPH STRUCTURES 231 for all a U 1, and µ Bk (a a j ) = µ B k (f(a )f(a j )), ν Bk (a a j ) = ν B k (f(a )f(a j )), for all a a j E k, k = 1, 2, 3, 4. Fgure 7. IFGS Ğs2 = (A 2, B 1, B 2, B 3, B 4) Remark 3.1. Identcal IFGSs are always somorphc but the converse s not necessarly true. As IFGS shown n Fg. 3 s somorphc to IFGS shown n Fg. 8 but they are not dentcal. Fgure 8. IFGS G s1 = (A 1, B 1, B 2) Defnton Let Ğs = (A, B 1, B 2,..., B n ) be an ntutonstc fuzzy graph structure of a graph structure G = (U, E 1, E 2,..., E n ). Let φ denote a permutaton on the set {E 1, E 2,..., E n } and the correspondng permutaton on {B 1, B 2,..., B n },.e., φ(b ) = B j ff φ(e ) = E j for all. If xy B r for some r and µ B φ(xy) = µ A (x) µ A (y) µ φbj (xy), j

14 232 M. AKRAM AND R. AKMAL ν B φ(xy) = ν A (x) ν A (y) ν φbj (xy), = 1, 2,..., n, j then xy Bm, φ whle m s chosen such that µ B φ m (xy) µ B φ(xy) and ν B φ m (xy) ν B φ(xy) for all. Then IFGS (A, B φ 1, B φ 2,..., Bn) φ denoted by Ğφc s, s called the φ-complement of IFGS Ğ s. Theorem 3.1. A φ-complement of an ntutonstc fuzzy graph structure s always a strong IFGS. Moreover, f φ() = r for r, {1, 2,..., n}, then all B r -edges n IFGS Ğ s = (A, B 1, B 2,..., B n ) become B φ -edges n Ğφc s = (A, B φ 1, B φ 2,..., Bn). φ Proof. Frst part s obvous from the defnton of φ-complement Ğφc s of IFGS Ğs, snce for any B φ -edge xy, µφ B (xy) and ν φ B (xy) respectvely have the maxmum values of (3.1) [µ A (x) µ A (y)] j That s, µ φbj (xy) and [ν A (x) ν A (y)] j ν φbj (xy). (3.2) µ φ B (xy) = µ A (x) µ A (y), ν φ B (xy) = ν A (x) ν A (y), for all edges xy n Ğφc s, hence Ğφc s s always a strong IFGS. Now suppose on contrary that φ() = r but xy s a B s -edge n Ğs wth s r, whch mples that φb B s. Comparng expressons (3.1) and (3.2), we get µ φbj (xy) = 0, ν φbj (xy) = 0, j whch s not possble because B s = φb j for some j {1, 2,..., 1, + 1,..., n}. So our supposton s wrong and xy must be a B r -edge. Hence we can conclude that f φ() = r, then all B r -edges n IFGS Ğs = (A, B 1, B 2,..., B n ) become B φ -edges n Ğ φc s = (A, B φ 1, B φ 2,..., Bn) φ for r, {1, 2,..., n}. Example Consder IFGS Ğs = (A, B 1, B 2 ) shown n Fg. 4 and let φ be a permutaton on the set {B 1, B 2 } such that φ(b 1 ) = B 2 and φ(b 2 ) = B 1. Now for a 1 a 2 B 1, µ φ B 1 (a 1 a 2 ) = µ A (a 1 ) µ A (a 2 ) j 1[µ φbj (a 1 a 2 )] = [µ φb2 (a 1 a 2 )] j = 0.3 µ B1 (a 1 a 2 ) = = 0, ν φ B 1 (a 1 a 2 ) = ν A (a 1 ) ν A (a 2 ) j 1[ν φbj (a 1 a 2 )] = [ν φb2 (a 1 a 2 )] = 0.7 ν B1 (a 1 a 2 ) = = 0, µ φ B 2 (a 1 a 2 ) = µ A (a 1 ) µ A (a 2 ) j 2[µ φbj (a 1 a 2 )] = [µ φb1 (a 1 a 2 )]

15 INTUITIONISTIC FUZZY GRAPH STRUCTURES 233 = 0.3 µ B2 (a 1 a 2 ) = = 0.3, ν φ B 2 (a 1 a 2 ) = ν A (a 1 ) ν A (a 2 ) j 2[ν φbj (a 1 a 2 )] = [ν φb1 (a 1 a 2 )] = 0.7 ν B2 (a 1 a 2 ) = = 0.7. Clearly, µ φ B 2 (a 1 a 2 ) = 0.3 > 0 = µ φ B 1 (a 1 a 2 ) and ν φ B 2 (a 1 a 2 ) = 0.7 > 0 = ν φ B 1 (a 1 a 2 ), so a 1 a 2 B φ 2. Smlarly for a 1 a 3 B 1, Fgure 9. IFGS Ğφc s = (A, B φ 1, B φ 2 ) µ φ B 1 (a 1 a 3 ) = 0, ν φ B 1 (a 1 a 3 ) = 0.4, µ φ B 2 (a 1 a 3 ) = 0.3, ν φ B 2 (a 1 a 3 ) = 0.7. Clearly, µ φ B 2 (a 1 a 3 ) = 0.3 > 0 = µ φ B 1 (a 1 a 3 ) and ν φ B 2 (a 1 a 3 ) = 0.7 > 0.4 = ν φ B 1 (a 1 a 3 ), so a 1 a 3 B φ 2. And for a 2 a 3 B 2 µ φ B 1 (a 2 a 3 ) = 0.5, ν φ B 1 (a 2 a 3 ) = 0.4, µ φ B 2 (a 2 a 3 ) = 0, ν φ B 2 (a 2 a 3 ) = 0.1, that s, µ φ B 1 (a 2 a 3 ) = 0.5 > 0 = µ φ B 2 (a 2 a 3 ) and ν φ B 1 (a 2 a 3 ) = 0.4 > 0.1 = ν φ B 2 (a 2 a 3 ), so a 2 a 3 B φ 1. Ths mples that B φ 1 = {(a 2 a 3, 0.5, 0.4)}, B φ 2 = {(a 1 a 2, 0.3, 0.7), (a 1 a 3, 0.3, 0.7)} and Ğ φc s = (A, B φ 1, B φ 2 ) shown n Fg. 9 s the φ-complement of Ğ s. Defnton Let Ğs = (A, B 1, B 2,..., B n ) be an IFGS and φ be a permutaton on the set {1, 2,..., n}. Then () Ğs s self-complementary, f t s somorphc to Ğφc s, the φ-complement of Ğ s. () Ğs s strong self-complementary, f t s dentcal to Ğφc s. () Ğs s totally self-complementary, f t s somorphc to Ğφc s, the φ-complement of Ğ s for all permutatons φ on the set {1, 2,..., n}. (v) Ğs s totally strong self-complementary, f t s dentcal to Ğφc s, the φ-complement of Ğ s for all permutatons φ on the set {1, 2,..., n}. Theorem 3.2. An IFGS Ğs s strong f and only f Ğs s totally self-complementary.

16 234 M. AKRAM AND R. AKMAL Proof. Let Ğs be a strong IFGS and φ be any permutaton on the set {1, 2,..., n}. If φ 1 () = j, then by Theorem 3.1, all B -edges n Ğs = (A, B 1, B 2,..., B n ) become -edges n Ğφc s = (A, B φ 1, B φ 2,..., Bn). φ Also Ğφc s s strong, so B φ j µ B (a 1 a 2 ) = µ A (a 1 ) µ A (a 2 ) = µ B φ(a 1 a 2 ), j ν B (a 1 a 2 ) = ν A (a 1 ) ν A (a 2 ) = ν B φ(a 1 a 2 ). j Then Ğs s somorphc to Ğφc s, under the dentty mappng f : U U and a permutaton φ [φ 1 () = j,, j = 1, 2,..., n], such that and µ A (a) = µ A (f(a)), ν A (a) = ν A (f(a)), for all a U µ B (a 1 a 2 ) = µ B φ j ν B (a 1 a 2 ) = ν B φ j (a 1 a 2 ) = µ B φ(f(a 1 )f(a 2 )), j (a 1 a 2 ) = ν B φ j (f(a 1 )f(a 2 )), for all a 1 a 2 E. Ths holds for all permutatons on the set {1, 2,..., n}. Hence Ğs s totally selfcomplementary. Conversely, let φ be any permutaton on the set {1, 2,..., n} and Ğs and Ğφc s be somorphc. From the defnton of φ-complement and somorphsm of IFGSs, we have µ B (a 1 a 2 ) = µ B φ(f(a 1 )f(a 2 )) = µ A (f(a 1 )) µ A (f(a 2 )) = µ A (a 1 ) µ A (a 2 ), j ν B (a 1 a 2 ) = ν B φ(f(a 1 )f(a 2 )) = ν A (f(a 1 )) µ A (f(a 2 )) = ν A (a 1 ) ν A (a 2 ), j for all a 1 a 2 E, = 1, 2,..., n. Hence, Ğ s s a strong IFGS. Remark 3.2. Every self-complementary IFGS s necessarly totally self-complementary. Theorem 3.3. If graph structure G = (U, E 1, E 2,..., E n ) s totally strong selfcomplementary and A s an IFS of U wth constant fuzzy mappngs µ A and ν A then a strong IFGS Ğs = (A, B 1, B 2,..., B n ) of G s totally strong self-complementary. Proof. Consder a strong IFGS Ğs = (A, B 1, B 2,..., B n ) of a graph structure G = (U, E 1, E 2,..., E n ). Suppose that G s totally strong self-complementary and that for some constants s, t [0, 1], A = (µ A, ν A ) s an IFS of U such that µ A (u) = s, ν A (u) = t, for all u U. Then we have to prove that Ğs s totally strong self-complementary. Let φ be an arbtrary permutaton on the set {1, 2,..., n} and φ 1 (j) =. Snce G s totally strong self-complementary, so there exsts a bjecton f : U U, such that for every E -edge a 1 a 2 n G, f(a 1 )f(a 2 ) (an E j -edge n G ) s an E -edge n (G ) φ 1c. Consequently, for every B -edge a 1 a 2 n Ğs, f(a 1 )f(a 2 ) (a B j -edge n Ğs) s a B φ -edge n Ğφc s. From the defnton of A and the defnton of strong IFGS Ğs µ A (a) = s = µ A (f(a)), ν A (a) = t = ν A (f(a)), for all a, f(a) U,

17 INTUITIONISTIC FUZZY GRAPH STRUCTURES 235 and µ B (a 1 a 2 ) = µ A (a 1 ) µ A (a 2 ) = µ A (f(a 1 )) µ A (f(a 2 )) = µ B φ(f(a 1 )f(a 2 )), ν B (a 1 a 2 ) = ν A (a 1 ) ν A (a 2 ) = ν A (f(a 1 )) ν A (f(a 2 )) = ν B φ(f(a 1 )f(a 2 )), for all a 1 a 2 B, = 1, 2,..., n, whch shows Ğs s strong self-complementary. Hence Ğ s s totally strong self-complementary, snce φ s arbtrary. Remark 3.3. Converse of Theorem 3.3 s not necessary, snce a totally strong selfcomplementary and strong IFGS Ğs = (A, B 1, B 2, B 3 ) as shown n Fg. 11, has a totally strong self-complementary underlyng graph structure but µ A and ν A are not constant fuzzy functons. Example The IFGS shown n Fg. 8 s self-complementary,.e., t s somorphc to ts φ-complement, where φ = (1 2). Also, t s totally self-complementary because φ s the only non-dentty permutaton on set {1, 2}. Example The IFGS Ğs = (A 1, B 1, B 2, B 3, B 4 ) shown n Fg. 10, s strong selfcomplementary,.e., t s dentcal to ts φ-complement where the permutaton φ s (1 2) (3 4). It s not totally strong self-complementary. P Fgure 10. IFGS Ğs = (A 1, B 1, B 2, B 3, B 4 ) Example The IFGS Ğs = (A 1, B 1, B 2, B 3 ), shown n Fg. 11, s totally strong self-complementary because t s dentcal to ts φ-complement for all the permutatons φ on the set {1, 2, 3}.

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19 INTUITIONISTIC FUZZY GRAPH STRUCTURES 237 [17] G. Pas, R. Yager and K. T. Atanassov, Intutonstc fuzzy graph nterpretatons of mult-person mult-crtera decson makng: generalzed net approach, In: Proceedngs of the 2nd IEEE Internatonal Conference on Intellgent Systems, Vol. 2, IEEE, pp , [18] A. Rosenfeld, Fuzzy graphs, fuzzy sets and ther applcatons to cogntve and decson processes, In: Proceedngs of the US-Japan Semnar on Fuzzy Sets and ther Applcatons, Academc Press, pp , [19] E. Sampathkumar, Generalzed graph structures, Bull. Kerala Math. Assoc. 3(2) (2006), [20] A. Shannon, K. T. Atanassov, A frst step to a theory of the ntutonstc fuzzy graphs, In: Proceedng of the Frst Workshop on Fuzzy Based Expert Systems, pp , [21] L. A. Zadeh, Fuzzy sets, Informaton and Control 8 (1965), [22] L. A. Zadeh, Smlarty relatons and fuzzy orderngs, Informaton Scences 3(2) (1971), Department of Mathematcs, Unversty of the Punjab, New Campus, Lahore, Pakstan E-mal address: m.akram@puct.edu.pk 2 Department of Mathematcs, Unversty of the Punjab, New Campus, Lahore, Pakstan E-mal address: raba.akmal@ymal.com