# Operations on Intuitionistic Fuzzy Graph StructuresAcademic research paper on "Mathematics"

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## Abstract of research paper on Mathematics, author of scientific article — Muhammad Akram, Rabia Akmal

Abstract An intuitionistic fuzzy graph structure (IFGS) is a generalization of an intuitionistic fuzzy graph. The concept of intuitionistic fuzzy graph structure is introduced and investigated in this paper. Some operations including union, join, Cartesian product, cross product, lexicographic product, strong product and composition on intuitionistic fuzzy graph structures are defined and elaborated with a number of examples. Some basic properties of these operations are also presented.

## Academic research paper on topic "Operations on Intuitionistic Fuzzy Graph Structures"

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Fuzzy Information and Engineering

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Fuzzy Information and Engineering

Operations on Intuitionistic Fuzzy Graph Structures

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Received: 16 September 2015 / Revised: 22 March 2016 / Accepted: 22 July 2016 /

Abstract An intuitionistic fuzzy graph structure (IFGS) is a generalization of an intuitionistic fuzzy graph. The concept of intuitionistic fuzzy graph structure is introduced and investigated in this paper. Some operations including union, join, Cartesian product, cross product, lexicographic product, strong product and composition on intuitionistic fuzzy graph structures are defined and elaborated with a number of examples. Some basic properties of these operations are also presented.

Keywords Intuitionistic fuzzy graph structure • Union • Join • Composition • Cartesian (Cross, Lexicographic, Strong) product

1. Introduction

A fuzzy set introduced by Zadeh [1] gives the degree of membership of an object in a given set. In 1983, Atanassov [2] widened an idea of fuzzy sets and introduced the concept of intuitionistic fuzzy sets. He added a new component, degree of non-membership, in the definition of a fuzzy set with the requirement that sum of two degrees must be less or equal to one. Research on the theory of intuitionistic fuzzy sets has been scoring an exponential growth in mathematics and its applications. Kauf-mann's initial definition of a fuzzy graph in [3], was based on Zadeh's fuzzy relations

Muhammad Akram (K) • Rabia Akmal

Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan email: makrammath@yahoo.com m.akram@pucit.edu.pk

Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research Society of China.

This is an open access article under the CC BY-NC-ND license

in [4]. The fuzzy relations between fuzzy sets were also considered by Rosenfeld who developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Later on, Bhattacharya [5] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by Mordeson and Peng [6].

Shannon and Atanassov [7] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties in [7]. Parvathi et al. [8] defined operations on intuitionistic fuzzy graphs. Parvathi and Thi-lagavathi [9] introduced intuitionistic fuzzy hypergraphs in 2009. The advantages of intuitionistic fuzzy sets and graphs are that they give more accuracy into the problems and reduce the cost of implementation and improve efficiency. Graph structures (or generalized graph structures), introduced by Sampathkumar [11] in 2006, are a generalization of graphs which is quite useful in studying structures including graphs, signed graphs and graphs in which every edge is labeled or colored. It helps to study various relations and the corresponding edges simultaneously. Every graph can be represented by a graph structure but a graph structure is not, simply a graph. It is the grouping of edges of graph, in case of a colored graph. Dinesh and Ramakrish-nan [12] introduced the notion of a fuzzy graph structure and discussed some related properties.

In this paper, we have introduced an intuitionistic fuzzy graph structure. We have defined some operations, including union, join, Cartesian product, cross product, lexicographic product, strong product and composition on intuitionistic fuzzy graph structures and elaborated with a number of examples. Some basic properties of these operations are also presented.

We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications which are not mentioned in the paper, the readers can refer to [10,13-20].

2. Preliminaries

We first review some definitions from [1,2,10-12].

A graph structure G* = (U, E1,E2, ■■ ■ , Ek) consists of a non-empty set U together with relations Ei, E2, ■ ■ • ,Ek on U, which are mutually disjoint such that each Ei is irreflexive and symmetric. If (u, v) € Ei for some i, 1 < i < k, it is usually written as "uv" and said to be an Ei-edge. A graph structure G* = (V, Ei, E2, ■■• , Ek) is complete, if (i) Each edge Ei, 1 < i < k appears at least once in G*. (ii) Between each pair of vertices uv in V, uv is an Ei-edge for some i, 1 < i < k.

A graph structure G* = (U, Ei, E2, ■ ■ ■ , Ek) is connected if the underlying graph is connected. In a graph structure, Et-path between two vertices u and v, is the path which consists of only Ei-edges for some i. If G* and are two graph structures given by (U, Ei, E2, ■ ■ ■ , Ek) and (V, E[, E'2, ■ ■ ■ , E'k), respectively, then Cartesian product of Gl and denoted by "G^ x G2", is defined by (U x V,Ei x E[,E2 x E'2,--- ,EkxE'k)-wheieEiXE'i = {(uiv,u2v) | v € V, uiu2 € Ei}U{(uvi,uv2) \ u G U, V1V2 £ .Ej}, i = 1,2, • • • ,k. Composition of G* and G\, denoted by "Gj o Gl", is defined by (U o V, Ei o E[, E2 o E'2, ■ ■ ■ , Ek o E'k) where U o V = U x V and EioE[ = {(mv, u2v) | v e V, U1U2 e E{} U {(uvi,uv2) | u e U, V1V2 e

El} U {(uivi, U2V2) I U1U2 S Ei, v\ ± V2}, ¿ = 1,2,--- , k. Union of and G2, denoted by U is defined by (UUV,E1UE'1,E2UE!2, ■ ■ ■ ,EkUE'k) and the join of G\ and G\ is defined by, GJ+G^ = (U+V,E1+E'1,E2+E'2, ■ ■ • ,Ek+E'k) such that U + V = UUV and Et + E[ = Ei \JE[ UE, i = 1,2, - - • ,k, where S is the set consisting of all edges which join vertices of U with the vertices of V.

Definition 2.1 A fuzzy set p, in a universe X is a mapping p : X —¥ [0,1]. A fuzzy relation on X is a fuzzy set uinX x X. Let /ibea fuzzy set in X and v fuzzy relation on X. We call v is a fuzzy relation on p ifv{x, y) < min{/i(:r), p(y)} Wx, y e X.

Definition2.2 AnintuitionisticfuzzysetisanobjectofformA = {(x, pa{x), va{x)) | x e V}, where mappings pa '-V —t [0,1] and va V —» [0,1] denote, respectively, the degrees of membership and non-membership of each element x Ç. V such that + va{x) < 1 for all x € V.

Definition 2.3 LetG* = (U,Ei,E2, - ■ • be a graph structure and v,py,p2,-■ • , Pk be the fuzzy subsets of U, E\, E2, ■ • ■ ,Ek, respectively, such that 0 < Piixy) < p(x) A p(y)forall x, y G V andi = 1,2, ■ ■ • , k. Then G = (f, pi,p2, • • • , Pk) is a fuzzy graph structure of G*.

Definition 2.4 If G\ = (p. pi, P2, ■ ■ ■ , Pk) and G2 = {v,Ti,T2, ■ ■ ■ ,Tk) are the fuzzy graph structures of graph structures G* = (U, E\, E2, • • ■ , Ek) and G\ = (V, E[, E'2, ■ ■ ■ ,E'k), respectively, then Cartesian product ofG\ and G2, denoted by G1 x G2, is defined by (p x u, p\ x n, p2 x T2, • • • , Pk x t*) such that

• (jBX v)(uv) = p(u) A v(v) Vuv eU xV,

• {pi x Ti)(uiv,u2v) = u(v) A pi(u 1U2) VveV andu\u2 € Ei:V i,

• (pi x Ti)(uvi, UV2) = p(u) An(viv2) Vu e U andv\v2 € E<,Vi.

Definition 2.5 If Gi = (p., pi, P2, ■ ■ ■ , Pk) and G2 = (v, ti,T2, ■ ■ • ,Tk) are the fuzzy graph structures of graph structures G\ = (U, E\, E2, ■ ■ • , Ek) and G2 = (V, E[, E'2, ■ ■ • , E'k), respectively, then composition of G \ and G2, denoted by G\ o G2, is defined by (p. o v, pi o T\,p2 o T2, ■ ■ ■ ,Pk°Tk) such that

• (/i o i>)(uv) = p,(u) A v(v) V uv et/ x V,

• (Pi 0 Ti)(uiv,u2v) = u(v) A Pi(u iu2) Vv€V andu\ii2 e Ei,Vi,

• (pi ° n)(uvi,uv2) = p(u) A n(viv2) Vu € U andv\v2 € Ei,Vi

• (Pi°T~i)(u ivi,u2v2) = v(vi)Av(v2)Api(uiu2) Vvi,v2 £ V such that v\ ^ V2 and U\U2 € Ei, V i.

Definition 2.6 IfGi = (p,, plt p2, ■ ■ ■ , pk) and G2 = {v,ti,t2,- ■ ■ ,tk) are the fuzzy graph structures of graph structures G* = (U, E\, E2, • • ■ , Ek) and G2 = (V,E[,E2,--- , E'k ), respectively, then union of G\ and G2, denoted by G\ U G2, is defined by (p, U v,p\ U n, p2 U r2, • ■ • ,pk Ur^) such that

(p, U v)(w) = p(w) V u(w) VweUUV,

(,Pi u Ti)(uv) = Pi(uv) V Ti(uv) V uv G Ei U E[, i = 1,2, ■ ■ ■ , k.

Note that fi and v are extended toU UV by defining \i as fi(w) in U and as 0 in V for w G U and v as u(w) in V and as 0 in U for w € V. Likewise, pi and Ti are extended on Ei U E[ by defining pi as Pi(uv) in Ei and as 0 in E[ for uv G Ei and Ti as Ti(uv) in El and as 0 in Ei for uv G E[.

Definition 2.7 If G\ = (fi,pi,p2,--- ,pk) and G2 = (u,ti,t2,-■ ■ ,Tk) are the fuzzy graph structures of graph structures G* = (U, E\,E2, ■ • ■ ,Ek) and G2 = (V, E[, E'2, ■ ■ ■ , E'k), respectively, then join of G\ and G2, denoted by G i + G2, is given by (fj, + v, pi + Ti, p2 + t2, ■ ■ ■ ,Pk + Tk) such that

(n + u)(u) = n(u) V u(u) VueUliV,

and fori = 1,2, • • • , k,

(Pi+Ti)(uv) = { (Pi u Ti)(™)'v UV e Ei u

\ (/i(u) A v(v), if uv joins a vertex u of U with a vertex v of V.

3. Operations on Intuitionistic Fuzzy Graph Structures

Definition 3.1 Let G* = (U, Ei, E2, ■■■, En) be a graph structure (GS). Then Gs = (A, B\,B2, ■•■ , Bn) is called an intuitionistic fuzzy graph structure (IFGS) of G* with underlying vertex set U if the following conditions are satisfied:

(i) A is an intuitionistic fuzzy set on U with fiA ■ U —> [0,1] and vA : U —» [0,1], the degree of membership and the degree of nonmembership of x G U, respectively, such that

o < (j.a(x) + vA(x) <lVxeU.

(ii) Bi is an intuitionistic fuzzy set on Ei for all i and the membership functions fiBi ■ Ei —> [0,1] and VBi ■ Ei —» [0,1] are restricted by

VBiixy) < Ha{x) A fiA{y) , vBi(xy) < vA(x) V vA(y)

such that

0 < UBiixy) +VBi{xy) < 1 M xy G Ei C U x U, i = 1,2,-•• ,n.

We illustrate the concept of an intuitionistic fuzzy graph structure with an example.

Example3.1 LetG* = (f/, i?i, i?2 ) be a GS such that U = {ai, a2, a-j, a^, a^}, E\ = {aia4,0,30,5} and E2 = {0103, aia2, a2a\$, 0,3(14, a2ai, «5(14}. Let A, B\ and B2 be intuitionistic fuzzy subsets of U, E\ and E2, respectively, such that A = {(01, .2, .3), (a2, .3, .3), (a3, .6, .4), (a4, .4, .6), (aB, .3, .3)}, B1 = {(aia4, .2, .6),

(o3a5, .3, .4)} and B2 = {(aia3, .2, .4), (oio2, .2, .3), (a2a3, .3, .4), (o3a4, .4, .6), («2£«4, .3, .2), (0504, .3, .2)}. It is easy to see from simple calculations that Gs = (A, B\, B2) is an IFGS of G* as shown in Fig.l.

ai (0.2,0.3)

Fig. 1 IFGS Gs = (A. B1,B2)

Definition 3.2 LetG„ 1 = {A1,Bn, B12, ■ ■ ■ , Bin) and Gs2 = (A2, B21, B22, ■ ■ ■ , B2n) be respective IFGSs of GSsG\ = (Uu En, E12, ■ ■ ■ ,Eln)andG*2 = {U2,E21, E22, ••• , E2n). The Cartesian product of Gs\ and Gs 2 is defined by Gs 1 x Gs 2 = (Ai x A2,BU x B2i,Bi2 x B22, ■■■ , Bln x B2n), where i AHA1xA2){xy) = (/Mi x flA2){xy) = ft Ax (z) A MA2(y), I v(A1xA2){xy) = {"At x VA2)(xy) = VAAx) V VAi(y), V Xy G Ui x U2,

iM(B„XBM) ((xyi)(xy2)) = (/lBli x P>B2i)((xyi)(xy2)) = fJ,Al (x) A VB2i (2/12/2), ^(SiiXSai) {{xy^xy^) = (uBli X VB2i){(xyi){xy2)) = vaAx) V VB2i(2/12/2),

Vi£i/i, 2/12/2 G E2i,

(i1(Bli-x.B2i){{x\y)(x2y)) = (pBli X liB2i){{xiy){x2y)) = HA2{y) Ap,Bli{xix2), i(xiy){x2y)) = [vBli x VB2i)({xiy){x2y)) = VA2{y) v VBuix^),

Vy G U2, xxx2 G En.

Example 3.2 Let Gsi = (A\,Bn,Bi2) and Gs2 = (A2, B2\, B22) be respective IFGSs of graph structures G* = (U\, En, E\2) and G2 = {U2, E2i, E22), where Ui = {ai,02,a3,a4}, U2 = {6i,62,63}, En = {0102}, Ei2 = {a3a4}, E21 = {6162,6163} and E22 = {6263}. IFGSs Gsi and Gs2 are shown in Fig.2 and their Cartesian product Gsl x Gs2 = {A\ x A2, Bn x S2i, B12 x B22) is shown in Fig.3.

Proposition 3.1 LetG* = (Ui x U2, En x E2i, E12 x E22, ■ ■ ■ ,E\n x E2n) be the Cartesian product ofGSs G\ = (Ui, En, Ei2, ■■■ , Ein) and G2 = {U2, E2i, E22, • ■ • , E2n). Let Gsi = {Ai,Bn,Bi2, ••■ , B\n) andGs2 = {A2, B2i, B22, ■■■ , B2n) be respective IFGSs o/G* and G\\. Then Gs 1 x Gs2 = (Ai x A2, Bn x B2i, B12 x B22, ■ ■ ■ , Bln x B2n) is an IFGS ofG*.

Proof It is quite clear that Ai x A2 is an intuitionistic fuzzy set of Ui x U2 and Bu x B2i is an intuitionistic fuzzy set of En x E2i, for all i. Therefore, to prove that Gsi x Gs2 = (Ai x A2, fln x B21,B12 x B22, ■ ■ ■ , Bln x B2n) is an IFGS

of G*, we have to show that for all i, Bu x B2z is an intuitionistic fuzzy relation on Ai x A2. For this, we discuss some cases as follows:

62(0.5,0.5)

011(0.1,0.2)

01 (0.2,0.2)

<12(0.1,0.2)

Bl2(0.2,0.3)

a3(0.3,0.3)

04(0.4,0.3) bi (0.2,0.3)

fe(0.2,0.1)

Ga 2 — (A2,B2l,B22)

Fig. 2 Two intuitionistic fuzzy graph structures

0162(0.2,0.5) B11 xB2i(0.1,0.5) 0262(0.1,0.5)

0362(0.3,0.5) B13X 022(0.2,0.5) 0,62(0.4,0.5)

Fig. 3 Cartesian product of two IFGSs We now formulate Cartesian product of two IFGSs as a proposition.

Fuzzy Inf. Eng. (2016) 8: 389-410_395

Case 1 When u e U-,, b\b2 £ E2i,

^BlixB2i)((ubi)(ub2)) = fj,Al(u) A Mfl2i (&i &2)

< fiAl (u) A \fiAi (61) A /¿a2 (62)] = \p,Al (w) A fiA„ (61)] A \p,Al (u) A pAl (i>a)] = M(AixA2)(«&l) A/i(AlXA2)(ii62), v(BlixBzi)({ubl)(ub2))=VAi{u) V ^2i(6l62)

<VAl(«)V[vA2(6l)VvA2(b2)] = [vAl (u) V va2 (61)] V [vAl («) V (62)]

= V(A1xAi)(ubl) V^AlxA2)(«&2)

/or ub\,ub2 € U\ X U2-

Case 2 When u e U2, 6162 e Eu,

Ai(SiiXB2i)((6lU)(62«)) = MA2(U) A fJ-B^b^)

< fiA2(u) A ^1(61) A /1^1(62)] = [/*A3(m) A ^(61)] A \p,A2(u) A ^(62)] = x A2) (bin) A X A2) (62«), ^(SiiXiJaOCi6!")^«)) =VAa(«) V fBn(&l&2)

= [l/A3(li) V I/Ax (6l)] V [l^A2(w) V VAl(b2)] = f(AxXA2)(6l«) V ^(Ai X A2) {b2tl)

for biu, b2u 6 U\xU2-Both cases hold for i = 1,2, • • • , n. Hence for all i, Bu x B2i is an intuitionistic fuzzy relation on A\ x A2 and so Gsi x Gs2 = (Ai x A2, Bu x -B21, B12 x B22, ■■■ , Bln x B2n) is an IFGS of G*. This completes the proof. We now define the concept composition of two IFGSs.

Definition 3.3 LetGsi = (Alt Blly B12, ■ ■ ■ ,Bln) and Gs2 = {A2, B21, B22, ■ ■ ■ , B2n) be respective IFGSs of GSsG\ = (Uu En, E12, ■ ■ ■ ,Eln)andG% = {U2,E21, E22, ■•■ , E2n). The composition GS1°GS2 ofGsi andGs2 is defined by Gsi°Gs2 = (Ai o A2, B11 o B21, B12 o B22, ••• , Bln o B2n), where f HM"A2)(xy) = {pAl ofj,A2){xy) = fj.Al(x)A /lA„(iO, l^(Al0A2)(a;y) = (VA! ° VAi){xy) = vaAx) V ^A2(y), Vxy e C/i X U2,

(HBuoB2i)(xyi)(xy2) = (HBu 0 fJ-B2i){(xyi){xy2)) = flAl(x) A HBM(2/13/2), ^(.BlioBlt){xy1)(xy2) = {"Bu ° vB2i){(xy{){xy2)) = vAl(x) V z/s2i (2/12/2),

VieUi, 2/12/2 6 E2i,

(HBu°B2i){xiy)(x2y) = (/iSu 0 IJ-B2i)({xiy){x2y)) = haM a PBu{xiX2), ^BliaB2i){xiy)(x2y) = (vBli O VBM)({xiy)(x2y)) = vA3(y)V VBU(X1X2),

Vy eU2, xix2 e Eu,

' ^BuoB2i){xiyi){x2y2) = (flBu ° HB2i){{xiyi){x2y2)) , = (yi) a ha2(y2) a pbli(Xix2),

Bli0B2i)(xm)(x2y2) = (UBli 0 vB2i)({xiyi){x2y2)) . = (2/1 ) v VA2 (2/2 ) v (xi X2 ),

V xix2 e Sii, 2/15 2/2 e t/2 such that 3/1 + y2. We illustrate the concept of composition of intuitionistic fuzzy graph structures with an example.

Let Gsi and Gs2 be the IFGSs shown in Fig.2. Their composition G„i o Gs2 = {Ai o A2, Bu o B2i,Bi2 o B22) is shown in Fig.4.

«MHi.il 0361(0.2,0.3)

Fig. 4 Composition of two IFGSs

Proposition 3.2 Let G* = (UioU2, EuoE2i,Ei2oE22, ' ■ ■ , ElnoE2n) he the composition of graph structures GI = (Ui, En, E\2, • • • , E\n) andG2 = {U2,E2\,E22, ■ • ■ , E2n).LetGs\ = (Ai,Bn,Bi2, ■ ■■ , B\n) andG32 = (A2, B2i,B22, ■ ■■ , B2n) be respective IFGSs ofG* and G2. Then Gs 1 o Gs2 = (-Ai 0 A2, Bu o B2i, Bj2 o B22,--- ,Blno B2n) is an IFGS ofG*.

Proof We have to prove that Gsi°Gs2 = {AioA2,Bn°B2i,Bi2°B22, - ■ ■ ,B\no B2n) is an IFGS of G*. Ai o A2 is clearly an intuitionistic fuzzy set of U\ o U2 and Bu o B2-i is an intuitionistic fuzzy set of Eu o E2i for all i, therefore, the proof requires to show that Bu o B2{ is an intuitionistic fuzzy relation on A\ o A2 for all i. For this, we discuss some cases as follows:

Fuzzy Inf. Eng. (2016) 8: 389-410_397

Case 1 When u e U-,, b\b2 £ E2i,

M(Bli°B2i)((u6l)(uft2)) a/ib2i(6i62)

< pAl{u) a [ma2(&i) a HA2(b2)]

= («) a Ha2 (6l)] a [fJ,Al («) a flAi (62)] = M(AioA2)(«&l) A^oAjjW,

^(Sli0S2i)((«6l)(«62)) («) V^B24(&1&2)

<^1(«)V[^2(6I)V^2(62)] = K» v ^2(6i)] v [^(w) V ^(62)] = f(AioA„)(u&l) V "(i4ioAa)(u62)

for ubj, ub2 £ C/i O f72.

Case 2 When « e [/2, 61&2 e ¿?ij,

HBii°B2i)((hu)(b2u)) = fj,A2{u) A pBlt(bib2)

< MA2 («) a [/UAj (61) a /mi (i>2>]

= ¡MAi(«) A mai (61)] a [ma2 (u) a /¿ai (62)]

= m(ai°a2) (biu) a 11{AioA2) (b2ll), ^(BliOfl2i)((&lw)(i>2u)) =vA2{U) V ¡^4(6162)

<1/Aa(w) V [1/^(61) Vi/Al (62)] = [i/Aa(w) V i/a^&i)] V K2(u) v 1/^(62)]

= ^(Ai0A2)(6im) V V(AlaA2){b2u)

for 61«, 62" e i7i o i/2-

Case 3 When 6162 G -Eli, «i,«2 £ such that uj ^«2,

tl(BlioB2i){{biu1)(b2u2)) = fiA2(ui) A fiA2(u2) A liBuihh)

<MA2(«I) AfiA2{u2) A [MA^&I) A/iAj(62)] = DiAa(ui) A/iAi(6l)] A [/iAa(«2) A/JAi(&2)]

= M(AioA2)(bl«l) A /i(Ai°A2)(&2U2),

J/(Sli°S2i)((bl«l)(62«2)) = fAa(«l) V VAa(«2) V i/Sli (&1&2)

< ^A2 ("l) V VA2 (u2) V \vM (61) V I^Ai (62)] = [fAa(«l) V ^(61)] V [vM{u2) V ^(62)] = aioaa)^!"!) v ¡"'(ajoaj)(&2u2)

for &1U1,62U2 e C/i o J72- All three cases hold for î = 1,2, • ■ • , ra. Hence ¿?ii o S2i is an intuitionistic fuzzy relation on Ai o for all i and so Gs 1 o GS2 = (v4i o

A2, Bn O B21,B12 o B22, ■■■ , B 1„ o B2n) is an IFGS of G*. This completes the proof.

Definition 3.4 LetGs\ = (Ai,Bn,Bi2, ■ ■ ■ ,Bi„)andGs2 = (A2, B2i, B22, ■ ■ ■ , B2n) be respective IFGSs of GSs G* = {U1: Eu, E12, ■ ■ ■ ,Eln)andG% = (U2,E2i, E22, ■•• , E2n) and let U\ H U2 = 0. The union Gs 1 U Gs 2 ofGs 1 and Gs 2 is given by

Gsi U Gs 2 = (Ai U A2, Bn U B2i,BI2 U B22, ■■ ■ , Bin U B2n),

where A\ U A2 is defined by

M(AiUA2)W = (/Mi U HA3)(x) = HAi(x) V Ha2(x), "(atua^ix) = {^ai u fA2)(a;) = va1 (x) A va2{x) V x e U-l U U2, assuming /j-a^x) = 0, vas{x) = 1 if x g Uj, j = 1,2. And Bu U B2ifor i = 1,2, • • ■ ,n,is defined by

^BuUB2i)(Xy) = (VBu U l*B2i){xy) = HBuMV flBiiixy), v(BriUB2i){xy) = (fBii u vBii){xy) = vBli{xy) A VB2i(xy) V xy e Eu \JE2i, assuming p.Bji{xy) = 0, vBji{xy) = 1 ifxy g Eji, j = 1,2.

Example 3.3 Let G„i and Gs2 be IFGSs shown in Fig.2. Their union Gs 1 U Gs2 = (Ax U A2, Bu U B2i, B\2 U B22) is shown in Fig.5.

63(0.5,0.5)

Fig. 5 Union of two IFGSs

Proposition 3.3 Let G* = (Z7i U U2, Eu U £21, £12 U E22, • ■ • , Eln U E2n) be the union of graph structures G\ = ifJi,E\\,E\2, • • • , Ei„) and G?I = (U2, E2i, E22, ■ • ■ ! E2n).LetGs 1 = (Ai,Bu,Bi2, ■■■ , B\n) andGS2 = (A2, B2i,B22, ■■■ , B2n) be respective IFGSs ofG\ and G2. Then Gsi U Gs2 is an IFGS ofG*.

Proof We have to prove that Gsi UG"s2 = (^iUA2, S11UB21, S12UB22, • • • ,Bln\J B2n) is an IFGS of G*. Clearly, Ал U A2 is an intuitionistic fuzzy set of U\ U U2 and

-Bif U B2i is an intuitionistic fuzzy set of Eu U E2l for all i. It is required to show that B\i U B2i is an intuitionistic fuzzy relation on Ai U A2 for all i. For this, we discuss the following cases:

Case 1 When Ui, u2 € Ui, then by Definition 3.2, fiA2{ui) =/M2(u 2) = AiB2,(uiu2) = 0, vA2{u 1) = 1,l>a2(u2) = vb2i{u 1W2) = 1, so we have

^BuUB2i)(uiU2) = HBU(U1U2) V /iB2i(«l«2) = PBu(uiu2) V 0 < \piAl (wi) A pAl (u2)] V 0 = [fiAl(u 1) V 0] A [fJ,Al(u2) V 0] = ¡PAtM V/M2(«l)] A \p,Al («2 ) V/M2(«2)]

= M(A1ua2)(«I) A^AJUA,)^),

"(BuUB^ji"!^) = VBli{u\U2) A ^B2i(«l«2) = ^li («1W2) A 1 < Vi/Ai^)] A 1

= K1(wi)A 1] V [vAl(u2)h 1] = [val (ui) A (wi)] V [ual (u2) A va2 (ii2)] = "(AtUA^iu 1) V V(AluA2)(U2)

for uiu2 € Eu u E2i.

Case 2 When ui,u2 € U2„ then by Definition 3.2,

KAl(ul) = MAi(il2) = HBlt (U1U2) = 0, UAl(u-i) = VAl{u2) = VBn(wi«2) = 1, so we have

/i(B„UB24)(ulu2) = MBn(«l«2) V MB2i(«l«2) = 0 V/isai(«i«2) <0 V [/M2(«i) A/xa2(U2)] = [0 V /iA2(ui)] A [0 V pAl(u2)] = [/Mifal) V/M2(ui)] A l/JAj {u2)v HA2{U2)\

= M(AiUA2)(«I) A/i(,tlLI,t2)(u2),

f(BnUB2i)(«l«2) = VBu(u 1W2) л VB2i(u 1И2) = 1 Л VB2i (щи2) < 1 Л \va2 (til) V ¡^A2(U2)] = [1 А ^A2 («l)] V [1 Л val («2)] = л va2{u 1)] V [^(«2) л г/А2(«г)]

= f(AlLjA2)(ul) V V(MuM){u2)

for Mi «2 G U i?2¿. In both cases, Si¿ U Дгг is an intuitionistic fuzzy relation on A\ U A2 and it holds for i = 1,2, • ■ • , ra. Hence Вц U B2i is an intuitionistic fuzzy relation on Ai U A2, for all i and so Gs 1 U Gs2 = {Ai U A2,Bu U B2i,B12 U B22, ■ ■■ , Bin U B2n) is an IFGS of G*. This completes the proof.

Proposition 3.4 If G* = {Ui U U2, Ец U E21,E12 U E22, ■ ■ • , Eín U E2n) is the union of GSsG* = {Ui,Eh,Ei2,--- ,Ein) and G2 = {U2, E2U E22, ■ ■ ■ ,E2„). Then every IFGS Gs = (Д Bu B2, ■ ■ ■ , Bn) ofG* is the union of an IFGS Gsl of G\ and an IFGS Gs2 ofG\ ■

Proof We define Ai, A2, Вц and B2i for i = 1,2, • • • , ra as:

hal(u) = ßa(u), vaí{u) = va{u), if и eUl, = pa(u), va2(u) = ua(u), if и G u2, /J-Bu(mu2) = ßB,(uiu2), "Вц(«1«2) = vbi{uiu2), if щu2 G exi, MB2i(U 1^2) = MBi(MlU2), VB2i{uiU2) = vBi(uiu2), ifuiu2 G e2i,

so that Ai, A2, Вц and B2i are intuitionistic fuzzy sets of Ui, U2, Ец and E2.t for all г, respectively and definitely, A = Ai U A2 and B¡ = Вц U B2i, i = 1,2, • • • , ra. Now for щи2 G Eji, j = 1,2 and i = 1,2, ■ ■ • , ra

¡¿Вц («i«2) = ßBi (щи2) < РА(Щ) Л MA(«2) = ßAj (ni) Л pAj («2), vbji {uiu2) = vBi (И1И2) < ^a(mi) V va{u2) = vaj (ui) V vaj (u2),

G si = (А0,Вц,В&, ■■■ , Bjn) is an IFGS of G*, j = 1,2. Thus, an IFGS of G* = G\\J G2 is the union of an IFGS of G* and an IFGS of G2. This completes the proof.

Definition 3.5 LetGs 1 = (Аь Вц, Bí2, ■■■ , Bín) andGs2 = {A2, B2Í, B22, ■■■ , B2n) be respective IFGSs of GSs Gl = (Ui, Ец, Ei2, ■ ■ ■ ,Eln)andG*2 = (U2, E2U E22, ■ ■ ■ , E2n) andletUiC\U2 = 0. The join of G si andGs2 is givenby Gsi+Gs2 = (Ai+A2, Вц + B2i,Bi2 + В 22, ■■■ , Вы + B2n) such that Ai + A2 is defined by

/i(A1+A2)(z) = M(A1UA2)(®),

V(Al+M){x) = V(Alua2)(z) Vz G Ui U U2,

Вц + B2i for i = 1,2, • • • , ti is defined by

HBii+B2i)(xy) = Нвыивъ){ху), V(Bli+B2i)(xy) = "(виивк)(ху)у ху G Eu и E2i,

and = + ) ("У) = MAi (ж) Л {у),

v(Bli+B2i){xy) = (VBU + VB2i)(xy) = val(x) V ual(у) Vж G ui, у G u2.

Example 3.4 Let Gsi = (A-i ,BU, B12) and Gs2 = (A2, B21, B22) be respective IFGSs of GFs G* = (UuEn,E12) and G\ = (U2, £21, E22) where C/i = {a1,a2,a3},U2 = {61,62,63}, En = {aia2}, E12 = {a2a3}, E21 = {6162} and E22 = {6163}. IFGSs G3i and Gs2 are shown in Fig.6 and their join Gsi + Gs2 = (Ai + A2,Bn + B2i,Bi2 + B22) is shown in Fig.7.

(>»] — (Ai, Bn,Ba) Gs 2 — (A2, B21, B22)

Fig. 6 Intuitionistic fuzzy graph structures

Fig. 7 Join of two IFGSs

Proposition 3.5 LetG* = (Ui + U2, En + E2U E12 + E22, ■ ■ ■ , Eln + E2n) be the joinof graph structures Gl = (Ui, En, E\2, ■ ■ ■ , Ein) andG2 = {U2,E2i,E22, - ■ ■ , E2n). Let Gs 1 = (Ai,Bu,Bi2, ■ ■ ■ , Bi„) and Gs2 = (A2, B2i,B22, ■ ■ ■ , B2n) be respective IFGSs ofG\ and G2. Then Gs 1 + G"s2 = (Ai + A2, Bn + B21,B12 + B22, ■ ■ ■ , Bln + B2n) is an IFGS ofG*.

Prvof We have to prove that Gsi+Gs2 = (Ai+A2, Bu+B21,B12+B22, ■■■ , Bin + B2n) is an IFGS of G*. By the définition, it is clear that Ai + A2 is an intuitionistic fuzzy set of Ui + U2 and Bu + B2l is an intuitionistic fuzzy set of Eu + E2l, for ail i, so we now prove that Bu + B2i is an intuitionistic fuzzy relation on A1 + A2 for ail i. For this, we discuss following cases:

Case 1 When u\,u2 e U\, then by Définition 3.3,

1) = MA2(«2) = /iB24(w 1«2) = 0, ua2(U 1) = UA2(U2) = uB2i(u 1U2) = 1,

so we have

= MBii («1W2) V (uiu2) = MBh (M1M2) V 0 < [/lAi(«l) AiiAi(ii2)] V 0 = [/Ui(m) V 0] A[/iAl(«2)V 0] = ¡MA! («1) v /¿a2 («1)] a [/Ml («2) v fj,A2 («2)]

= fBii (iil«2) A Vfl2i (ui«2) = Vflii («1M2) A 1 <[^(«i)v 1/^(112)] A 1 = A 1] y[uAl(u2) A 1]

= [i/Aj(Ml) A fyi2(«i)] V [vAi(ti2) A VA2{U2)}

= I/(A1+Aa)(Ml) V "(,A1+A2)(u2)

foriii,u2 G f/i + U2.

Case 2 When ui,u2 G U2, then by Definition 3.3,

/M^W 1) = 2) = MBii(«l«2) = 0, I/AtCwi) = ^(«2) = ^„(«l^) = 1, consequently, we have

M(Bli+B2i)(ul"2) =MBn(«l«2) V/iB2i(uiU2)

= 0 V flB2i (U!U2)

<0 V lfiA2{ui) A iia2(u2)]

= [0V/m2(ui)1 A[0VmaiM1

= [/Mi («1) V ha2 («1)] A \p,Al («2) V /¿a2 (u2)]

= HA!+A2){UI) A H(Ai+A2)(U2),

"(Bii+B„) («IW2) = fBii («IM2) A ^B2i (ttl«2) = 1 AvB2i(«ili2) <1 A [VA2 («1) V VA2 («2)] = [lA^(m)] V[lA^i(ii2)] = [^(«1) A va2{ui)\ V [fAiM A ¡^(«2)] = f(Ai+A2)(ui) V f(Al+A2)(u2)

foriii,U2 G Ul + i/2-

Case 3 When ui G U\, u2 G [/2, then by Definition 3.3,

MA! («2) = /M2(M 1) = 0, I/Ai(«2) = VA2{UI) = 1,

HBii+B2i)(UlU2)

v(Bn+B2i){uiU2)

Fuzzy Inf. Eng. (2016) 8: 389-410_403

and we have

/»(Bu+BM) (Ul«2) = MA! (Ul) A /iA2 (U2)

= lfiAlMV 0]A[0V/m2M] = [MA! (Ui) V fiA2 (wi)] A [//Ai (U2) V MA2 («2)] = M(Ai+A2)(«I) A M(AI+A2)(«2), "(Bii+B2i)("i«2) = ^(«l) V vM{u2)

= K(ui)A 1] VflA^M] = [VA1 («1) A VA2 («1)] V [vAl («2) A ua.2 («2)] = ¡/(A1+A2)(«I) V V(A1+A2)(u2)

for tii, «2 G C/i + U2. All of three cases, hold for i = 1,2, • • • , n. Hence Bu + B2i is an intuitionistic fuzzy relation on Ai + A2, for all i and so Gs 1 + Gs2 = (A\ + A2, Bu + B2UB12 + B22, ■■■ ,Bln + B2n) is an IFGS of G*. This completes the proof.

Proposition 3.6 IfG* = (Ux + U2,En + E21,E12 + E22, ■■■ ,Eln + E2 „) is the join ofGSs Gl = (Uu Ellt E12, ■■■ , Eln) and G\ = {U2,E21,E22, ■■■ , E2„) and Gs = (A, B\,B2,--- , Bn) is a strong IFGS of G*, then Gs is the join of Gs 1, the strong IFGS of G\, and Gs2, the strong IFGS ofG%.

Proof We define Aj, and Bji fori = 1,2, ■■ ■ , n and j = 1,2 as

MAj(u) = Ha(m), vAj(u) = vA(u) if u&Uj, MB3i(«l«2) = MBi(^l«2), VBjAu 1U2) = VBi{u 1U2) ifu 1«2 £ Eji.

So that A1: A2, Bu and B2i are intuitionistic fuzzy sets of Ui, U2, Elt and E2i for all i, respectively.

Likewise the proof of Proposition, for u\u2 G E]%, j = 1,2 and i = 1,2, ■ • ■ , n,

MB3i («1^2) = MBi(« 1«2) = MA(«1) A MA(«2) = MAj(«l) AMAj(«2), 1'Bjt (uiu2) = vBi (U1U2) = va{u 1) v vA(u2) = uAj («1) V vAj (u2).

So Gsj = (Aj,Bji,Bj2, ■■ ■ , Bjn) is a strong IFGS of G*, j = 1,2.

Now we shall show that Gs is the join of Gs 1 and Gs2. Using Definitions 3.2 and 3.3 , we have A = AxUA2 = A1+A2 andB* = BuliB2i = Bu + B2i, V«iu2 G En U E2{. Whereas for uiu2 G (Bij + E2i) \ (En U E2i), i.e., for ui G Ui and u2 G U2,

MB;(«1«2) = MA(«l) A Ma(«2) = MAi("l) A MA2("2) = MBii-(-B2i(iilM2), ^(«1^2) = i/A(ttl) V ¡^A(«2) = ^Ai(til) V = VBli+B2i{uiU2).

There are similar calculations when u\ G U2 and u2 G U\. This is true for i = 1,2, • • • , n. Hence a strong IFGS of G* is the join of a strong IFGS of Gj and a strong IFGS of G2. This completes the proof.

Definition 3.6 Let G si = (Ai, Bu, Bi2, ■■■ , B\n) andGs2 = (A2,B2\,B22, ■■■ ,B2, be respective IFGSs of GSsG\ = (Uu En, E12, ■ ■ ■ ,Eln)andG*2 = (U2,E21,E22, ■ ■

E2n). The cross product Gsi * Gs2 ofGs 1 and Gs2 i'.s defined by

Gsi * Gs 2 = (Ai * A2,B11 * B21,B12 * B22, ■■■ , Bln * B2n),

such that

UKA^A^ixy) = (/MA 1 *PA2)(xy) = HAl(x) A PaM,

I"(A^A^ixy) = (vAl * vM){xy) = vAl(x) V vA2{y), Vxy € !7l X [/2, M(Bli»B2i)((a;iyi)(a;2i/2)) = (MB^ *Mfl2i)((zm)0E22/2)) = W?14 («1^2) A iis2i (¡/13/2), ' ^SniBaijC^iyOC^ite)) = ("Bu * VB2i)((xiyi){x2V2)) = VBu(xix2) v ^B2i{ym)

Vxis2 g En, yiy2 e E2i.

Example 3.5 Let Gsi and GS2 be IFGSs shown in Fig.6. Their cross product Gsi * GS2 = (Ai * A2,Bn * B2i, B12 * B22) is shown in Fig.8

ai6l(0.4,0.3)

Fig. 8 Cross product of two IFGSs

Proposition3.7 LetG* = (U\*U2, E11*E21,E12*E22, ■ ■ ■ , Eln*E2n) bethecross product ofGSs Gî = [UUE1UE12, ■■■ , Eln) and G2 = {U2,E2l,E22, ■■■ , E2n). LetGs\ = (Ai, Bn, B12, • • • , Bin) andGs2 = (A2, B21, B22, • • • , B2n) be respective IFGSsofGl andGl-ThenGsl*Gs2 = (A1*A2,Bn*B21,B12*B22, ■ ■ ■ ,Bin* B2n) is anIFGS ofG*.

Proof Wehave to prove that Gsi*Ga2 = (Ai*A2,Bn*B2i,Bi2*B22, ■ ■ ■ , B\n* B2n) is an IFGS of G*. By définition, Ai * A2 is clearly an intuitionistic fuzzy set of Ui * U2 and Bu * B2i is an intuitionistic fuzzy set of Eu * E2i for ail i, by définition, so we have to prove only that Bu * B2i is an intuitionistic fuzzy relation on Ai * A2

Fuzzy Inf. Eng. (2016) 8: 389-410_405

for all i. We see that for all (aibi)(a2b2) G Elt * E2l

М(Вц»в24)((°1б1)(а2Ь2)) =Мв14(а1а2) Лдв^Мг)

< \fiA1(ai) A pAl (02)] Л [/xa2(&i) Лда2(&2)] = [/Wai) AnA2(bi)] А \j^Al(a2) Л/м3(&2)] = М(Л!*А2)(аl^i) Л fi(A1*A2)(a,2b2),

^Bli*B2i)((aibi)(a2b2))=i'Bii(a1a2) V vB2i(b-J>2)

< [i/Al(ai) V 1^(02)] V [uAl{h) V vA2(b2)] = [i/Al (ai) V и a 2 (61)] V [i/Al (a2) V j/a2 (62)] = f(A1«A2)(ai6i) V i/(A!»A2)(a2b2).

This holds for i = 1,2, • • • , n. Hence Вц * B2i is an intuitionistic fuzzy relation on Ai * A2 for all i and so G3l * Gs2 = (Ai*A2,Вц*B21,Bi2*B22, ■ ■ ■ , Bln*B2n) is an IFGS of G*. This completes the proof.

Definition 3.7 LetGs 1 = (Ab _Bn. B12, ■ ■ ■ ,Bi„)andGs2 = {A2,B21,B22, ■ ■ ■ , B2n) be respective IFGSs of GSsG\ = (Uu Elu E12, ■ ■ ■ ,Ein)andG% = (U2,E2i, E22, ■■■ , S2n). The strong product G si H Gs2 ofGs 1 and Gs2 is defined by

G3i И GS2 = (Ai И A2,Bu H B2I,B12 И B22,..., Bln И B2„),

Ы(,А^А2)(ху) = (jtAl SI ЦЛ2)(ху) = РАг(х) А ЦА2(у), \f(aiba2)(®l/) = ("ai №vA2)(xy) = VAl(x)y VA2(y), Уху 6 (/] x U2,

!HB1MB2i(xyi)(xy2) = Мви И fiB2i(xVi)(xV2) = ЦаЛх) A/iB„(ym), VB1MB2i(xyi)(xy2) = i>Blt E fB2i(xyi)(xy2) = vaAx) V VB2i(y\V2),

VxeUu У1У2 eE2i,

itJ-B1MB2i(xiy)(x2y) = HBU № HB2i(xiy)(x2y) = HA2(y) AnBli(xix2), VBu®b2i(x-iV)(x2y) = ¡'в» В ^в^(хгу)(х2у) = vm(y) Vvbu(x-ix2),

Vy€U2, Х1Х2 € Eli,

(1*Ви®В2ЛХ1У1)(х2У2) = p,B2i(xiy\)(x2y2) = МВ14 (xiX2) A p,B2i (ут), vBuSBu (xiyi)(x2y2) = ¡'Вн И fB2i(x1yi)(x2y2) = (xix2) V 1>В21(У1У2)

Чх!х2 е Ен, У1У2 е E2i.

Proposition 3.8 LeiG* = ([/1И U2, Ец И E2i, £12 И £22, ■ •' , Еы И Е2п) be the strong product of GSsGl = (Ui, En,Ei2, ■ ■ ■ , Eln) andG2 = (U2, E2i, E22, ■ ■ ■ , E2n)- Let Gsi = (Ai,Bu,Bi2, • • • , Bi„) and Gs2 = (A2, B21, B22, • • • , B2n) be respective IFGSs ofG\ and G*2. Then G3i H Gs2 = (Ax И A2, Bn И B2i,B12 И B22, ■ ■ ■ , Вт И B2n) is an IFGS ofG*.

Proof We have to prove that G"siHGs2 = (А1ИА2, B11&B21, S12KIS22, ■ ■ ■ , Bln И B2n) is an IFGS of G*, clearly, Ai И A2 is an intuitionistic fuzzy set of Ui И U2 and Вц И B2i is an intuitionistic fuzzy set of Ец H E2i for all i. So the proof requires only showing that Вц И B2i is an intuitionistic fuzzy relation on A\ И A2, for all i. For this, we discuss some cases as follows:

Case 1 When ueUu &i&2 e E2i

M(biiHS2i)((ubl)(ub2)) = /mi (u) A /¿B2i(&i&2)

< /mi(") a [/u2(&i) a fJ,A,{b2)]

= [/mi (m) a /IA2 (&i)l a [/i^! («) a /¿a2 (62)]

f(blibb2i)((«bl)h>2)) = val(u) v 1^(6162)

V[^2(6i)V^2(b2)] = [vAl («) v (61)] V [uAl (u) V va2 (62)]

for ub\, «62 G C/l Kl [/2.

Case 2 When u€U2, bjb2 € Eu

M(BiiKls2i)((6iu)(62u)) = /M2(") AMBn(6l62)

< /¿a2 («) a [/iai (61) a fial (62)]

= [/m2 (u) a nal (61)] a [/m2 (w) a ¡j,al (62)] = M(AiEM2)(&1«) A/¿(AiEL42)(i>2^),

t,(BliHS2i)((6lu)(62«)) = VA*(u) V ^„(6162)

<uA,{u) V[vAl{bi)VvAl{b2)] = [uAi («) v uAl (61)] V \va2 (U) V uAl (62)] = f(AiEU2)(6l«) V ^A^A^ihu)

for bju, b2u eUjM U2.

Case 3 Whenaia2 G Eu, bib2 € E2i

M(BnHB2i)((aibi)(a2i>2)) = /1314(1102) A /iB2i(bi&2)

< [/mi(ai) a /mi(a2)] a [ma2(6i) a /m2(&2)1 = [/mi(ai) A //a2(bi)] A [//^1(02) a /m2(&2)] = M(AiEM2)(ai&i) A l^A^A^fah),

^BiiElB^Uai&iX0^)) = ^„(1102) V vB2i{b-ib2)

< [fAi(ai) V vAl(a2)] V K,(&i) V ^2(fc2)] = ["Ai(ai) V ^a2(6i)] V h,(a2) V uA2(b2)] = ^(AiKIA2)(ai6i) V v^MArffafo)

for a\b\,a2b2 € TJ\ U2.

All cases hold for i = 1,2, • • • ,n. Therefore, Bu El B2.t is an intuitionistic fuzzy relation on IE A2, for all i and Gs 1 El Gs2 = (A\ E3 A2, Bn B2\,B\2 El B22,-- ,Blnm B2n) is an IFGS of G*. This completes the proof.

Example 3.6 Let Gsi and Gs2 be IFGSs shown in Fig.2 . Their strong product Gli El Gs2 = (Ax El A2, Bu El B2i,B12 El B22) is shown in Fig.9 .

0362(0.3,0.5) Bl2 3^22(0.2,0.5) 0462(0.4,0.5)

Fig. 9 Strong product of two IFGSs

Definition 3.8 Let G si = (Ai, Bu, B12, ■ • ■ , Sin) andGs 2 = (A2,B2i,B22, ■■■ , B2 „ be respective IFGSs of GSsGl = (Uu Bu, E12, ■ ■ ■ ,Eln)andG*2 = (U2,E21,E22,-■ ■ E2n). The lexicographic product of Gsi andGs2 is defined by Gsi»Gs2 = (A-i *A2, B11 • B2i, B12 • B22, ■■■ , Bin • B2n), where f m(ai.a2)(®2/) = (mai • VA2){xy) = VaAx) a Ma2(2/), 1 t,(ai«a2)(x2/) = ("ai • va2)(xy) = vai(x) v 1/aa(y), v xy g t/'i x U2

(^(Bu.B^iixyiXxy^) = (jiBli • tiB2i){{xyi){xy2)) = ha1 [x) A(2/12/2), "(bh.baoi^j/i)^)) = (fs„ • "b„)(0e!/i)0es/2)) = ^(z) v ^(2/12/2),

VxeUu 2/12/2 e E2i,

(HBii'B2i)({xiyi){x2y2)) = (iJ-Bu • HB2i)((xiyi){xm)) = W5n(zi®2) A/iBM(i/r ^(Bii.B2i)((:El2/l)(:E22/2)) = (fflii • ^fl2i)((:El2/l)(®22/2)) = ^H^l^) V VB2i (yiy.

Vx-lx2 g Eu, ym e E2i.

Example 3.7 Let Gsi and Gs2 be IFGSs shown in Fig.6. Their lexicographic product Gsi • Gs2 = (Ai • A2,Bll • S2i, S12 • B22) is shown in Fig. 10.

Fig. 10 Lexicographic product of two IFGSs

Proposition 3.9 LetG* = (Ui»U2, E11»E2i, Ei2»E22, ■ ■ ■ ,Eln»E2n) be the lexicographic product of GSs G* = (Ui,Eu,Ei2, - ■ ■ ,Ein)andG2 = {U2,E2i,E22, ■■■ ,E2n).LetGs\ = (Ai,Bu,Bi2, - ■ ■ ,B\n)andGs2 = (A2, B21, B22, ■ ■ ■ ,B2n) be respective IFGSs ofG\ and G2. Then G„\ • Gs2 = (Ai • A2,Bu • B2i,B12 • B22, ■ ■ ■ , Bln • B2n) is an IFGS ofG*.

Proof We have to prove that Gsi*Gs2 = (Ai»A2, Bu»B2i, Bi2»B22, ■ ■ ■ , Bin* B2n) is an IFGS of G*. A\ • A2 is clearly an intuitionistic fuzzy set of U\ • U2 and Bu • B2i is an intuitionistic fuzzy set of En • E2i for all i. So the proof requires only to show that Bu • B2i is an intuitionistic fuzzy relation on A\ • A2 for all i. For this, we discuss some cases as follows:

Case 1 When u € U1} 61 b2 € E2i,

HBu'B^dubi^ubz)) = p,Al(u) A fis^ihbn)

< fiAl (U) a [p,M (61) a HA2 (62)] = [mai («) a (61)] a \p,Al (it) a pA2 (62)] = M(A!.A2)(w&i) A p,(Al.M){ub2), "(Bu.B^iiublXuh)) = VAl(u) V I>B2i(6l62)

<vAl{u)y[vM{b 1) Vi/A2(b2)] = [vAl (u) v vM(61)] v [vAi (u) v va2 (62)] = V(AimAl)(ubl) V V(Al.A3)(ub2)

for ub\,ub2 e U\ • Ü2.

Fuzzy Inf. Eng. (2016) 8: 389-410_409

Case 2 When a\a2 £ Ец, b\b2 £ E2i,

М(В14.В24)((°161)(а2б2)) =МВ14(°1°2) A/iB2i(6l&2)

< IpiA^Ui) A fiAl (a2)] Л \p,A2(bi) A HA2(b2)} = [/iA^ai) A/iA2(bi)] л [/Mi («г) Л/м2(Ьг)]

= HM'A^iflib-C) А щАг.А2){а2Ъ2), V(B-ii.B2i){.{a>\b-i)(a2b2))=uBli{a-ia2) V иВз%(M2)

< [vaAo.i) V vM(a2)\ V v ua2(b2)] = [г/Л1 (ai) V va2 (bi)] V [vAl Ы V uM (62)] = ^(Ai.A2)(oi6l) V г/(Л1.А2)(о2&2)

for aibi, 02&2 £ Ui • U2. All cases hold for i = 1,2, • • • , n. Hence Bis • B2j is an intuitionistic fuzzy relation on Ai • A2, V i, and so Gs\ • Gs2 = (A\ • A2, В и • B2i, Bi2 • B22, ■ ■ • , Bin • B2n) is an IFGS of G*. This completes the proof.

4. Conclusion

Graph theoretical concepts are widely used to study and model various applications in different areas. However, in many cases, some aspects of a graph-theoretic problem may be vague or uncertain. It is natural to deal with the vagueness and uncertainty using the methods of fuzzy sets. Since an intuitionistic fuzzy set has shown more advantages in handling vagueness and uncertainty than fuzzy set, we have applied the concept of intuitionistic fuzzy sets to graph structures. We have discussed some operations on intuitionistic fuzzy graph structures. We will extend our work to: (1) m-fuzzy graph structures, (2) Soft graph structures, (3) Fuzzy soft graph structures, and (4) Roughness in fuzzy graph structures.

Acknowledgments

The authors are highly thankful to Managing Editor, Zhen-yu Zhuo and the referees for their valuable comments and suggestions for improving the paper.

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