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ORIGINAL ARTICLE

Intuitionistic Fuzzy Graphs with Categorical Properties

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Hossein Rashmanlou - Sovan Samanta - Madhumangal Pal - Rajab Ali Borzooei

Received: 15 May 2014/ Revised: 31 March 2015/ Accepted: 31 July 2015/

Abstract The main purpose of this paper is to show the rationality of some operations, defined or to be defined, on intuitionistic fuzzy graphs. Firstly, three kinds of new product operations (called direct product, lexicographic product, and strong product) are defined in intuitionistic fuzzy graphs, and some important notions on intuitionistic fuzzy graphs are demonstrated by characterizing these notions and their level counterparts graphs such as intuitionistic fuzzy complete graph, cartesian product of intuitionistic fuzzy graphs, composition of intuitionistic fuzzy graphs, union of intuitionistic fuzzy graphs, and join of intuitionistic fuzzy graphs. As a result, a kind of representations of intuitionistic fuzzy graphs and intuitionistic fuzzy complete graphs are given. Next, categorical goodness of intuitionistic fuzzy graphs is illustrated by proving that the category of intuitionistic fuzzy graphs and homomorphisms between them is isomorphic-closed, complete, and co-complete.

Keywords Rationally • Intuitionistic fuzzy graph • Strong intuitionistic fuzzy graph © 2015 Fuzzy Information and Engineering Branch of the Operations Research Society

Hossein Rashmanlou (/ ') - Rajab Ali Borzooei (> .)

Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran email: rashmanlou@gmail.com

borzooei@sbu.ac.ir Corresponding Author: Sovan Samanta (/ ')

Department of Mathematics, Joykrishnapur High School (H.S.), Tamluk-721649, India email: ssamantavu@gmail.com Madhumangal Pal (,- .)

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, India email: mmpalvu@gmail.com

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

© 2015 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

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

(http://creativecommons.org/licenses/by-nc-nd/4.0/). http://dx.doi.org/10.1016/jfiae.2015.09.005

of China. Hosting by Elsevier B.V. All rights reserved. 1. Introduction

In 1983, Atanassov [4] introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Atanassov added a new component (which determines the degree of non-membership) in the definition of fuzzy sets. The fuzzy sets give the degree of membership of an element in a given set, while intuitionistic fuzzy sets give both the degree of membership and the degree of non-membership which are more-or-less independent from each other, the only requirement is that the sum of these two degrees is not greater than 1. Intuitionistic fuzzy sets have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine, chemistry, economics, etc. Applications of graphs in the mentioned fields are shown in [7-10, 12, 20, 23]. In 1975, Rosenfeld [21] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [11] in 1973. The fuzzy relation between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. And the complement of a fuzzy graph was defined by Mordeson [14].

A fundamental part of intuitionistic fuzzy graphs is, as in the case of classical graph theory, operations on intuitionistic fuzzy graphs. Mordeson and Peng [15] defined some useful operations such as Cartesian product, composition, union and join on fuzzy graphs and gave several deeper and interesting results. Akram and Dudek [2] generalized these operations from fuzzy graphs to interval-valued fuzzy graphs, they also introduced some related notions on interval-valued fuzzy graphs (such as interval-valued fuzzy complete graph and self-complementary interval-valued fuzzy graph). Talebi and Rashmanlou [31] studied properties of isomorphism and complement on interval-valued fuzzy graphs. Likewise, they defined isomorphism and some new operations on vague graphs [32,33]. Rashmanlou and Jun [18] defined complete interval-valued fuzzy graphs. Samanta and Pal introduced fuzzy planar graphs [24], fuzzy tolerance graph [25], irregular bipolar fuzzy graphs [29], fuzzy ¿-competition graphs and p-competition fuzzy graphs [27], bipolar fuzzy hypergraphs [26] and investigated several properties. Pal and Rashmanlou [16] studied lots of properties of irregular interval-valued fuzzy graphs. For further details, reader may look into [3,5, 13, 17-19, 21, 28, 30, 31, 34, 35]. The remaining part of the paper is organized as follows. In Section 2, we introduce some useful preliminary notions and define three kinds of new operations (called direct product, lexicographic, and strong product) on intuitionistic fuzzy graphs. In Section 3, we demonstrate the rationality of some important notions (mainly, operations) on intuitionistic fuzzy graphs, such as intuitionistic fuzzy graphs, intuitionistic fuzzy complete graph, and Cartesian product, direct product, lexicographic product, strong product, composition, union and join of intuitionistic fuzzy graphs, by characterizing these notions by their level counterparts graph. As a result, we give a kind of representation of intuitionistic fuzzy graphs, intuitionistic fuzzy complete graphs. In Section 4, we illustrate categorical goodness of intuitionistic fuzzy graphs by proving that the category of intuitionistic fuzzy graphs and homomorphism between them is isomorphic-closed, complete and co-complete. The final section is concluding remarks.

2. Preliminaries

Definition 2.1 Let X be a set, and II = {[b,c] | 0 < b < c < 1} (i.e., the set of all closed intervals in [0,1]). Every mapping A : X —> II is called an intuitionistic fuzzy set on X. The value A(x) is written as [/¿a,i/a]. For the sake of simplicity we write A =< ¡¿a, i>a >• Obviously, ha and va are mapping from X to [0,1] satisfying 0 < pA(x) + va(x) < 1 for all xeX. For a given set V, define an equivalence relation onVxV - {(x, x) | x e V] as follows: (xi.yi) ~ (x2,y2) <=> {xi,yi) = {x2,y2}-The quotient set obtained in this way is denoted as V2, and the equivalent class that contains the element (x,y) is denoted as [(x,y)], xy or yx.

Definition 2.2 An intuitionistic fuzzy graph (IFG) is a pair G = (A,B), where A and B are intuitionistic fuzzy sets on V and V2 respectively which satisfy ¡iB(x, y) < liA(x)ApA(y) andvB(x,y) > vA (x) AfiA (y) for all (x, y) G V1 and ¡iB(x,y) = vB(x,y) = 0 for all (x, y) e V2 - E (E is the edge set). Here ¡1r(x, y) and vB(x, y) represent the membership and non-membership values of the edge (x, y). Also, ha(x) and vA(x) represent the membership and non-membership values of the vertex x.

Definition 2.3 Let G* = (V\,E\) (resp., G\ = (V2,E2)) be a given graph, and Gi = (Ai, Bi) (resp., G2 = (A2,B2)) be an intuitionistic fuzzy graph of G\ = (Vi,E\) respectively, G*2 = (V2, E2)).

A homomorphism from Gi to G2 is a mapping / : V1 —» V2 that satisfies the following two conditions:

(i) pal(xi) < pa2(j(xl)),l/Al(*l) > va2(f(.x0) (Vx, e V,),

(ii) psMm) < Ats2(/(xi)/Oi)), VBMiyi) > vBl(J(xi)f(yi)) (V*m e V2).

G\ is the underlying crisp graph of the intuitionistic fuzzy graph GJ. The category of intuitionistic fuzzy graphs and homomorphism between them are denoted as IFG.

An isomorphism between G\ and G2 is a bijective mapping f . V1 -> V2 that satisfies the following two conditions:

(i) ma,(*i) = ^A2(f(xi)),vA,(xi) = vAl(f(xi)) (Vx 1 e Vi),

(ii) pBl(.xiyi) = pB2(f(.x1)f(yl)), vBl(xiy{) = fB2(/(x,)/0>i)) (Vx,);, 6 V2).

Definition 2.4 A fuzzy graph G = (cr,fi) of a graph G* = (V, E) is called a complete if pixy) = min(cr(x), cr(y)) (Vxy 6 V2).

Definition 2.5 Let A : X —> II be an intuitionistic fuzzy set on X. Then A[i,f] = jx 6 X | pa(x) > b, va(x) > c] is called a [b,c]-level set of A (V[6, c] e II).

Definition 2.6 Let G* = (V, ,£,) and G\ = (V2, £2) be two graphs. The graph GJ x Gj = (V, E) is called the cartesian product of GJ and G*v where V = Vi x V2 and E = {(x,x2)(x,y2) I x e V,, x2;y2 e E2} U {(x,, z)0>,, z) | z e V2,x1yl e £1}.

Definition 2.7 The graph G\ X G\ = (V, E) is called the direct product ofG\ and G*v where V = Vi x V2 and E = {(xi,x2)(yi,y2) \ xiyi e E,, x2y2 e E2).

Definition 2.8 The graph GJ • G*2 = (V, E) is called the lexicographic product of G* and G*2, where V = Vi X V2 and E = {(x,x2)(x,y2) \ x 6 Vi,x2y2 e E2} U {(xi,x2)(yuy2) I x\yi e Eux2y2 e E2}.

Definition 2.9 The graph G\ • G2 = (V, E) is called the strong product ofG\ and G2, where V = V\ x V2 and E = {(x,x2)(x,y2) \ x e Vi,x2y2 e E2} U {(xi,z)(yi,z) I z e V2,x,y, e £|}U (xl,x2)(yt,y2) I x,y, e Eux2y2 e E2}.

Definition 2.10 The graph GJ o G*2 = (V, E) is called composition of Gj and G*v where V = Vi x V2 and E = ((x,x2)(x,y2) \ x e V,, x2y2 e E2} U {(x,, z)Cyi, z) \ z e V2,xiyi e £i}U {(x\, x2)(yi,y2) \ x2y2 e V2,x2 + y2,xiyi e £i}.

Definition 2.11 The graph G\ U G\ = (V, E) is called the union ofG\ and G\, where

V = Vi x V2 and E = Ei U E2.

Definition 2.12 The graph Gj + G*2 = (V, E) is called the join ofG\ and G2, where

V = V] U V2 and E = Ei U E2 U E',E' is the set of all edges joining the vertices Vi and V2.

Definition 2.13 Let Gi = (Ai,B{) (resp., G2 = (A2,B2)) be an intuitionistic fuzzy graphs o/Gj = (Vi ,Ei) (resp., G* = (V2, E2)). The Cartesian product Gi X G2 of Gi and G2 is defined as a pair (A, B), where A = (jia, i>a) and B = (pB, i>b) are intuitionistic fuzzy sets onV = Vi XV2 and E = {(x, x2)(x,y2) \ x € V], x2y2 € E2] u {(*i,z)(yi,z) I z e V2, xiyi e Ei} respectively, which satisfies the fallowings:

(Mxi,x2) = nun^x,), ,Al(x2)) 6 x [vA(xi,x2) = max(vA](xi), vA2(x2)),

jM(x,x2)(x,y2)) = mm(pAl(x), uB2(x2y2)), , .

( > \vB({x,x2){x,y2)) = max(vAl(x), vB2{x2y2)\ (X VuX2n *2>'

(iii) /^«(^I'^tVi.z)) = min(//fll(xi;yi), fiAl(z)), g ^ £ . lwB((*i,z)(yi,z)) = max(i/fll(j:iyi), vM(z)), 2' 1 1 1 '

Definition 2.14 The composition G\G2 ofGi and G2 defined as a pair (A, B), where A = (p.a , vA) and B = (jib,vb) are intuitionistic fuzzy sets onV = Vi xV2 and E = \(x,x2)(x,y2) | * e Vux2y2 e E2\ u {(x,,z)(Ji,z) I z e V2,xiyi e E,} u {(xi, x2)(yi,y2) I x2y2 e V2 # y2, xiyi e Ei} respectively, which satisfies the fallowings:

<Mxi,x2) = min^O, ,a2(x2)) 6 x

[vA(xi,x2) = max(vAl(x,), vM(x2)),

(ii) iM(x,^2)(x,y2)) = minO/Al(x), pB2(x2y2)), £ ^ g \vB((x,x2)(x,y2)) = max(vAl (x), vBl(x2y2)), u 2 2 2 '

ran jVB(ixi<z)(yi,z)) = rmn(fiBl(xiyi), fiM(z)),

(U1) \vB((xi,z)(yi,z)) = max(i>B] (xi_yi), (z 6 6

(iv) /^я^ь^СУьй)) = ™п(Ma2(x2), (*i.yi)),

\vB((xi,X2)(yUy2)) = шЫрАг(х2), VA2(y2),VBl(x1y1)),

(x2,y2 e V2,x2 * y2, Xiyi e £1).

Definition 2.15 The union Gi U G2 ofG\ and G2 is defined as a pair (A, B), where A = (¿¡a,va) and В = (pB,vB) are intuitionistic fuzzy sets on V = Vi U V2 and E = Ei U E2 respectively, which satisfies the fallowings:

IpA(x) = /Xa, (x), if x eVt and x <t V2, Ma(x) = Ma2 (x), if xeV2 and x $ Vu Ma(x) = max (jxAl (x), ца2(х)), if xeVifl V2.

IVa(x) = vAl(x), if x e Vi and x t V2, vA(x) = vAl(x), if x e V2 and x g Vi, vA(x) = max (vA, (x), vAl(x)), if xeVif\V2.

1Ив(ху) = pBl (xy), if xye Ei and xytE2, Рв(ху) = fiBl (xy), if xye E2 and xy f E\, Цв(ху) = max (pBl (xy), Цвг(ху)), if xye Eif) E2.

IvB(xy) = vB, (xy), if xye Ei andxy t E2, vB(xy) = vBl(xy). if xy e E2 and xy t Ei, vB(xy) = max (vBl (xy), vBl(xy)), if xye Eif) E2.

Definition 2.16 The join Gi + G2 of Gi and G2 is defined as a pair (A, B), where A = (jjla, va) and В = (jiB, vB) are intuitionistic fuzzy sets on V = Vi U V2 and E — EiUE2UE' (E' is the set of all edges joining the vertex of Vi andV2) respectively, which satisfies the followings:

IHa(x) = fiai (x), if xeVi and x<tV2, pA(x) = Pa2 (x), if x e V2 and x$V\, Мл(х) = max (jiAl (x), Pa2(x)) , if x e Vi f| V2.

Iva(x) = vAt (x), if xy e Ei and xy <t E2, va(x) = vAl(x), if xy e E2 and xy $ Ei, vA(x) = max (vAl (x), Al(x)), if x e Vi f| V2.

1Мв(ху) = fiBl (xy), if xy e Ei and xy t E2, Мв(ху) = рв2 (xy), if xye E2 and xy <t E,, ¡iB(xy) = max Од, (xy), Цв2(ху)), if xy e Ei f| E2.

IvB(xy) = vBl (xy), if xye Ei and ху$Е2, vB(xy) = vB2(xy), if xy e E2 and xy t Ei, vB(xy) = max (vBt(xy), vBl(xy)), if xy e Ei П E2.

м (Ыху) = min (jiM(x),pM(y)), W \ Ув(ху) = max (vAl(x), vM(y)), 7 ^

Finally, we define three kinds of new operations called direct product, lexicographic product, and we strong product on intuitionistic fuzzy graphs.

Definition 2.17 The direct product Gi * G2 of two intuitionistic fuzzy graphs Gi = (/t i, fli) andG2 = (A2, B2) o/Gj = (Vi, E{) and G"2 = (V2, E2) respectively is define as a pair (A,B), where A = (p.A, vA) and B = (jiB, 1iB) are intuitionstic fuzzy sets on V = Vi x V2 and E = {(xi,x2)(yi,;y2) I *\yi e Ei,x2y2 e E2} respectively, which satisfies the fallowings:

, x2) = mm ) nAl(xij), M e x ^

(vA(x,,x2) = max (vAl(xi), vA2(x2)),

iMB((xi,x2)(yi,y2) = ioiii(fiBl(x1y1),fiB1(x2y2)), (") <, ., » , , . . ... №)>, e ti ,x2y2 6 £.2) ■

(i/B((xi,x2)(yi,y2) = max (vBl(x^i),uS2(x2;y2)),

Definition 2.18 The lexicographic product G\ • G2 of two intuitionistic fuzzy graphs Gi = (Ai,Bi) and G2 = (A2,B2) of G\ = (VuEi) and G* = (V2,E2) respectively is defined as a pair (A,B), where A = (jxA,vA) and B = (pB,vB) are intuitionistic fuzzy sets on V = Vi X V2 and E = {(x, x2j(x,y2) \ x € Vi,x2y2 € E2j U {(xi, x2)(yi, y2) I xiyi e Ei, x2y2 e £2} respectively, which satisfies the fallowings:

(i) {¿(a(xi,x2) = min (fial(xi), jUa2(x2)) ^a(xi,x2) = max (kai(xi), i/a2(x2)) for all({xi,x2)),

(ii) iA'B ((x,x2)(x,y2)) = min (pAi(x),iiBi(x2y2)), , g ^

((x,x2)(x,y2)) = max (vai(*)> Wft(*2y2))» 2 2 2 '

(iii) i^®((xi' ^l,»)) = min (jisMiy 1). ^{xiyi)), . g g £ , \t/B((xi,x2)(yi,y2)) = max (t/Bl(xiyi), uft(x2y2)), 11 u 2 2 2'

Definition 2.19 The strong product GiG2 of two intuitionistic fuzzy graphs Gi = (Ai, Bj) and G2 = (A2, B2) o/GJ = (Vi, Z?i) and G*2 = (V2> £2) respectively is defined as a pair (A,B), where A = (jiA,vA) and B = (p.B,vB) are intuitionistic fuzzy sets on V = Vi x V2 anrf £ = {(x,x2)(x,y2) I x e Vi,x2>'2 6 E2} U {(xi,z)Cvi,z) | z 6 V2,xiyi e Ei) U{(xi,x2)(yi,y2) | xiyi e E\,x2y2 e £2} respectively, which satisfies the fallowings:

U^.x^mm^ixO^fe)) |.i/a(xi,x2) = max (vAl(xi), vA2(x2)),

(ii) ^^ I ^ (xeV„ x2y2eE2),

[vB(.(.x,x2)(x,y2)) = max (vAi(xi),vB2(x2y2)),

(1U) KCta.zX,,,;)) = max K(xm), vAl(z)), 6 ^J"3" 6 Elh

\nB((xi,x2)(yi,y2)) = min (pBl(xiyi), nBl(x: ^KB((xi,x2)(yi,y2)) = max (nBl(xiyi), vfl2(x2,y2)),

av) •b=(w €EU x2y2 eE2). [1

Fuzzy Inf. Eng. (2015)7: 317-334_323

3. Rationality of Some Defined Notions on Intuitionistic Fuzzy Graphs

In this section, we demonstrate the rationality of some notions (i.e., intuitionistic fuzzy complete graph, Cartesian product, direct product, lexicographic product, strong product, composition, union, and join of intuitionistic fuzzy graphs) on intuitionistic fuzzy graphs by characterizing them by their level counterpart graphs. As a result, we obtain a kind of representation of intuitionistic fuzzy graphs (respectively, intuitionistic fuzzy complete graphs).

We first show the rationality of intuitionistic fuzzy graphs and intuitionistic fuzzy complete graphs.

Theorem 3.1 Let V be a set, and A = (jia,va) and B = <jiB,vB) be intuitionistic fuzzy sets on V and V2 respectively. Then G = (A, B) is an intuitionistic fuzzy graph (respectively, intuitionistic fuzzy complete graph) if and only if (A^], B^]) (called [a, b]-level graph ofG = (A,B)) is a graph (respectively, a complete graph) for each [a, b] e II.

Proof We only show the case of intuitionistic fuzzy graph.

Necessity. Suppose that G = (A, B) is a fuzzy graph. For each [a, b] 6 II and each xy e B[aM, we have pB(x,y) > a, vB(xy) > b, fiB(xy) < min (ma(x), pA(y)), and vB(xy) > max (vA(x),vA(y)) since G - (A,B) is an intuitionistic fuzzy graph. It follows that iua(x) > a, vA(x) > b, fiA(y) ^ a. and vA(y) > b which means x,y e A\a,b\-Therefore < A[a^], B[aj,] > is a graph for all [a, b] e II.

Sufficiency. Assume that (A[aji,], is a graph (V [a, b] e II). For each xy e V2, let pB(xy) = a and vB(xy) = b. Then xy e B[aJ>1. Since (A[aM,B[aM) is a graph for each [a,b] e II, we have x,y e A[a^, which means pa(x) > a and va(x) > b, MA(y) > a and vA(y) > b. Therefore pB(xy) = a < min (j*a(.x), pA(y)), vB(xy) = b> max (pa(x), va(y)), i.e., G = (A, B) is an intuitionistic fuzzy graph.

Next we show the rationality of four kinds of product operations on intuitionistic fuzzy graphs.

Theorem 3.2 Let G\ = (A,, B,) (respectively, Gi = (A2, B2)) be an intuitionistic fuzzy graph of G* = (Vi.Ei) (respectively, G* - (V2,E2)). Then Gi X G2 = (A, B) is the Cartesian product of G\ and G2 if and only if (AB[a^) is the Cartesian product of((At)[aM, (Si)[o,i]) and ((A2)[aM, (B2\aM)for each [a, b] 6 II.

Proof Necessity. Suppose that Gi x G2 = (A, B) is the Cartesian product of Gi and G2. Firstly, we show A^^j = (Ai)^] x (A2)[a,b] for each [a, b] e n. Actually, for every x,y e Af0iil, we have min (jj.Ai(x),pA2(y)) = PA(x,y) > a and max (ua,W, VA2(y)) = i>A(x,y) > b since (A, B) is the Cartesian product of Gi and G2. It follows that x e (A{)[aM and y e (A2)[aM (i.e., (x,y) e (A{)[aM x (A2)[aM). Therefore, A[aM c (A{\aM x (A2)[aM.

Conversely, for every (x,y) e (Ai)[o>i,] x (A2)[aM, we have x e (Ai)[aij)] and y e (A2)[a,b\ which implies min(pA) (x), ¡.iAl(y)) > a and max(vAl(x), vAl(y)) > b. Thus we have pA(x,y) > a and vA(x,y) > b since (A, B) is the Cartesian product of Gi and G2. Therefore, (Ai)[aj,] x (A2\a^] c A^j. Secondly, we prove B[a^ = E(a,b) for each (a, b) e n, where E(a,b) = {(x,x2)(x,y2) \ x e (Ai)[aii,], x2y2 6 (B2)[aM] u

{(•*i>z)(yi,z) I z e (A2\a,bb Xiyi e №)[«,«(. For every (x\,x2)(yuyi) e B[aM (which means fiB ((xi, x2)(yl, y2)) > a and vB ((xi, x2)(yi, y2)) > b), either xt = yi and x2y2 e E2 hold or x2 = y2 and x\yi e Ei hold since (A, B) is the Cartesian product of G\ and G2. For the first case, we have fiB (.(xi,x2)(yi,y2)) = min(//Al(xi), fiB2(x2y2)) > a and vB({xux2)(y\,y2)) = max(vAl(xi), vBl(x2y2)) > b, which implies /¿A,(*i) > a, vAl(xi) > b, nBr(x2y2) > a and vBl(x2y2) > b.

Therefore x1 =yi e (Ai)[a,4j, x2y2 e (B2)iaM, i.e., (xlx2)(yl,y2) e E(a, b). Analogously, for the second case, we have (xt x2)(y\, y2) e E(a, b).

Conversely, for every (x, x2)(x, y2) e E(a, b) (i.e., fiAl (x) > a, vAl (x) >b,p.Bl (x2y2) > a and vBl(x2y2) > b), as (A, B) is the Cartesian product of Gi and G2, we have HB ((x, x2)(x, y2)) = min (jiAl (x), HB2 (x2y2) > a and vB ((x, x2)(x, y2)) = max (vAl (x), vBl (x2y2) > b, which implies (x, x2)(x, y2) e 8M. Analogously, for every (xi, z)(yi, z) e E(a, b), we have (xi, z)(yi, z) t .

Sufficiency. Suppose that (A[a.b], B\a,b\) is the Cartesian product of ((Ai)[a.j,], (B\)\a t,\) and {(A2\a.bh{B2\a.bi) (V \a, b\ e II). Foreach (x,, x2) e V, x V2, let min (jiA, (x,), fiAl(x2)) = a and max (vAl (xi), vAl(x2)) = b (which implies x\ e (Ai\a,b] and x2 e (A2)[a,i,]). Then (xi,x2) e A[aM since (A[aM, B[aM) is the Cartesian product of ((Ai)[a.6],

(Bi)[a.t]) and ((A2)[a.b],(B2)ia.bd< thus nA(xi,x2) > a = min {nAt(xi),HAl(x2)) and vA(xi,x2) > b = max (vAl(x\), vAl(x2)). Again, let fiA(x],x2) = c and vA(xi,x2) = d (which implies (xi,x2) e A^). Then x\ e (Ai)^ and x2 e (A2)[Cilg since (A[lvfl, B[Cjd]) is the Cartesian product of ((Ai)[tvfl, (Bi)[c,fl) and ((A2\cA, (B2)[cA), thus ¡iM(xi) > c =fiA(xi,x2), vM(xi) >d = vA(xux2), liAl(x2) > c = fiA(xi,x2), vAl(x2) > d = vA(x\,x2), which implies min(¡iAi(x\),nAl(x2)) > pA(x\,x2) and max(DA,(xi), vAl(x2)) > vA(xi,x2). It follows that

\vA(xux2) = max (vAl(xi), vAi(x2)),

Analogously, for each x e Vi and each x2y2 e E2, let min (¿iAl(x), ¡iB2(x2y2)) = a, max (uAl(xi), i/ft(x2y2)) = b,[iB ((x,x2)(x,y2)) = candi/B((x,x2)(x,y2)) = d. Then

nil flJB(ix,X2)(x,y2)) = mm(iiAl(x),fiB2(.x2y2)), w ((x,x2)(x,y2)) = max (i/Al(x), vBl(x2y2)), U

For each z g V2 and each xiyi e E\, let min(//A2(z), fiBl(xiyi)) = a, max(i/A2(z), vBl(xlyl)) = b,nB((xi,z)(yi,z)) = c and vR ((x,,z)(yr,z)) = d. Then

¡Ma ((x,,z)(y,,z)) = min (pBl(x,y,I Ma2(z)), (m) < ,, ., „ „ / , . , (z e V2,xiyi e hi).

lKa((^i,z)Cyi,z)) = max (^(xi^i), vAl(z)),

Theorem 3.3 Let Gi = (Ai,Bi) (respectively\ G2 = (A2,B2)) be an intuitionistic fuzzy graph of G\ = (Vi,£i) (respectively, G* = (V2,E2)). Then Gi * G2 = (A,B) is the direct product of G] and G2 if and only if (A[aj,], B[a,&]) is the direct product of ((AiWj, (Bi\aM) and {{A2\aM, (B2\aM)for each [a, b] e II.

Proof Necessity. Suppose that Gi * G2 = (A, B) is the direct product of Gi and G2. Firstly, we show A[aj,] = (Ai)^ x (A2)[a,i)| for each [a, b] e II. Actually, for every

(x,y) e A{aM, wehavemin (fiAi(x), PA2(y)) = fiA(x,y) > aandmax (vAl(x),vAl(y)) = vA(x, y)>b since (A, B) is the direct product of G\ and G2. It follows that x £ (Ai)^] andy e (A2)[o,&] (i.e., (x,y) e x (A2)ia,b])•

Therefore A[a>i)] c (Ai)[a,s] x (A2)[a,s]. Conversely, for every (x,y) e (AO^j x (a2XaM<we have * e (A\)\aM and y e {A2\aM' which implies min (pA,(x), /JAl(y)) > a and max(i/a, (x), vAl(y)) > b.

Thus we have fiA(x, y)> a and vA(x, y)>b since (A, B) is the direct product of Gi and G2. Therefore (Ai)^ x (A2\a^ c AlaM. Secondly, we prove B^ = E(a, b) for each [a,b\ £ II, where E(a,b) = {(xux2)(yi,y2) I xlyl £ (Bi\aM, x2y2 £ (B2\aM}. For every (x,, x2)(yi, y2) e B[aM (which means jub((jci , x2)(yi, y2)) > a and vB((x\, x2) (yi,y2)) ^ b), then xiyi e (Bi)[a,6] and x2y2 6 (B2)[aj>i hold since (A,B) is the direct product of Gi and G2. This implies (xj, x2)(yi,y2) e E(a,b).

Conversely, for every (xi,x2)(yi,y2) e E(a,b) (i.e., pBl C*i.yi) > a , ^(Jtiyi) > t*b2(x2y2) > a and vBl(x2y2) > b) as (A,B) is the direct product of Gi and G2, we have pB ((*i> x2)(yi,y2)) = mia(jiBl(x\y\),pB2(x2y2)) > a and vB ((x\,x2)(yuy2)) = max(vB, Uiyi), vB2(x2y2)) > b, which implies (xux2)(yi,y2) £ B[a,b].

Sufficiency. Suppose that (A[ain,%n) is the direct product of ((Ai)[0ii,], (Bi)[0,j,]) and ((A2)[flij], (B2)[aj,]) (V [a, b] £ II). For each (*i, x2) e Vi x V2, letmin(/iAl(xi), Ma2(x2)) = a and max(i/A, (*i), vAl(x2)) = b (which implies x\ e (Ai)iaM and x2 e (^2)[a,i]» then (x!, x2) e A[aM since (A[aJ>], B[aJ>]) is the direct product of ((Ai)[ai&], №)[<.,«) and (,(A2)[aM,(B2\aM), thus fiA(xi,x2) > a = min(juAl(*i), Ma2(x2)) and vA(xi,x2) >b = max (vAl(xj), vAl(x2)).

Again, let pA(xi,x2) = c and vA(xi,x2) = d (which implies (xi, x2) £ A^). Then x\ e (Ai)^ and x2 e (A2)[c,rf] since (Aic^, BicA) is the direct product of (iAi\c4\<(P\\c4\) and ((A2)[cj],(B2\cA), thus ¡j.A](xi) > c = fiA(xi,x2),vA,(xi) ^ d = vA(xi,x2), /ha2(x2) > c = pA(x\,x2),vA2(x2) > d = vA(xi,x2), which implies min^Ct!), ¡ia2(x2)) > [iA(x\,x2) and max(uA,(^i), vM(x2)) > vA(x{,x2). It follows that

Analogously, for each x\y\ e E\ and eachx2y2 e E2, let min (fiBl(xiyi), fiBi(x2y2)) = a, max (vBl (xtyi), vBl(x2y2)) = b, fiB ((ati,x2)(yuy2)) = c and vB ((xi, x2)(yl,y2)) = d. Then

Theorem3.4 LetG\ = (Ai,B{)(respectively, G2 = (A2,B2))beanintuitionisticfuzzy graph of G\ = (Vi,E{) (respectively, G* - (V2, E2). Then Gl • G2 = (A,B) is the lexicographic product of G\ and G2 if and only if(AiaB[a,i>]) is the lexicographic product of((Ai\aM, (B,)[aM) and ((A2)[aM, (B2)iaj,]) for each [a, ti\ £ II.

Proof Necessity. Suppose that G\ • G2 = (A, B) is the lexicographic product of Gi and G2. Firstly, we show A[aj,j = (Aj)^] x (A2\a^] for each [a,b] £ n by definition of lexicographic product and the proof of Theorem 3.3. Secondly, we prove

ha(xi,x2) = min (pal (xi), fiA2(x2)), vA(xux2) = max(vAi(xx), vM(x2)),

' ((xi, x2) e Vi x V2).

HB((x\,x2)(yi,y2)) = min(/j5] (x^i), pBl(x2y2)), vB((xi,x2)(yi,y2)) = max(i/Bl(xiyi), vBl(x2y2)),

' (xiy\ 6 E\, x2y2 e E2).

B\ajs\ = E(a, b) U F(a, b) for each [a, b] e n, where E(a, b) is as that defined in Theorem 3.3 and F(a, b) = {(x,x2)(x,y2) \ x e {Ai\aJii, x2y2 e By the proof of Theorem 3.3, we have E(a,b) c For every (x, x2)(x,y2) e F(a,b)(i.e.,pal(x) > a, i>a1 (x) >b, fify(x2y2) > a, vB2(x2y2) > b), as G\»G2 = (A, B) is the lexicographic product of Gi and G2, we have fiB ((x, x2)(x, y2)) > a and vB ((x, x2)(x, y2)) > b which imphes (x, x2)(x, y2) e B[a^.

Therefore, E(a,b) U F(a.b) c B\aj,]. Conversely, for every (xi,x2)(yi,y2) e B[ajj\ (i.e., Hb ((x^x2)(yt,y2)) > a and vB ((x\,x2)(y\,y2)) > i as G, • G2 = (A,B) is the lexicographic product of Gy and G2, we have (xi,x2)(y\,y2) e E U F, where E = {(xi, x2)Cyi, y2) | xiyi e Ei, x2y2 eE2},F = {(x, x2)(x, y2)\xeVu x2y2 e E2). If (xi,x2)(yi,y2) e E, then (xi,x2)(y\,y2) 6 E(a,b) by the proof of Theorem 3.3. If (xi,x2)Cyi,>>2) e F, i.e., xi = yi, x2y2 e E2, then mm(fiAt(xl),iiBl(x2y2)) = /xB((xi,x2)(yi,y2)) > a and max(uAl(x,), vB2(x2y2)) = KB((xi,x2)(yi,;y2)) > b, which imphes x\,y\ e (Ai)[0,i.|, and x2y2 e (B2)iaM.

Therefore, (xi, x2)Cyi, y2) e F(a, b). It follows that B[aM c E(a, b) U F(a, b). Sufficiency. Assume that (A^j, B^) is the lexicographic product of ((Ai), (•BiWl) and ((A2)[a,i.], (B2)[aj,]) (V [a, b\ e II). By the proof of Theorem 3.3, we know

Wx.x^min^^^fe)), ((X|i;C2)eV|XV2)i \vA(xi,x2) = max(i>Ai(xi), uA2(x2)J,

(ii) iMB ((M,x2)(yi,y2)) = imn(pBl(xiyilMB2(x2y2)), ^ g ^ _ g \ng((xi,x2)(yi,y2)) = max(uBl(xiyi),Kg2(x2y2)), 11 u 2 2 2

For each x e Vi and each x2y2 £ E2, let min (jj-a, (x), nBl (x2y2)) = a, max (vAl (x), vb2 (x2y2)) = b, hb ((x, x2)(x, y2)) = c and vB ((x, x2)(x, y2j) = d. Then

(iii) {^{^,x2)(x,y2)) = mai{1iA1(x)'liB2{.x2y2)), ^ \ vB ((x, x2)(x, y2)) = max (vAl (x), vBl (x2y2)),

Theorem 3.5 Let G\ = (Ai,Bi) (respectively, G2 = (A2,B2)) be an intuitionistic fuzzy graph of G\ = (VUE\) (respectively, G*2 = (V2tE2). Then GiG2 = (A,B) is the strong product ofG\ and G2 if and only if (A^¡.j, B[aj,\) is the strong product of ((Ai)[a,fc], (B\\aM) and ((A2\aM, (B2\aM)for each [a, h] e II.

Proof It is similar to that of Theorem 3.2 and Theorem 3.3.

Now we show the rationality of operations composition, union and join of intuitionistic fuzzy graphs.

Theorem 3.6 Let G\ = (Ai,Bi) (respectively, G2 = (A2,B2)) be an intuitionistic fuzzy graph of G\ = (Vi, Ei) (respectively, G* = (V2, E2)). Then Gi [G2] = (A, B) is the composition o/Gi andG2 if and only if (A\aj,],B[aM) is the composition product of{(A\\aM, (B, )[„,«) and ((A2)[0j^], (B2)[a,b])for each [a, b\ e II.

Proof Necessity. Suppose that Gi [G2] = (A, B) is the composition of Gi and G2. Firstly, we know = (Ai)^] x (A2)[aii,\ for each [a, b] e n by definition of composition and the proof of Theorem 3.2. Secondly, we prove B[aj>] = E(a, b) U F(a, b) for each [a, b] e n, where E(a, b) is as that defined in Theorem

3.2 and F(a,b) = {(xl,x2)(yi,y2) I x2y2 e (M\a,bb x2 * y2, x,yi e )[„,(,]}. By the proof of Theorem 3.2, we have E(a,b) c b[aji\. For every (xi,x2)(yi,y2) e F(a,b) (i.e., /¿Bl(xiyi) > a, vR](x\y\) > b, pa2(x2) > a, vAl(x2) > b, ha2(y2) > a, and va2(y2) > b), as Gi [G2] = (A,B) is the composition of Gi and G2, we have Mb ((xux2)(y\,y2)) > a and vB ((xi, x2)(yi,y2)) > b, which implies (xux2)(yi,y2) £ . Therefore E(a,b) U F(a,b) c b[aj,]. Conversely, for every (x],x2)(y],y2) e b\a,b\ (i e., Hb ((*i ■ x2)(yi, y2)) > a and vB ((xi ,x2)(yuy2)) > b, as Gi [G2] = (A, b) is the composition of G\ and G2, we have (x\, x2)(y\, y2) e E u F, where

E = {(x,x2)(x,y2) I ac 6 Vi,x2y2 e E2} U ((xi,z)0<i,z)z e V2, xiyi e Ei)

F = [(xuxrfiyuyi) | x2,y2 e V2,x2 *y2,xiyi € Ei).

If (Jti,Jt2)(yi.y2) e then (xi,x2)(yi,y2) £ E(a,b) by the proof of Theorem 3.2. If (xux2)(yi,y2) e F, then mm(pA2(x2), nA2(y2),pBi(x{y{)) = ¡iB ((x{,x2)(yuy2)) > a and max.(vA2(x2),VA2(y2),vBl(xiyi)) = vB((xi,x2)(yi,y2)) > b, which implies x2,y2 e (A2)laM, x2 * y2, andx\y\ e (b\)[am. Therefore, (xux2)(yuy2) e F(a,b). If follows that b[am c E(a, b) U F(a, b).

Sufficiency. Assume that (A^, biab]) is the composition of ((Ai)[a,f>|, (Bi)[a,i|) and ((A2)[a,i>], (B2)[a,6]) (V [a, b] e II). By the proof of Theorem 3.2, we know

(pA(M,x2) = min(pAj(M) ^A2(x2)), ((xuX2)eV]XVlh \vA(x\,x2) = max (vAl(xi), vA2(x2)),

(ii) i^8 ((x'X2>'yi)) = 111111 № (X2>,2))' 6 yj 6 £2)

,iin (VB((xi,z),(yi,z)) = min(pBl(xi^i), fiAl(z)),

W> | // \ r vi / / v , y, (Z e V2,Xiyi£E1).

For each x2,y2 e V2, (x2 + y2) and each x\y\ e E\, let min (jxa2(x2), ha2(y2), MB^xm)) = "> max(vA2(x2, vAl(y2), vBl(x\yi)) = b, ¡iB((xi,x2)(yi,y2)) = c, and vB ((M,x2)(yuyi)) = d. Then

(iv) /^««i.^Cw.»» = min ha2(y2),mb,Urn)),

((x\,x2),{y\,y2)) = max (vM(x2), vM(y2), vBl(x^,)),

(x2,y2 e V2,x2 * y2, x\y\ e Ei).

Theorem 3.7 Let Gi — (Ai,fii) (respectively, G2 = (A2,B2)) be an intuitionistic fuzzy graph ofG\ = (Vi, E-\) (respectively, G*2 = (V2,E2), and Vif]V2 = <j>. Then Gi U G2 = (A, B) is the union ofG\ and G2 if and only if (A[ab], B[aj>]) 's union of ((Al)\aM> (Bl)[a,«) and ((M)\a,b\' (B2)[a,b]) for each [«'b] £ 11 ■

Proof Necessity. Suppose that Gi U G2 = (A, B) is the union of Gi and G2. Firstly, we show A[aii)] = (Ai)[aj,] U (A2)[a,d] for each [a, b\ £ II. Actually, for every x £ either x e Vi and x i V2 hold or x e V2 and x £V\ hold by Vip\V2 = <j>. For the first case, we have¡iAl (x) = pA(x) > a and vAl (x) = vA(x) > b, which implies x £ (Ai)^. For the second case, we have pa2(x) = p(x) > a and vAl(x) = va(x) > b, which implies x e (A2)[aM. Therefore, * e (A{)[aM U(A2)[aM and A[aM c (Aj),^, u (A2\aM■

Conversely, for each x e (A\\aM u (A2)[o?i)], either x e (^i)[<m>] and x t (A2\ai>] hold or x f (A2)[a,&] and x £ hold by V\ Q V2 — For the first case,

we have Ma(.x) = (¿a,(.x) > a and va(x) = fAl(x) > b, i.e., x e For the

second case, we have //A(x) = fia2(x) > a and vA(x) = va2(x) > b, i.e., x e A^j and (Ai\aM u (A2)[aM c AWM. Secondly, we prove BlaM = (Bi)[a,i,] U (B2\aJb] for each [a, b\ e n. Actually, for each xy e B[ajb], either xy e E\ and xy t E2 hold or xy e £2 and xy i E\ hold since E\ H E2 = <f>. For the first case, we have (xy) = pB(xy) > a and vB,(xy) = vB(xy) > b, i.e., xy e (B]\a^. For the second case, we have piBl(xy) = ¡iB(xy) > a and vBl(xy) = vB(xy) > b, i.e., xy 6 (B2\aj,]-Therefore xy e (B-i)iaM u (B2\aM. Conversely, for each xy e (Bi\aM U (B2\aM, either xy e (Bi\aJb] and xy i (B2)[aM hold or xy e (B2)[<M>] and xy i (Bi)laJ>] hold since Ei(~)E2 = <t>. For the first case, we have nB(xy) = pBl(xy) > a and vB(xy) = vBl(xy) > b, i.e., xy e B[aj>]- F°r the second case, we have fiB(xy) = fiBl(xy) > a and vB(xy) = vBl(xy) > b, i.e., xy e BWM.

Sufficiency. Assume that (A[a>t], B^j) is the union of ((A])[a,b], (¿ii)[a,t.]) and ((A2)[aM, (B2\aM) (V [a, b] e II). For each x€ V, (i = 1,2), let fiAl(x) = a, vAi(x) = b, iua(x) = c and va(x) = d. Then¡xa(x) = Pa,(x) and va(x) = vA,(x), i.e.,

/iA(x) = /iAl(x), if X e Vi,

For each xy £ Et (i = 1,2), let nB,(xy) = a, vBi(xy) = b, nB(xy) = c and vB(xy) = d. Then fiB(xy) = fiBi(xy) and vB(xy) = vBl(xy), i.e.,

i^(xy) = Mb, (xy), if xy e Ej,

Theorem 3.8 Let Gi = (Ai,Bi) (respectively, G2 = (A2,B2)) be an intuitionistic fuzzy graph of G\ = (VuEi) (respectively, G* = (V2,E2)), and Vi f| V2 = 0. Then G\ + G2 = (A,B) is the join ofG\ and G2 if and only if (A[aj,],B[aj^) is the join of ((¿i)[<M>], (SiWl) and ((A2\aM, (B2\aM)for each [a, b] € II.

Proof Necessity. Suppose that G\ + G2 = (A, B) is the join of Gi and G2. Firstly, we know A[a,b\ = (Ai)[0if,] U (A2)[a>s] for each [a, b] e II by definition of union and the proof of Theorem 3.7. Secondly, we prove B[aj>] = (B\)\a,b\ U (B2)[a,i>] U E'(a, b) for each [a, b\ 6 II, where E'(a, b) is the set of all edges joining the vertices of (M)\a,b\ and (A2)[aj,]. (B\\aM U (B2)[<i,4| £ (V [a, b] 6 n) follows from definition of union and the proof of Theorem 3.7. For each xy e E'(a,b) (i.e., Ma,(x) > a, VA, (x) > b, ha2(x) > a and vAl(x) > b), we have nB(xy) = min (¿iA, (x), //A2(y)) > a and vB(xy) = max (nAl (x), vA2(y)) > b since Gi +G2 = (A, B) is the join of Gi and G2, it follows that xy e B[aj,y Therefore, B^j 2 (Bi)[aj,]U(B2)[aj,]UE'(a,b). Conversely, for each xy e B^j, if xy e E\ U E2, then xy e (Bi)[<,,/> 1 U (B2)\a,b\ by the proof of

Ha(x) = Ha2(x), if x e V2.

>i«(xy) = jus2(xy), if xy e E2.

vB(xy) = vBl (xy), if xyeEu vB(xy) = vBl(xy), if xy g E2.

Fuzzy Inf. Eng. (2015)7: 317-334_329

Theorem 3.7. If xy e Ell)E2,E' is the set of all edges joining the vertex of V\ and V2, then min (jiA,(s),MA2(y)) = MB(xy) > a and max (vAl(x),vAl(y)) = vB(xy) > b since Gi + G2 = (A,B) is the join of Gi and G2, it follows that x e (Ai)^] , y e (A2\a^, i.e., xy e E'(a,b). Therefore B[aM c (B,)[aM U (B2)laM U E'(a,b).

Sufficiency. Assume that (A[fl,i], B[a>6]) is the join of ((Ai)^], №)[<,,&]) and ((A2)[ath], (B2)[aM) (V[a, b] e II). By the proof of Theorem 3.7, we know

(ma(x) = Ma, (x), if x 6 Vu \pA(x) = pAl(x), ifxeV2.

ivA(x) = vAl(x), if xcVu ViW = vM(x), ifxe V2.

(mB(xy) = tiBi (xy), if xy e E,, = MB2(xy), if xy e E2.

{ vB(xy) = vBl (xy), if xy e Ei, \us(.xy) = vB2(xy), if xy e E2.

For each xy e E', let min(//Al(x),MA2(y)) = a, max^^x),^^)) = b, fiB(xy) = c and vB(xy) = d, Then

(v) [VB(xy) = rmn(MAl(x),fiAl(y)), ,f E, \vB(xy) = max(uAl (x), vAl(y)),

4. Categorical Properties of Intuitionistic Fuzzy Graphs

Many real-world problems can be very effectively described by a graph (e.g. a network), a fuzzy graph, or an intuitionistic fuzzy graph, but efficient methods to solve such problems often rely on our understanding of the structure of these graphs.

There have been some deeper and untraditional approaches to graph theory which are benefit for our understanding of the structure (including limit structure) of graphs and may allure capable pure mathematician in other areas. Research also indicates that category theory may provide a realistic platform on which inter-imitations and inter-inspirations between some field of mathematics some true [12]. We will show the categorical goodness of intuitionistic fuzzy graphs.

Theorem 4.1 The category IFG of intuitionistic fuzzy graphs and homomorphisms between them is isomorphic-closed, complete, and co-complete.

Proof Proposition 39 of [3] implies that IFG is an isomorphic-closed category [1].

Next we prove that IFG is both complete and co-complete.

Step 1: IFG has equalizers. Let Gi = (A\,B]) (respectively G2 = (A2,B2)) be

an intuitionistic fuzzy graph of GJ = (Vi,£i) (respectively G*2 = (V2, E2)). Then

Gi,G2 e object(IFG). Assume that ViZ?V2 are IFG-morphism from Gi to G2. Let

E = [x e Vi | f(x) = g(x)}, and e : E Vi be an inclusion mapping. We will show that E —» is an equalizer of / and g.

(i) (ii) (iii) (iv)

Firstly, let A = Ai\e and B = Bi\&. Then, G = (A,B) is an intuitionistic fuzzy graph on E. As e : E -* Vi is an inclusion, we have \iax (eM) = MAi (x) = jua(x), vA](e(x)) = vA,(x) = vA(x), iiB,(e(x)e(y)) = /¿B, (xy), vR,(e(x)e(y)) = vR] (xy) = vB(xy) for each x,y e E. Therefore, e e Morphism (IFG). Obviously, f ° e = g o e by definition of E.

Secondly, let «':£'—> vi be an IFG-morphism from Ge> to Gi which satisfies fo

e' = goe', where G& = (C, D) is on intuitionistic fuzzy graph of (£', E' ). Define e : E' -> E by e(x) = e'(x)(Vx e E'). As foe' = goe', we have/(e'(x)) = g(e'(x))(Vx e E'), thus e is well-defined. For each x e E', we have e o e(x) = e(e'(x)) = e'(x), thus e' = e o e. As e' e Morphism (IFG), we have fic(x) < fiAl(e'(x)) = ¡iA(e'(x)) = fiA(e(x)), vc(x) > vAl(e'(x)) = vA(e'(x)) = vA(e(x)), nD(xy) < fiB, (e'(x)e'(y)) = Hn(e'(x)e'(x)) = nB(e(x)e(y)), ¡ic(x) < nAl(e'(x)),pA(e'(x)) = /xA(e(x)), vn(xy) > vB, (e'(x)e'(y)) = vB(e'(x)e'(xj) = vB(e(x)e(y))fic(x) < fiAl(e'(x)\ ¡iA(e'(x)) = fiA(e(x)), for each x,y e E', which implies e e Mor (IFG). Clearly, such an IFG-morphism e is unique.

Suppose that e' :£'—»£ is an IFG-morphism from G^-to G\ satisfying e' = e ° e', then e o e = e o e', thus we have e'(x) = e o e'(x) = e o e~(x) = e(x) (Vx e E') since e is an inclusion, which implies e' = e.

Step 2: IFG has products (this together with Theorem 12.3 in [1] and Step 1, implies that IFG is complete). Assume that G,- = (A;, Bt) is an intuitionistic fuzzy graph of G* = (Vi,Ei)(Vi g /), then G; eobject (IFG) (V, e /).

Let V = n Vi, nj : V -» Vj be the projection (Vj e /), and define intuition-

istic fuzzy graph sets A = (41A, vA) and B = (jiB, vB)on V andV2 respectively by

^a((x«),e/) = h flAt(Xi), VA((Xi)ie[) = V VA(Xi), llB((Xi)iEl(yi)«il) = AUBt(Xiyi), VB((Xi)ieI ie/ je7 1 €/

(y/)ie/)) = v vBXx{yi) for any (Xi)iel, (yi)ieI e n Vi. Denote G = (A, B), we will show

that (G, (;r,-)je/) is the product of (G,),e/.

Firstly, as gi = (at, bi) is an intuitionistic fuzzy graph of G* = (V,, £;)(V, e /),

Mli((Xi)ier(yi)iel) = A tiBt(Xiyi) < A mm(jiA.(Xi),IXA,(yi)) = (A UA,(Xi)) A (AJlAfyi)) = iel if / jg/ ¡i /

/^a(,(.xi)i€i)A//a((Vi)ie/)- Analogously, uB(fe)te/Cy.)i€/) > faifewivuaicy.k/)- Therefore, Ge object (IFG).

Secondly, as Kj: V -» Vj is a projection (Vj e /), we have fiA((xi)iei) = =

A WA,(Jr>((*i)iE/)) <ilA,(^((*i)iE/)),^B((*i)iei(yi)ie/) = = A

(6/ ' ¡e/

((yi)iel) < MB^jdxdiel^jVydiel))-Analogously, uA((Xi)ie/) > ^.(^(teW),

VB(iXi)i^(yi)M) > VBj(njdXi)^)?!j((yi)i€,)).

Therefore, ^ e Morphism (IFG) (V, e /).

Now, we finally suppose that if = (C, £>) is an intuitionistic fuzzy graph of H* =

(X,R) and fj : X -» Vj is an IFG-morphism from H to Gj (Vj e /). For each x £ X,

let /(x) e Vwith7r//(x)) = /;(x)(Vy £ /). Then /: X -> V is a mapping. Since X —^ Vj is an IFG-morphism (Vj e /), /icM < A fiA,(fi(x)) = uA(f,(x)iel) = fiA(f(x)),

Mn(xy) < AMB,(fi(x)fi(y)) = LiB((fi(x))ie,(fi(y))ie,) = tiB(f(x)f(y)). Analogously,

fc(x) > vA(f(x)) and vo(xy) > vB(f(x)f(y)). Therefore, f is an IFG-morphism with

7tj o / = fj(yj e f). Such an IFG-morphism / is unique. In fact, if / : X -> V is an IFG-morphism satisfying nj a f = fjQfj 6 /), thennjof = xjof. As nj is a projection (V/ e I), we have / = /.

Step 3: IFG has co-equalizers. Let Gi = (A],B\) (respectively, G2 = (A2,B2)) be

an intuitionistic fuzzy graph of GJ = (Vi,£i) (respectively, G"2 = (V2,E2)). Then

G\,G2 e object (IFG). Assume that ViZ?V2 are IFG-morphism from Gi to G2,

is the smallest equivalence relation on V2 such that f(s) ~ g(s) for all s £ Let Q = V2I ~= {[x]x e V2} and q : V2 —> Q be the mapping defined by q(x) = [x] for

each x 6 V2. We will show that V2 —> Q is a coequalizer of / and g.

Firstly, define intuitionistic fuzzy sets A = (fia< va) and B = (jib, vb) on Q and Q2 respectively by jUa(W) = V uAl(z),vA(W) = A vAl(z), //b(WLvJ) = V

Ze[x] Ze[x] Z€[x]Jce\y]

fiih(zk), vB([x\\y\) = A vRl(zK). Since Gi = (A,,B,) and G2 = (A2,B2) are

z€[x],ie [y]

intuitionistic fuzzy graphs of GJ = (Vi, £1) and G*2 = (V2,E2) respectively, we have ft(WM) < min(iiA(W),^A(ty])),ys(WM) > maxii/ACUD.i'Aib'])), which implies G = (A, B) is an intuitionistic fuzzy graph on Q. As fiAl(x) < v fiAl(z) = //a(W) =

z£[l]

¡iA(q(x)),vAl(x) > A vAl(z) = vA(\x\) = vA(q(x)),/iB2(xy) < V Un2(zk) = Ze[x] Ze[x]Miy]

/isCWM) = mb(q(x)q(y)), vBl(xy) > a vBl(zk) = i/j([i]|jD = vB(q(x)q(y))

for each x,y e V2,q e Morphism (IFG). By definition of q(f(s)) = [/(s)l =

= l(g(s))(Vs 6 Vi), i.e., q°f = q°g.

Secondly, let : V2 -* Q' be an IFG-morphism from G2 to Gq which satisfies

q' 0 f = q' 0 g> where Gq = (C, D) is an intuitionistic fuzzy graph of (¡2', Q'2)- For

each [xj e Q, let ?([x]) = q'(x). Then we define a mapping q : Q -> Q. In fact, as

R = {(x,y) e V2 | q'(x) = q'(y)} is an equivalence relation on V2 and q' ° f = q' ° g,

we have (/(x), g(x)) e R, thus ~ eg. For any x, y e V2 satisfying (x, y) e~, we have

[x] = [y] and (x,y) e R, thus q [x] = q'(x) = q'(y) = q(\y\). As q o q(x) = q({x\) =

q'(x) (Vx e V2), q ° q = q'. As q' eMorphism (IFG), we have ¡J.A([x\) = V fiA2(z) <

y Uc(q'(z)) = V uc(q([z])) = Mc(q([x\)) ^a(W) = A vAl(z) > V vc(q'(z)) = ze[x] ze[x] ze[x] ze[x]

A, vc(q{[zi)) = vc(qdx\)),tiB([x\ [>]) = V UBl{zk) < V uD(q'(z)q'(k) =

zeW ze[x], ze[x],te[y]

V md (?(M )?(№])) = MD (<7(M )?(M)) , VB ( [x] [y]) = A VB2(zk) >

A r n VD (q'(z)q'(k)) = A vD ®[z] )) = vD {q([x\)?([y])). For each Z€[i]^€[y] zeW M[y\

[x], [y] e Q, which implies q e Morphism (IFG).

Clearly, such an IFG-morphism q is unique. Suppose that q' : Q —> Q is an IFG-morphism from G to Gq satisfying q' o q = q', then q' o q = q o q, thus ?'([*]) = q' o q(x) = q o q(x) = q(\x]) (V [x] 6 Q) by definition of q , which implies q' = q.

Step 4: IFG has coproducts (this, together with Theorem 12.3 in [1] and Step 3, implies that IFG is co-complete). Assume that G, = (Ai; B,) is an intuitionistic fuzzy graph of G* = (Vi,Ei) (Vi e I), then G,- e object (IFG) (Vi e I).

Let V = ffi Vi = U (Vi x {/"}), qi : V, -» V is a mapping satisfying qAxt) = (x,-, f)

iel i€ 1

for all Xj e Vj (V/ e I), and define intuitionistic fuzzy sets A =< ¡iA,vA > and B = (ftB,vB) on V and V2 respectively by fiA(Xi, i) = ¡iA:(Xi), vA(Xi, i) = vA.(x,),

(, ¡HBl(Xi,yj), if I = j,

MB{(Xi,i)(yj,j)) = |0;

1, {vBi(Xi,yj), if i = j,

vB{(xt,i)(yjJ)) = ^

For each (x,, i), (yj,f)e V. Denote G = (A, B), we will show that ((<?,),£/, G) is the coproduct of (G,)i£/.

Firstly, as Gi = (A„ B,) is an intuitionistic fuzzy graph of G* = (V,-, £,) (Vi 6 /), G = (A, B) e object (IFG).

Secondly, by definition of qj(Vj e /), we have

MAjiXj) = (iA(Xj,j) = iiA(qj(Xj)), vaj(xj) = vA(xjJ) = vA(qj(xj)),

tiB^Xjyj) = Ha ((Xj, j)(yj, j)) = n„ (qj(Xj) qjiyj)), VBjixjyj) = vB ((Xj, f)(yj, j)) = vB (qjixj) q/yj)). Therefore q} e Morphism (IFG) (Vj e I).

Finally, suppose that H = (C, D) is an intuitionistic fuzzy graph of H* = (X, R) and qj : Vj -» X is an IFG-morphism from Gj to H (Vj e I). Define g : V —> X by g(xj, i) = gi(xi) for each (x,-, i) e V. Obviously, goqj = gjQ/j e /). Since Vj X is an IFG-morphism (Vj g I), we have fiA(Xj, j) = fiAj(xj) < juc (gj(Xj)) = fic (g(xj, /)), Mb ((xj, j)(yj, j)) = MBfayj) < Md (gj(xj)gj(yj)) = fiD(g(xj, j)g(yj, j)). Analogously, va(xj,j) > vc{g(xj,j)), vB[(xj,j)(yj,j)) > vD{g(xj,j)g(yj,j)). Therefore, g is an IFG-morphism with g 0 qj = gj(Vj e I). Such an IFG-morphism g is also unique. In fact, if g : V -* X is an IFG-morphism satisfying g o = gj(Vj e /), then 8 ° 1j = 8 ° y definition of qj(V j € /), we have g = g.

5. Conclusion

Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. In computer science, graphs are used to represent networks of communication, data organization, computational devices and the flow of computation. One practical example in the link structure of a website could be represented by a directed graph. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. It is known that intuitionistic models give more precision, flexibility and compatibility to the system as compare to classical and fuzzy models. In this paper, we demonstrate rationality of some defined important notions and some newly defined notions on intuitionistic fuzzy graphs where some skills or ideas of fuzzy set theory and lattice theory are applied and thus very closed connections between those intuitionistic fuzzy graphs and their level counterparts are established. We also show some categorical goodness of intuitionistic fuzzy graphs by proving that IFG is isomorphic-closed, complete, and co-complete. We believe that category theory may provide a possible platform on which inter-imitations and inter-inspirations between graph theory and some linked fields of mathematics come true. In our future work, we will focus on energy and hyperenergetic of intuitionistic fuzzy graphs which are very useful in physics and chemistry.

Acknowledgments

The authors would like to thank the referees and Editor-in-Chief for their useful comments and suggestions to improve the presentation of the paper.

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