Pacific Science Review A: Natural Science and Engineering xxx (2016) 1—10

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Pacific Science Review A: Natural Science and Engineering

journal homepage: www.elsevier.com/locate/PSRA

Different types of products on intuitionistic fuzzy graphs

Sankar Sahoo*, Madhumangal Pal

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India

ARTICLE INFO ABSTRACT

In this paper, we define three operations on intuitionistic fuzzy graphs, viz. direct product, semi-strong product and strong product. In addition, we investigated many interesting results regarding the operations. Moreover, it is demonstrated that any of the products of strong intuitionistic fuzzy graphs are strong intuitionistic fuzzy graphs. Finally, we defined product intuitionistic fuzzy graphs and investigated many interesting results.

Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University. Production and hosting by Elsevier B.V. This is an open access article under the CC

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Keywords:

Intuitionistic fuzzy graphs Direct product Semi-strong product Strong product

Product intuitionistic fuzzy graphs

1. Introduction

Graph theory has applications in many areas of computer science, including data mining, image segmentation, clustering, image capturing, and networking. An intuitionistic fuzzy set is a generalisation of the notion of a fuzzy set. Intuitionistic fuzzy models provide more precision, flexibility and compatibility to the system compared to the fuzzy models.

In 1983, Atanassov [6,7] introduced the concept of intuitionistic fuzzy set as a generalisation of fuzzy sets. Atanassov added new components that determine the degree of non-membership in the definition of fuzzy set. The fuzzy sets give the degree of membership, 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 one. Intuitionistic fuzzy sets have been applied in a wide variety of fields, including computer science, engineering, mathematics, medicine, chemistry, and economics.

In 1975, Rosenfeld [19] discussed the concept of the fuzzy graph, the basic idea of which was introduced by Kauffman [9] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld; he developed the structure of fuzzy graphs and obtained analogues of several graphs theoretical concepts. Atanassov introduced the concept of the intuitionistic fuzzy relation. Different types of intuitionistic fuzzy graphs and their applications can be found in several papers. In addition, Sahoo and Pal [20] discussed the concept of the intuitionistic fuzzy competition graph. In this

* Corresponding author. E-mail addresses: ssahoovu@gmail.com (S. Sahoo), mmpalvu@gmail.com (M. Pal).

Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University

study, the direct product, semi-strong product and strong product of two intuitionistic fuzzy graphs are defined, and many interesting results involving these operations are investigated. Moreover, we defined product intuitionistic fuzzy graphs and investigated their many interesting properties.

2. Preliminaries

2.1. Graphs

A graph is an ordered pair G = (V,E), where V is the set of all vertices of G (which is non empty), and E is the set of all edges of G. Two vertices x,y in a graph G are said to be adjacent in G if (x,y) is an edge of G. A simple graph is a graph without loops and multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has n(n-1)/2 edges.

An isomorphism of graphs G1 and G2 is a bijection f between the sets of vertices of G1 and G2, such that any two vertices v1 and v2 of G1 are adjacent in Ref. G1 if and only iff(v1) and f(v2) are adjacent in Ref. G2. Isomorphic graphs are denoted by Ref. G1 y G2.

Let G1 = (V1,E1) and G2 = (V2,E2) be two simple graphs. Next, the direct product of G1 and G2 is a graph G1 nG2 = (V, E) with V = V1 x V2 and E = {((u1,v1),(u2,v2))|(u1,u2)eE1,(v1,v2)eE2}.The union of graphs G1 and G2 is defined as G1uG2 = (V1uV2, E1uE2) and join is the simple graph G1 + G2 = (V1 uV2, E1 uE2uE/), where E' is the set of all edges joining the vertices of V1 and V2, also assume that. V1 nV2 = f.

2.2. Intuitionistic fuzzy graphs

An intuitionistic fuzzy set A on the set X is characterised by a mapping m:X/[0,1] (which is known as a membership function) and n:X/[0,1] (which is called a non-membership function). An

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intuitionistic fuzzy set is denoted by Ref. A = (X,mA,nA). The membership function of the intersection of two intuitionistic fuzzy sets A = (X,mA,nA) and B = (X,mB,nB) is defined as mAnB = min{mA, mBg and the non-membership function nAnB = max{nA, nBg. We write A = (X,mA,nA) 4 B = (X,mB,nB) (intuitionistic fuzzy subset) if mA(x) < mB(x) and nA(x) > nA(x) for all xeX.

Definition 1. The support of an intuitionistic fuzzy set A = (X,mA,nA) is defined as Supp(A) = {x eX:mA(x) s 0 and nA(x) s 1}. In addition, the support length is SL(A) = |Supp(A)|.

Definition 2. The core of an intuitionistic fuzzy set A = (XA,mA,nA) is defined as Core(A) = {xeX:mA(x) = 1 and nA(x) = 0}. In addition, the core length is CL(A) = |Core(A)|.

Now, we defined the height of an intuitionistic fuzzy set below:

Definition 3. The height of an intuitionistic fuzzy set A = (X,mA,nA) is defined as h(A) = (supmA(x), inf nA(x)) = (hm(A), hn(A)).

Here, an intuitionistic fuzzy graph is defined below:

Definition 4. An intuitionistic fuzzy graph is of the form G = (V,E,s,m) where s = (s1,s2), m = (m1,m2) and

(i) V = {vo,vv..,vn} suchthat s1:V/[0,1] and s2:V/[0,1] denote the degree of membership and non-membership of the vertex vieV, respectively, and 0 < s1(vi) + s2(vi) < 1 for every vieV (i = 1,2,...,n).

(ii) m1:VxV/ [0,1] and m2:VxV/ [0,1], where m1(vi,vj) and mi(vi,vj) denote the degree of membership and non-membership value of the edge (vi,v)), respectively, such that m1(vi,vj) < min {s1(v,),S1(vj)} and M2(v,-,vj) < max{s2(vi),s2(vj)}, 0 < m^v,-) + m2(vi,v,) < 1 for every edge (v;,v,).

Iow, we give an example of the intuitionistic fuzzy graph: Example 1 Let G = (V,s,m) be an intuitionistic fuzzy graph, where s(v) = (s1(v),s2(v)), m(u,v) = (m1(u,v),m2(u,v)). Let the vertex set be V = {v1,v2,v3,v4} and s(v1) = (0.3,0.6), s(v2) = (0.8,0.2), s(v3) = (0.2,0.8), s(v4) = (0.5,0.4); m(v1,v2) = (0.25,0.45), m(v2,v3) = (0.18,0.75), m(v3,v4) = (0.15,0.52), m(v4,v1) = (0.3,0.25), m(v1,v3) = (0.2,0.8), m(v2,v4) = (0.4,0.35). The corresponding intuitionistic fuzzy graph is shown in Fig. 1.

Definition 5. [1] An intuitionistic fuzzy graph G = (V,E,s,m) is said to be complete if m1(vi,v)) = s1(vi)As1(v)) and m2(vi,vj) = s2(v,)vs2(vj) for all vi,vjeV.

Definition 6. [11] The complement of an intuitionistic fuzzy graph G = (V,E,s,m) is an intuitionistic fuzzy graph G = (V, E,s, m) where

Fig. 1. An intuitionistic fuzzy graph.

S = s, m1(vi, vj)= S1(vi)AS1(vj) - m1(vi, vj) and m2(v;, vj)=S2(vi) vs2 (vj) - m2 (v; , vj) for all vi,vj e V.

2.3. Review of the literature

After Rosenfeld [19], the fuzzy graph theory increases with its various types of branches, such as fuzzy tolerance graph [25], fuzzy threshold graph [24], bipolar fuzzy graphs [17,18,30], highly irregular interval valued fuzzy graphs [12,14], isometry on interval-valued fuzzy graphs [16], balanced interval-valued fuzzy graphs [10,15], fuzzy k-competition graphs and p-competition fuzzy graphs [28], fuzzy planar graphs [23,31], bipolar fuzzy hypergraphs [26,27], and m- step fuzzy competition graphs [22]. The fuzzy graph theory is used in telecommunication system [29]. A new concept of fuzzy colouring of fuzzy graph is given in Ref. [32]. Ramaswamy and Poornima [13] discussed product fuzzy graphs.

Akram and Davvaz [1] defined strong intuitionistic fuzzy graphs. They also discussed intuitionistic fuzzy hypergraphs with applications [3]. A novel application of intuitionistic fuzzy digraphs is given by Akram et al. [2]. In addition, Akram and Al-Shehrie [4] defined intuitionistic fuzzy cycles, intuitionistic fuzzy trees and intuitionistic fuzzy planar graphs [5]. Balanced intuitionistic fuzzy graphs are discussed by Karunambigai et al. [8]. Moreover, Parvathi, Karunambigai [11] defined intuitionistic fuzzy graphs. Sahoo and Pal [20] discussed the concept of intuitionistic fuzzy competition graph. They also discussed intuitionistic fuzzy tolerance graphs [21].

2.4. Our contribution

In Section 3, we define the direct product of two intuitionistic fuzzy graphs and demonstrate that the direct product of two strong intuitionistic fuzzy graphs is strong; consequently, if the direct product is strong, then any one of two intuitionistic fuzzy graphs is strong. Next, we define semi-strong, strong product of two intui-tionistic fuzzy graphs and many interesting properties. Finally, in Section 4 we discuss the product intuitionistic fuzzy graph and investigated many interesting results.

3. Products on intuitionistic fuzzy graphs

In this section, we define three operations on the intuitionistic fuzzy graphs, viz. direct product, semi-strong product and strong product.

3.1. Direct product of two intuitionistic fuzzy graphs

First, the direct product of two intuitionistic fuzzy graphs is defined.

Definition 7. The direct product of two intuitionistic fuzzy graphs G' = (V,E',Sm) and G" = (V",E",s",m") such that VnV" = f, is defined to be the intuitionistic fuzzy graph G' nG'' = (V, E, S n&'', m' nm'') where V = VxV'', E = {((u1,v1),(u2,v2))|(u1,u2)eE',(v1,v2)eE''}. The membership and non-membership values of the vertex (u,v) in G nG are given by

(s'1ns'1) (u, V) = s'1(u)Asi(v) and

(s2nS2) (u, v) = s2(u)vs'2(v).

In addition, the membership and non-membership values of the edge ((ui,vi),(u2,v2)) in GnG'' are given by

(mi nmi) ((ui, vi), (U2, v2)) = mi(ui, u2)Ami(vi, V2)

(m2nm2)((ui, vi), (u2, V2)) = m2(ui,u2)vm2>i, V2).

Example 2. Let G' = (V,E',s',m') and G" = (V',E'',s'',m'') be two intuitionistic fuzzy graphs, such that V = {a,b}, V = {c,d}, E' = {(a,b)}, E = {(c,d)}, s'(a) = (0.7,0.3), s'(b) = (0.5,0.4), m'(a,b) = (0.4,0.3), s''(c) = (0.5,0.5), s' (d) = (0.6,0.4), and m''(c,d) = (0.5,0.4). Thus, G', G'' and G'nG'' are shown in Fig. 2.

Definition 8. An intuitionistic fuzzy graph G = (V,E,s,m) is called strong intuitionistic fuzzy graph if mi(vi,v)) = si(vi)Asi(v)) and m2(Vi,Vj) = s2(vi)vs2(vj) for all (Vi,Vj)2E.

Theorem 1. If G' = (V,E',s',m') and G' = (V',E'',s",m") are strong intuitionistic fuzzy graphs, then G' nG'' is also strong.

Proof. Because G' = (V,E',s',m') and G = (V,E",s",m") are strong intuitionistic fuzzy graphs, then m'i(ui, u2) = CT'i(ui)ACT'i(u2), m2(ui,u2) = s2(ui)vs2(u2), mi(vi,V2) = si'(vi)Asi'(V2) andm2(vi,V2) = ct2'(vi)vct2'(v2) for all (ui,u2)eE' and (vi,v2)eE .

(minmi) ((ui, vi), (u2, V2)) = mi (ui, u2)Ami(vi, = [si(ui)As'i(u2 )]A[si(vi)Asi(v2)] = [si(ui)Asi(vi)]A[si(u2)Asi(v2)] = (s'i ns'i) (ui, vi) A si ns'i) (u2, V2) In addition,

(m2nm2) ((ui, vi), (u2, v2)) = m2(ui, u2)vm2(vi, v2) = [s2(ui)vs2(u2 )]v[s2(vi)vs2(v2)] = [s2(ui)vs2(vi)]v[s2(u2)vs2(v2)] = (s2ns2) (ui, v1)v(s2ns^ (u2; v2) Hence, G' nG'' is a strong intuitionistic fuzzy graph.

Theorem 2. If G' = (V,E',s',m') and G = (V ,E ,s ,m ) are two intuitionistic fuzzy graphs such that G' nG'' is strong, then at least one of G or G must be strong.

Proof. Suppose that G' and G are not strong. Thus, there exists at least one (ui,u2) e£',(vi,v2) gE such that mi (ui, u2) < Si (ui )As'i(u2), m2(ui,u2)<s2(ui)vs2'(u2), mi(vi,v2)<s'i(vi)As'i(v2) and m2(vi,v2) < s2(vi)vs2(v2). Thus,

(minmi)((ui, vi), (u2, v2)) = mi (ui, u2)Ami(vi, v2)

< [si (ui)Asi(u2)] A[si(vi)Asi(v2)] = [si(ui)Asi' (vi)]A[si'(u2)As'i (v2)] = (s'i ns'i) (ui, vi )a(si ns'i) (u2, v2) Therefore,

(minmi) ((ui, vi), (u2, v2)) < (sinsi) (ui, v')A(sinsi) (u2,

Again,

(m2nm2)((ui, vi), (u2, v2)) = m2(ui,u2)vm2(vi, <[s2(ui)vs2(u2)]v[s2(vi)vs2(v2)] = [s2(ui)vs2(vi)]v[s2(u2)vs2(v2)] = (s2ns^ (ui, vi)v(s2ns^ (u2, v2) Therefore,

(m2nm2) ((ui, vi), (u2, v2)) < (s2ns2) (ui, vi)v(s2ns2) (u2,

This equation shows that G'nG' is not strong, a contradiction. Hence, at least one of G or G must be strong.

3.2. Semi-strong product of two intuitionistic fuzzy graphs

Next, the semi-strong product of two intuitionistic fuzzy graphs is defined.

Definition 9. The semi-strong product of two intuitionistic fuzzy graphs G' = (V,E',s',m') and G" = (v",eVV) such that V'n V = f is defined to be the intuitionistic fuzzy graph G'»G = (V,E,s'»s ,m'»m ), where V = VxV' and E = {((u, v'), (u, v2 ))|ugE', (v', v2)gE''g u{((ui, vi), (u2, v2))|(ui,u2)2E', (vi, v2)gE'g. The membership and non-membership values of the vertex (u,v) in Ref. G »G are given by

(si » s'i)(u, v) = si(u)As'i(v)

(s2 » s2)(u, v) = s2(u)vs2(v).

In addition, the membership and non-membership values of the edge in G »G are given by

J (m'l » mi) ((u, vi), (u, v2)) = s'i (u)Ami'(vi, v2) 1 (mi » mi) ((ui, vi), (u2, v2)) = mi (ui, u2)Ami (vi, v2)

(ii)í * m2)((u, vi), (u v2 h ^ (m2 • m2')(("i, vi), (u2,

), (u, V2)) = s2(u)vm2'(vi, V2) ), (U2, V2)) = m2(ui, U2)vm2(vi, V2).

Fig. 2. Direct product of two intuitionistic fuzzy graphs.

Example 3 Let G = (V,E',S,m') and G' = (V',eVV) be two intuitionistic fuzzy graphs, such that V = {a,b}, V = {c,d}, E' = {(a,b)}, E = {(c,d)}, S (a) = (0.7,0.3), S(b) = (0.5,0.4), m'(a,b) = (0.4,0.3), a (c) = (0.5,0.5), a (d) = (0.6,0.4), and m''(c,d) = (0.5,0.4). Thus, G', G and G'»G are shown in Fig. 3.

Theorem 3. If G' = (V,E',S,m') and G = (V ,E ,m ) are strong intuitionistic fuzzy graphs, then G'»G is also strong.

Proof. If ((u,Vi),(u,v2))eE, then (mi • mï)((u, vi), (u, v2)) = S1(u)Amï(v1,

= si (u)A[sï(vi)As'i(v2)] = [si(u)As'ï (vi)]A[s'ï(u)Asï(v2)] = (si • s'D (u, vi )a(si • s'D (u, V2). Similarly, we can show that

(m2 • m2)((u, vi); (u v2)) = (s2 • s2)(u vi)v(s2 • s2)(u v2)-Again, if ((ui,vi),(ui,v2))eE, then

(mi • mi')((ui, vi), (u2, V2)) = mi(ui,u2)Amï(vi, V2)

= [si (ui ) as' (u2 )] a [s'Ï (vi ) as'' (V2 )] = [s'i (ui ) As'' (vi )] A [s'' (u2 ) As'' (v2 )] = (s'i • s'ï)(ui, vi)A(s'i • s'ï)(u2, v2). Similarly, we can show that

(m2 • m2)((ui, vi), (u2, v2)) = (s2 • s2) (ui, vi)v(s2 • s^ (u2, v2).

Therefore, G'»G is also strong intuitionistic fuzzy graph.

Theorem 4. If G' = (V,E',S,m') and G'' = (V',eW) are two intuitionistic fuzzy graphs, such that G'»G is strong, then at least one of G' or G must be strong.

3.3. Strong product of two intuitionistic fuzzy graphs

Finally, we defined strong product of two intuitionistic fuzzy graphs.

Definition 10. The semi-product of two intuitionistic fuzzy graphs G' = (V'EE,S,m') and G = (V',E'W) such that V'n V'' = f is defined to be the intuitionistic fuzzy graph G' 5 G = (V,E,S 5 a ,m' 5 m ), where V = V'xV' and E = {((u, vi), (u, »2))|ueE', (vi, eE"}u{((ui, v),

(u2, V))|(u1, u2)eE', veE}u{((u1, V1), (u2, v2)) |(u1, u2) eE', (V1, v2) eE''}. The membership and non-membership values of the vertex (u,v) in Ref. G' 5G are given by

(si®si')(u, v) = si(u)Asi'(v)

(s2®s^ (u v) = s2(u)vs2(v).

In addition, the membership and non-membership values of the edge in G' 5G are given by

( (mi 5 mi) ((u, V1), (u, V2)) = s1 (u)Am/1/(v1, V2) (iW (mi 5mi) ((u, v), (u2, v)) = mi (u1, u2)as/ (v)

{ (mi®mi)((ui, vi), (u2, V2)) = mi(ui,u2)Ami(vi, V2) C (m25m2,)((u1 Vi),(u V2)) = s2(u)vm2(vi V2)

(iiW (m2®m2)((ui>v), (u2,v)) = m2(ui,u2)(u)vs2(v)

I (m2®m^((ui, vi), (u2, V2)) = m2(ui,u2)vm2(vi, V2)

Example 4 Let G' = (V',E',S,m') and G'' = (V',E'',sV) be two intuitionistic fuzzy graphs, such that V = {a,b}, V = {c,d}, E' = {(a,b)}, E'' = {(c,d)}, s'(a) = (0.7,0.3), s'(b) = (0.5,0.4), m'(a,b) = (0.4,0.3), s''(c) = (0.5,0.5), s''(d) = (0.6,0.4), and m''(c,d) = (0.5,0.4). Thus, G', G'' and G' 5G are shown in Fig. 4.

Theorem 5. If G = (V ,E ,s ,m ) and G = (V ,E ,s ,m ) are complete intuitionistic fuzzy graphs, then G' 5G is complete.

Proof. As a strong product of intuitionistic fuzzy graphs is an intuitionistic fuzzy graph, and every pair of vertices is adjacent. If ((u,v1),(u,v2))eE, then

(mi ®mi)((u, vi), (u, V2)) = si(u)Ami(vi, V2))

= si(u)A[si(vi)A s/(v2)] = [si(u)Asi(vi)] A [si(u)Asi(v2)] = (s/ 5 s/) (u, Vi )a(s/ 5 s/) (u, V2).

(m25m2)((u, Vi), (u, V2)) = s2(u)vm2(vi, V2))

= s2(u)v[s2(v/)vs2(v2)] = (s2(u)vs2(vi)] v (s2 (u)vs2(v2)] = (s25s2) (u, V/)v(s2 5s2) (u, V2). If ((u1,v),(u1,v))eE, then

Fig. 3. Semi-strong product of two intuitionistic fuzzy graphs.

Fig. 4. Strong product of two intuitionistic fuzzy graphs.

(mi 5MÏ)((ui , v), (U2, v)) = mi (Ui, U2)As"(v))

= [si(Ui)Asi(U2)]Asi'(v)] = [si(Ui)^si(v) A [s'i (U2)As'i(v)] = (si 5 si) (Ui, v)A(si 5 si) (U2, v).

Similarly,

(m25m^((Ui,v), (U2,v)) = (s25s2) (Ui, v)V(s25s^ (U2,v).

Again, if ((ui,vi),(ui,v2))e£, then

(mi 5mi) ((ui, vi), (U2, v2)) = mi (ui, U2)Ami(vi, v2)

= [si(Ui)Asi(U2)]A[si(vi)Asi(v2)] = [si(Ui)Asi(vi) A[si(U2)Asi(v2)] = (si 5 si ) (Ui, vi ) A (si 5 si ) (U2, v2 ).

Similarly,

(m20((Ui, vi), (U2, v2))= (s2 5s2') (Ui, vi)v(s25s2) (U2, v2).

Hence, G 5 G" is complete.

Theorem 6. If G' = (V,E',s',m') and G = (V,E',s'',m'') are two intuitionistic fuzzy graphs, such that G' 5G is strong, then at least one of G or G must be strong.

4. Product intuitionistic fuzzy graphs

Here, we define product intuitionistic fuzzy graphs as follows.

Definition 11. Let G = (V,E,s,m) be an intuitionistic fuzzy graph. If mi(vi,vj)<si(v,)xsi(vj) and m2(vi,vj)<s2(v,)xs2(vj) for all (vi,vj)eV, where "x" represent ordinary multiplication, then the intuitionistic fuzzy graph G is called the product intuitionistic fuzzy graph.

Remark 1 If G = (V,E,s,m) is an intuitionistic fuzzy graph, because s1 and s2 are less than or equal to 1, it follows that

mi (vi, ) < si (vi) x si (j < min{si (vi), si (vj)} and

m2 (vi,Vj) < S2(vi) x S2 (vj) < max{s2(vi),^2 (v/)}fora//(vb J eV.

Thus, every product intuitionistic fuzzy graph is an intuitionistic fuzzy graph.

Example 5 Let G = (V,E,s,m) be a product intuitionistic fuzzy graph. Let the vertex set be V = {v1,v2,v3} and s(v1) = (0.7,0.3), s(v2) = (0.5,0.4), s(v3) = (0.4,0.6), m(vi,v2) = (0.35,0.10), m(v2,v3) = (0.i9,0.20), and m(vi,v3) = (0.25,0.i5). This example is shown in Fig. 5.

Definition 12. A product intuitionistic fuzzy graph G = (V,E,s,m) is said to be complete if mi(vi,vj) = si(vi)xsi(vj) and m2(vi,vj) = s2(v,)

x s2(vj) for all (vi,vj) e V.

Definition 13. The complement of product intuitionistic fuzzy graph G = (V,E,s,m) is an intuitionistic fuzzy graph G = (V, E, s, m) where s = s, mi(vi, Vj) = si(vi) x si(Vj) -mi(vi, vj) and m2(vi, vj) = s2(v,-) xs2(vj) - m2(v,-, vj) for all (vi,vj)eV, where "x" and "-" represent ordinary multiplication and subtraction, respectively.

Remark 2 It follows that, if G is a product intuitionistic fuzzy graph,

then. G = G.

Definition 14. Consider the product intuitionistic fuzzy graphs G = (V,E',S,m') and G" = (V,E ,s",m"), the isomorphism between two product intuitionistic fuzzy graphs G ,G is a bijective mapping h:V/V such that si(u) = S{(h(u)), S2(u) = s"(h(u)) for all ueV'

and mi(u,v) = m'i(h(u), h(v)),i2(u, v) = m2'(h(u),h(v)) for all u,veV. Thus, we can write G y G .

Definition 15. The union of two product intuitionistic fuzzy graphs G' = (V,E',S,m') and G" = (V',eVV) is defined as G = G'uG" = (V, E, s, m), where V = V'uV'' and E = E'uE''. The membership and non-membership values of the vertex ueG are given by

Fig. S. Product intuitionistic fuzzy graphs.

fs/(u), ifueV' - V''

s1(u) = (s/ us/) (u) = s/ (u), ifueV'' - V'

(s/ (u)vsi(u), ifueV' nV''

fs2 (u), ifueV' - V''

s2(u) = (s2us2')(u) = s2 (u), ifueV'' - V'

(s2 (u)As2(u), ifueV' nV''

In addition, the membership and non-membership values of the edge (u,v)eG are given by

( mi (u, v), if (u, v)eE' - E''

mi (u, v) = (m/umi)(u, v) = mi(u, v), if (u, v)eE'' - E'

( mi(u, v)vmi(u, v), if (u, v)eE'nE''

f m2(u, v), if (u, v)eE' - E''

m2(u, v) = (m2um2') (u, v) = m2(u, v), if (u, v)eE'' - E'

t m2(u, v)лm2(u, v), if (u, v)eE'nE''

Definition 16. The join G'+G = (V,E,s'+s ,m' +m ) of two product intuitionistic fuzzy graphs G' = (V',E',s',m') and G" = (V,E",s",m") is defined as follows:

(si + si)(u) = (siusi)(u), (s2 + s2')(u) = (s2ns2')(u), if u eV uV''; (mi + mi)(u, v) = (miumi)(u, v), (m2 + m2')(u, v) = (m2nm2) (u, v), if (u,v)eE'nE'' and (m/ + mi)(u,v) = s1(u)vs/(v) (m2 + m2)(u, v) = s2(u)as2'(v), if (u,v)eE1, where E1 is the set of all edges joining the vertices of V ,V .

Theorem 7. If G' = (V',E',s',m') and G'' = (V',E'',s'',m'') are the product intuitionistic fuzzy graph, then

(i)G' uG' 'sG' + G''

(ii)G' + G' 'sg' ug' '.

Theorem 8. The direct product of two intuitionistic fuzzy graphs is a product intuitionistic fuzzy graph.

Proof. Let G = (V",E,s",m") and G" = (V",£",s",m") be two product intuitionistic fuzzy graphs. We shall prove that G"nG"" is a product intuitionistic fuzzy graph.

If ((ui,v/),(u2,v2))eE, then

(mi nmi) ((ui, vi), (u2, V2)) = mi (ui, u2)Ami (v/ , V2)

< [si(ui) x si(u2)]A[si(vi) x s/(v2)] = (si(ui)Asi(vi^ x [si (u2)Asi(v2)] = (s/nsi)(ui, Vi) x (s/nsi)(u2, V2). Therefore,

(minmi)((ui, Vi), (u2, V2)) < (s/nsi)(ui, v/) x (s/ns/)(u2, V2). Again,

(m2nm2) ((ui, V/), (u2, V2)) = m2(ui, u2)vm2(v/, V2)

< [s2(ui)xs2(u2)]v[s2(vi)xs2(v2)]

= [s2(ui)vs2(vi)] x [s2(u2)vs2(v2)]

= (s2ns2)(ui, Vi) x (s2ns^ (u2, V2). Therefore,

(m2nm^((ui, Vi), (u2, V2)) < (s2ns2)(u/, v/) x (s2ns2)(u2, V2).

This shows that G'nG'' is a product intuitionistic fuzzy graph.

Corollary 1. Let G' = (V',E',s',m') and G'' = (V,E",s",m") be two product intuitionistic fuzzy graphs. If G' and G are complete then G nG is not necessarily complete.

The above statement is justified in the following example.

Example 6. In this example G' = (V,E',s',m') and G' = (V',E',s'',m'') are product intuitionistic fuzzy graphs that are complete, while G nG is not complete. This example is shown in Fig. 6.

Definition 17. Let G' = (V',E',s',m') and G'' = (V,E'',s'',m'') be the product intuitionistic fuzzy graphs. Thus, the ring sum of two product intuitionistic fuzzy graphs G' and G is G' 4G = (V,E,s' ©s ,m' ©m ) where (s/ ©s/)(u) = si(u)us/(u), (s2 ©s2')(u) = s2(u)us2(u) for all ue V V uV and

Fig. 6. Direct product of two product intuitionistic fuzzy graphs.

mi(U, v), if (U, v)eF - E' (mi ® mi) (U, v) = mi(U, v), if (u, v) eE' - E 0, otherwise

f m2(u, v), if (u, v)eE - E" (m2®m2) (u, v) = m2(u, v), if (u, v)eE' - E'

0, otherwise

Example 7 Let G' = (V,E',S,m') and G' = (V',E",s",m') be two product intuitionistic fuzzy graphs. Thus, the ring sum of G ,G is shown in Fig. 7.

Theorem 9. Let G' = (V,E',s',m') and G'' = (V'EW) be two product intuitionistic fuzzy graphs. Thus, G' ©G is also a product intuitionistic fuzzy graph.

Proof. Let G = (V ,E ,s ,m ) and G = (V ,E ,s ,m ) be two product intuitionistic fuzzy graphs. Now, we have to show that the ring sum G' ©G = (V,E,s' © s ,m' ©m ) is a product intuitionistic fuzzy graph.

Case 1: If (u,v)gE'-E' and u,vgV'-V', then (mi ©mi) (u, v) = mi(u, v) < [si(u) x si(v)] = (si us'') (u) x (sius')(v)

Fig. 7. Ring sum of two product intuitionistic fuzzy graphs.

= (si®si')(u)x (si®si')(v) and

(M2 ®m2) ("; v) = m2(U, v) < [®2(u) x s2(v)] = (s2us2)(u) x (s2us2)(v)

= (s2®s^ (U) x (s2®s2') (v).

Case 2: If (u,v)eE'-E', ueV-V' and veV'nV'', then (mi 4 mi) (u, v) = mi(u, v)

< (siusi)(u) x (si(v)vsi(v))

= (sius'ï)(u) x (siusi)(v)

= (si®s/Ï)(u) x (si®s/Ï)(v). Similarly,

(m24m2)(u,v) < (s2®s2)(U) x (s2®s2)(v).

Case 3: If (u,v) eE'-E andu, ve V'nV'', then

(mi 4 mi) (u, v) = mi(u, v)

(m2 + m2)(u v)= ( s2 + s2)(u)x ( s2 + s2) (v)- (m2 + m2)(uv).

(s'us')(u) x (s'us')(v) - (miumi)(u, v), if (u, v)eE'uE''

(mi + mi}(u'v) = \ (si(ui x si(v)) - (s'i(u) x si(v)), ifueV', ve V'

< (si(u)vsi'(u^ x (si(v)vsi'(v))

s2ns2)(u) x (s2nc^(v) - (m2nm2) (u v) if (u v)eE'uE''

(m2 + m2)(u'v) I (s2(u) x *2(v)) - (s2(u) x s2(v)), ifueV', ve V'

= (siusi) (u) X (siusi) (v)

= (si 4 s'i) (u) x (si 4 s'i) (v). Similarly,

(m24m2)(u, v) < (s24s2)(u) x (s2 4s2) (v). Similarly, we can show that if (u.vjeE —E" then

(mi 4 mi) (U; v) < (si 4 si) (u) x (si 4 si) (v)

(m24m2)(U; v) < (s24s2)(u) x (s24s2)(v).

Hence, G 4 G is a product intuitionistic fuzzy graph.

Theorem 10. Let G' = (V,E,s",m') and G = (V",EW) be two product intuitionistic fuzzy graphs with E nE"' = f, then G" + G"" yG 4G.

Proof. Let G" = (V"E",s",m") and G" = (V",E",s",m") be two product intuitionistic fuzzy graphs with EnE" = f. Now

(si + *i) (u)= (si + si) (u) = (siusi)(u)

= _(si®sUiu)

= si(u)®si(u)

Similarly,

(s2 + s2) (u) = ^s2 4 s^ (u). Again,

(mi + mi)(u, v) = ( si + si)(u) x (si + si and

—mi + mi) (u, v)

Now, we discuss different cases:

Case 1: If ueV' ,veV, then (m/ + m/)(u, v) =(s/(u)

xs/(v)) - (s/(u) x s/(v)) = 0. Therefore, (m/ + m/)(u,v) = 0.

Similarly, (m2 + m2)(u, v) = 0.

Case 2: If (u,v)eE-E and u,veV'-V', then

(mi + mi ) (u, v) = (si (u) x si (v)) - mi (u, v) = M(u) x s'(v)) - mi(u,v) = mi(u, v)'

Similarly,

(m2 + m2) (u, v) = m2(u, v).

Case 3: If (u,v)e£'-E'' and u, veV'nV'', then

(mi + mi ) (u, v) = (si us^ (u) x (si us^ (v) - mi (u, v) = mi (u, v). Similarly,

(m2' + m2) (u, v) = m2 '(u, v).

Case 4: If (u,v)eE-E, ueV'-V' and ve V'nV'', then

(mi + mi)(u, v) = si(u) x (si(v)usi(v)) - mi(u, v)

= (si(v)usi(u^ x (si(v)usi(v^ - mi(u, v) = (s'ius')(u) x (siusi)(v) -mi(u, v) = mi (u, v).

Similarly,

(m2 + m2) (u, v) = m2(u, v).

Case 5: If (u,v)eE-E and u,veV-V', then it is obvious that

(mi + m'1)(u,v) = m'1(u,v), (m2 + m2)(u,v) =m2(u,v).

Case 6: If (u, v) e£" - E' and u, veV'nV0, then it is obvious that

(mi + m'1)(u,v) = m'(u,v), (m2 + m2)(u,v) = m2(u,v).

Case 7: If (u,v)gE'-E', ugV'-V and vGV''nV', then it is obvious

that (mi + m')(u, v) = m'(u, v), (m2 + m2)(u, v) =m2(u, v). Combining all the cases we have

m'(u, v), if (u, v)eE' - E'' _

(mi + ml)(U;v) = < m1(u,v), if(u,v)eE" -E' = 4 ml)(uv)

otherwise

Case 3: If (u,v)eE'-E and u, veVnV'', then

(m1 ® mO (u, v) = (s1 (u)us1 (u)) x (v)us1 (v)) - m1 (u, v) = (u) x (SiUsD (v) m1 (u, v)

= m1 (u, v).

Similarly, (m2®m2)(u, v) = m2(u, v).

Case 4: If (u,v)eE'-E', ueV-V and veVnV'', then

m2(u, v), if (u, v)eE' - E" _

(m2 + m2)(u v)= m2(u, v), if (u, v)eE" - E' = 044m2) (u v)

otherwise

Hence, G' + G ' sG' ©G''.

Theorem 11. Let G' = (V',E',s',m') and G'' = (V'EW) be two product intuitionistic fuzzy graphs with E' nE' = f, then G' ©G'' sG + G.

Proof. Let G' = (V', E', s', m') and G '' =(V'', E', s'', m'') be two product intuitionistic fuzzy graphs with E nE = f. Now,

(s'1®sj(u) = (u)

=_(siusi)(u)

= (u) + (u)

= (si + ) (u).

Similarly,

(s2 4 s2) (u) = ^s2 + s^ (u).

(m14mO (u, v) = (s14 S1 ) (u) x (s 14 S1 ( (v) - (m14mO (u, v) = (s1 + s1) (u) x (s1 + s1) (v) - (m14m1) (u, v) i M + s1)(u) x (s1 + s1)(v) -m1(u, v), if (u, v)ef - E'' \ (s1 + s1)(u) x (s1 + s1) (v) - m1 (u, v), if (u, v)eE - E'

(m24m2)(u v) = '(s24s2)(u) x (s24s2((v) - (m2 4m^ (u v)

= (s2 + s2)(u) x (s2 + s2)(v) - (m24m^(u, v) = f (s2 + s2)(u)x (s2 + s2)(v)-m2(u, v), if (u, v)2E'- E" 1 (s2 + s2) (u)x (s2 + s2) (v) - m2(u, v), if (u, v)eE - E'

Now, we discuss different cases:

Case 1: If ugV',vgV , then (m'©mi)(u,v)= (s'1©s1)(u)x (s'l © si )(v) = (si)(u) x (si)(v)_ _ Similarly, (m2 ©m2)(u,v) = (s2)(u) x (s2)(v).

Case 2: If (u,v)gE'-E' and u,vgV'-V', then (m' ©m')(u, v)

= s'(u) x s'(v) - mi(u,v) = mi(u,v). Similarly, (m2©m2)(u,v) = m2(u, v).

(m14 mO (u, v) = s1 (u) x (s1 (v)us1 (v)) - m1 (u, v) = (s'1(u)us1(u^ x (s1 (v)us1(v^ - m!^ v)

= (s1us^ (u) x (s1us^ (v) - m1 (u, v)

= m1 (u, v).

Similarly, (m24m2)(u, v) = m2(u, v).

Case 5: If (u,v)eE'-E' and u,veV'-V, then it is obvious that

(mi©m')(u,v) = m'(u,v), (m2©m2)(u,v) =m2(u,v).

Case 6: If (u,v)gE'-E' and u, vGV'nV', then (mi©m')(u,v) = m'(u,v), (m2©m2)(u,v) =m2(u,v).

Case 7: If (u,v)gE'-E', ugV'-V and vgV 'nV', then (mi©m')(u,v) = m'(u,v), (m2©m2)(u,v) =m2(u,v). Case 8: If (u, v) gE'nE', then

(m14 m 0 (u, v) = (s14 s^ (u) x (s14 s 1 ) (v) - (m14 m J (u, v) = (s1us^ (u) x (s1us^ (v) - (m1um^ (u, v) = (s1us^ (u) x (s1us^ (v) - (m^u, v)um1(u, v)) = m^u, v)um1(u, v).

Similarly,

(m2 (u, v) = m2(u, v)um2(u, v). Combining all the cases we have

m1(u, v), m1(u, v), _ m1 (u, v)um1(u, v),

(m14m1) (u, v) =

if (u, v)2E - E if (u, v)2E - E if (u, v)eE'nE''

s^ (u) x s1 )(v), ifueVf, v2V

= + mD (u, v)

mP (u, v), m2(u, v), _ M27 (u, v)um2(u, v), si?) (u) x fe

(m2 '4m^ (u, v) =

= (m2' + m2) (u, v) These equations complete the proof.

if (u, v)eE/ -E if (u, v)eE" - E' if (u, v)eE/nE" ifu2V , v2V

5. Conclusions

In this paper, we introduced the concepts of direct product, semi-strong product, and strong product of any two intuitionistic fuzzy graphs and investigate many interesting properties. Finally, we define the product intuitionistic fuzzy graph and the isomorphic properties on it. In our future work, we will focus on intuitionistic fuzzy threshold graphs and attempt to investigate many properties. The concept of intuitionistic fuzzy graphs can be applied in various areas of neural networks, signal processing, pattern recognition, computer networks and expert systems.

Acknowledgements

Financial support of the first author offered by the Council of Scientific and Industrial Research, New Delhi, India (Sanction no. 09/599(0057)/2014-EMR-I) is gratefully acknowledged.

The authors are thankful to the reviewers for their valuable comments and suggestions to improve the presentation of the paper.

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