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ELSEVIER

Pacific Science Review A: Natural Science and Engineering

journal homepage: www.journals.elsevier.com/pacific-science-review-a-natural-science-and-engineering/

Properties of interval valued intuitionistic (S,T) - Fuzzy graphs

Hossein Rashmanlou a' *, R.A. Borzooei b, Sovan Samanta c, Madhumangal Pal d

a Department of Computer Science, University College ofRouzbahan, Sari, Iran b Department of Mathematics, Shahid Beheshti University, Tehran, Iran

c Department of Mathematics, Joykrishnapur High School (H.S.), Simulia, Tamluk, West Bengal 721649, India d Department of Applied Mathematics, Vidyasagar University, Midnapore 721102, India

ARTICLE INFO

Article history: Available online xxx

Keywords:

Interval valued intuitionistic (S,T)—fuzzy graph

Busy vertices Lexicographic product Direct product

ABSTRACT

The basis of the concept of interval valued intuitionistic fuzzy sets was introduced by K. Atanassov. Interval valued intuitionistic models provide more precision, flexibility, and compatibility to a system than do classic fuzzy models. In this paper, three new types of product operations (direct product, lexicographic product, and strong product) of interval valued intuitionistic (S,T)—fuzzy graphs are defined. One of the most studied classes of fuzzy graphs are regular fuzzy graphs, which appear in many contexts. For example, r-regular fuzzy graphs with connectivity and edge-connectivity equal to r play a key role in designing reliable communication networks. Hence, we introduced the concepts of regular and totally regular interval valued intuitionistic (S,T)—fuzzy graphs. Likewise, we defined busy vertices and free vertices in interval valued intuitionistic (S,T)—fuzzy graphs and studied their images under an isomorphism, which are highly important in fuzzy social networks.

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

BY-NC-ND license (http://creativecommons.org/Iicenses/by-nc-nd/4.0/).

1. Introduction

Graph theory has several interesting applications in systems analysis, operations research, computer applications, and economics. Since the introduction of fuzzy sets by Zadeh [37], there have been numerous generalizations of this fundamental concept. One generalization among the sets is the notion of intuitionistic fuzzy sets introduced by Atanassov [6]. For more details on intui-tionistic fuzzy sets, please refer to [6—8]. In 1975, Zadeh [36] introduced interval-valued fuzzy sets as an extension of fuzzy sets [37] in which the values of the membership degree are intervals of numbers, rather than numbers. Rosenfeld [21] introduced fuzzy graphs in 1975 and proposed other definitions (e.g., paths, cycles, and connectedness). The complement of a fuzzy graph was defined by Mordeson and Nair [19] and further studied by Sunitha and Kumar [30]. The concept of weak isomorphism, co-weak isomorphism and isomorphism between fuzzy graphs was introduced by Bhutani [9]. Akram et al. [1—5] introduced interval-valued

* Corresponding author. E-mail addresses: rashmanlou.1987@gmail.com (H. Rashmanlou), borzooei@sbu. ac.ir (R.A. Borzooei), ssamantavu@gmail.com (S. Samanta), mmpalvu@gmail.com (M. Pal).

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

fuzzy graphs, strong intuitionistic fuzzy graphs, bipolar fuzzy graphs, as well as certain types of vague graphs and vague hyper-graphs. Borzooei et al. [10—16,25—27] studied domination, degree of vertices, new concepts of vague graphs, and bipolar fuzz graphs. Complete interval-valued fuzzy graphs were investigated by Rashmanlou and Jun [22]. Pal and Rashmanlou [20] studied irregular interval-valued fuzzy graphs, defined antipodal interval-valued fuzzy graphs [23], and balanced interval-valued fuzzy graphs [24]. Samanta et al. [31—34] introduced fuzzy planar graphs, m-step fuzzy competition graphs, fuzzy k-competition and p-competition graphs, and showed some results on bipolar fuzzy sets and bipolar fuzzy intersection graphs. In this paper, three new types of product operations (direct, lexicographic, and strong) of interval-valued intuitionistic (S,T)—fuzzy graphs are defined. We introduce the concept of regular and totally regular interval valued intuitionistic (S,T)—fuzzy graphs. Busy vertices and free vertices in interval valued intuitionistic (S,T)—fuzzy graphs are defined, and their image under an isomorphism is studied.

2. Preliminaries and notations

An interval number, is defined as an interval, [a~,a+], where 0 < < a+ < 1. The set of all interval-numbers is denoted by D[0,1]. The interval [a,a] is identified with the numbers ae[0,1].

http://dx.doi.org/10.1016/j.psra.2016.06.003

2405-8823/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 BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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For interval numbers ai = [a(- ,a+]eD[0,1] andiel, we define infai = [AieJar, Aie/a+] and sup ai = [vieJar, vie/a+] and set:

(1) al < ajoa^ < a— and a+ < a+

(2) a, = ajoa^ = a— and a+ = a+

(3) a,<52^51 < a2 and a,sa2

(4) ka = [ka—, ka+]

whenever 0 < k < 1.

(D[0,1],<,v,a) is a complete lattice with 0 = [0,0] as the smallest element and 1 = [1,1] as the largest element. An interval number fuzzy set (F on X) is defined as the set, F = {(x, [mF (x),mF+(x)])|xeX)}, where mF and mF+ are two fuzzy subsets of X such that, mF (x)<mF+(x), for all xeX. By stating, mF(x) = [mF (x),mF+(x)], we see that F = {(x,mF(x))|xeX)}, where mF(x):XiD[0,1].The function 5:[0,1] x [0,1] / [0,1] is often referred to as a t" norm (s— norm) if 5 satisfies the following conditions:

(i) <5(x,1) = x (<5(x,0) = x),

(ii) 5(x,y) = 5(y,x),

(iii) 5(5(x,y),z) = d(x,d(y,z)),

(iv) d(x,u) < (x,w), for all x,y,z,u,we[0,1]

where u < w.

A t— norm (s— norm), 5, is called an idempotent t— norm (s— norm) if 5(x,x) = x for all xe[0,1] [22]. If 5 is an idempotent t— norm (s-norm), then the mapping D:D[0,1] x D[0,1] / D[0,1] as defined by D(a1,a2) = [<5(a—,a—), 5(a+,a+)] is an idempotent t— norm (s— norm) and called an idempotent interval t— norm (s— norm). According to Atanassov [7,8], an interval valued intuitionistic fuzzy set on X is defined as an object of the form:

Example 3.2. Consider a graph G = (V,E), such that V = {u,v,z} and E = {uv,vz,uz}. Let A be an interval valued intuitionistic fuzzy set of E4V x Vdefined by:

A = j (x,MA(x),NA(x^j |xexj,

where MA(x) and Na(x) are interval valued fuzzy sets on X(e.g., MA : X/D[0,1], na : X/D[0,1] such that 0 < sup MA(x)+ sup NA(x) < 1 for all xeX). Interval valued intuitionistic fuzzy sets will be denoted by A = (MA, NA) in this paper.

Let G = (V,E) be a graph, where V is the non-empty finite set of vertices of G, and E is the set of edges of G. The fuzzy set V is a mapping, a, from V to [0,1]. The fuzzy graph G is a pair of functions G = (s,m), where a is a fuzzy subset of a non-empty set V, and m is a symmetric fuzzy relation on a (i.e., m(uv) < s(u)As(v)). The underlying crisp graph of G = (s,m) is denoted by G* = (V,E), where E4V x V.

3. Interval-valued intuitionistic (S,T)-fuzzy graphs

In this section, we define interval-valued intuitionistic (S,T) fuzzy graphs and introduce three types of new product operations (direct, lexicographic, strong) of interval valued intuitionistic (S,T)— fuzzy graphs.

Definition 3.1. An interval-valued intuitionistic (S,T) fuzzy graph with underlying set V is defined as ordered pair (A,B), where A = (MA, Na), is an interval-valued intuitionistic fuzzy set in VandB = (PB, QB) is an interval valued intuitionistic fuzzy set in E such that for all u,veV

pB({u, v}) < ^MÎa(u), Ma(v)J, QB({u, v}) > s|na(u), Na(v)J.

' V[0.3, 0.5]' [0.4, 0.5]' [0.4, 0.6]/ uvz

[0.2, 0.4] [0.1, 0.3] [0.1, 0.3]

[0.3, 0.5]' [0.2, 0.4]' [0.3, 0.4]/ uv vz uz

[0.2, 0.4]'[0.1, 0.3]' [0.2,0.5]

From this graph, it can be determined that (A,B) is an interval-valued intuitionistic (S,T)—fuzzy graph. Fig. 1. Q2

Definition 3.3. Let G1 = (A1,B1) and G2 = (A2,B2) be two interval valued intuitionistic (S,T) fuzzy graphs on G1 =(V1, E1) and G2 = (V2, E2), respectively.

(i) A homomorphismg is a mapping g:V1/ V2 such that:

(1) Ma, (x1) < M^gx)), Na, (x^ Na2 (¿(x^J^fdr all x1 e V1

(2) Pb,(x,y,) <Pb2(g(xi)g(y,)), Qb,(x,y,) > QB2(g(x,)g(y,)), for all xiyieEi.

(ii) A bijective homomorphism g:G, / G2 is called a weak isomorphism if

(3) Ma,(x,) = Ma2(g(x,)), Na,(xi)=^(g(x,)),forallx,eV

(iii) A bijective homomorphism g:G, / G2 is called a co weak isomorphism if

(4) Pb,(x,y,)=PB2(g(xi)g(yi)), Qb,(x,y,) = Qb2(g(x,)g(y,)), for all xiyieEi.

A bijective homomorphism g:G1 called an isomorphism.

G2 satisfying (3) and (4) is

Definition 3.4. Let G, = (V,, E,) and G2 = (V2, E2) be two graphs.

(i) [28,29] The graph G, x G2 = (V, E) is the Cartesian product of G1 and G*2, where V = V, x V2 and

E ={(x, x2)(x, y2)|xeVi, x2y2eE2 }u{(xi, z)(yi, z)| ze V2, xiyieEi}.

(ii) [35] The graph G, x G2 = (V, E) is the direct product of G, and G2, where V = V, x V2 and

E = {(x,, x2 )(y 1, y2 )|x,y 1 e Ei, x2y2 e E2 }.

Fig. 1. Interval-valued intuitionistic (S,T)—fuzzy graph, G.

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(iii) [18] The graph G1 -G*2 = (V,E) is the Lexicographic product of G1 and G*2, where V = V1 x V2 and

E = {(x, x2)(x,y2)|xeV1, x2y2 2E2}u{(x1, x2)(y1;y2) x |x1y12E1, x2y2 2E2}.

(iv) [17] The graph G1uG2 = (V, E) is the union of G1 and G*2, where

V = V1 x V2 and E = E1uE2.

(v) [17] The graph G1 + G2 = (V, E) is the join ofG'1 and G*2, where

V = V1uV2 and E = E1uE2uE', E is the set of all edges joining the vertices V1 and V2.

Definition 3.5. Let G1 = (A1,B1) (resp. G2 = (A2,B2)) be an interval valued intuitionistic (S,T)—fuzzy graph of G1 = (V1, E1) (resp. G2 = (V2, E2)). Thus, the Cartesian product G1 x G2 ofG1jand G2 is defined as a pair (A,B), where A = (MA, NA) and B = (PB, QB) are interval valued intuitionistic fuzzy sets on V = V1 x V2 and E = {(x,X2)(x,y2)|xeV1,X2y2eE2)u{(x1,z)(y1,z)|zeV2,X1y1eE1} respectively which satisfies the following:

MA(X1, X2)=T (Ml (Xi), Ma2 (X2))((X1 ; X2)2VI X V2 ), Na(Xi, X2)=S(Na, (Xi), NA2 (X2))KK 1 ^ 1 2;

Pb((X, X2 )(X, y2))= T (Ma (x),PBi(X2y2)) (x2y x y 2E ) Qb((x, X2 )(x, y 2)) — S(Na, (X), QB2 (X2y2)) 122 2

Pb((x1;z)(y1;z)) — T(^(x^),l\(z)) (zeV2 XiVi 2E , Qb((X1, z) (y1,z))— S(Qb, (X1y1 ), Q/2 (z)L 2 11 ^

Remark 3.6. LetG1 = (A1,B1) andG2 = (A2,B2) be two interval valued intuitionistic (S,T)-fuzzy graphs. Thus, the Cartesian product G1 x G2 is an interval valued intuitionistic (S,T)-fuzzy graph.

Definition 3.7. Let G1 = (A1,B1) (resp. G2 = (A2,B2)) be an interval valued intuitionistic (S,T)-fuzzy graph of G1 = (V1, E1) (resp. G2 = (V2, E2)). The union G1uG2 ofG1 and G2 is defined as a pair (A,B), where A = (MA, NA) and B = (PB, Qb) are interval valued intuitionistic fuzzy sets on V = V1uV2 and E = E1uE2 respectively which satisfies the following:

Ma (X) — Ma, (X) _ Ma (X)=Ma2 (X)_ Ma (X) — S (Ma, (X), Ma2 (X))

if xeV, and x;V2, if xeV2 and x;V,, if xeV1nV2.

NA (X) — Na, (X) __ NA (X)=NA2 (Xl. Na (X) — T(NA, (X), NA2 (X))

if xeV, and x;V2, if xeV2 and x;V,, if xeV1nV2.

Pb (xy) — P, (xy) __ Pb (xy)=PB2 (xy) PB(xy) — S(PB, (xy),PB,(xy))

if xyeE, and xy;E2, if xy2E2 and xy;E,, if xyeEi nE2.

Qb (xy) — Qbi (xy) _ Qb (xy))—Qb2 (xyl Qb(xy) — T(Qb, (xy), Qb2 (xy))

if xyeE, and xy;E2, if xy2E2 and xy;E,, if xyeEinE2.

Proposition 3.8. Let G, — (A,,B,) and G2 — (A2,B2) be two interval valued intuitionistic (S,T)-fuzzy graphs. Thus, the union G,uG2 is an interval-valued intuitionistic (S,T)-fuzzy graph. Proof. Let xy eE,nE2. Thus,

f uf (xy) — S^PB, (xy), pb2 (xy) j

< S (T(mQ (x), Q (V)), T (mL2 (x), q(y)

— ^S (MAI (X), Mq (X^, S^MAl (y), Mq (v)

TttMAi uML) (X)J mA

( qb, uQb2j(

al uma}j (y)j , ))

(xy) = T^0,1 (xy), q (xy)

> T(S(NA1 (X),Nq(V)),s(nQ(X),Nq(y)

— S^T(nq (x), Q (x^ , (^Q (y), Q (y)

— ^(nLiuq (x), (nLiuQ Similarly, if xyeE, and xy;E2 or xyeE2 and xy ;E,, then

P, uPB | (xy) < Tl (mQi umQ\ (x), [mai uMQ

2) v) -

r xyeE2 <

; uf (xy) < uM/A^J (X), (ML, uM/A^J (V)j ,

uO,2 j (xy) > ^ (n/i uq (x), (n/i uq (y)j .

Definition 3.9. The join G1+G2 of two interval valued intuitionistic (S,T)-fuzzy graphs Gt_ = (A1,B1) and G2 =(A2iB2) is defined as a pair (A,B), where A =(MA, NA) and B =(PB, Qb) are interval-valued intuitionistic fuzzy sets on V = V1uV2 and E = E1uE2uE' (E is the set of all edges joining the vertex ofV1 and V2), respectively, which satisfies the following:

ma (x) — mai (x) _ M/ (x)—m/2 (Xl, M/ (x) — S (M/, (x), M/2 (x))

if xeV, and x;V2, if xeV2 and x;V,, if xeV1nV2.

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NA(x)=NAl(x) if xeV, and x;V2,

NA(x) = NA2(x) if xeV2 and x;V;,

NA(x) = T(NA1 (x2,N?(x)) if xeVinV2.

PB(xy)=PB1 (xy) __ PB(xy)zPB, (xy) PB(xy)=S(PB1 (xy), PB2 (xy))

if xyeE1 and xy;E2, if xy2E2 and xy;E1, if xy 2 E1nE2.

Qb (xy) = Qb; (xy) _ Qb (xy^Qg? (xy) QB(xy) = T(QB1 (xy), Qb2 (xy))

if xyeE1 and xy;E2, if xy2E2 and xy;E1, if xy 2 E1nE2.

Pjs(xy) = T(Ma, (x),Ma? (y)) if xy2E'

PB(xy)=S(NA1 (x), Na? (y))

Remark 3.10. Let G, = (A,,B,) and G2 = (A2,B2) be two interval valued intuitionistic (S,T)-fuzzy graphs. Thus, the join G,+G2 is an interval-valued intuitionistic (S,T)-fuzzy graph.

Definition 3.11. The direct product G,*G2 of two interval valued intuitionistic (S,T)-fuzzy graphs G, = (A,,B,) and G2 = (A2,B2) of G, = (V,, E,)and G2 = (V2, Ej) respectively is defined as a pair (A,B), where A = (MA, Na) and B = (PB, QB) are interval-valued intuitionistic fuzzy sets on V = Vi x V2 and E = {(xi,x2)(yi,y2)|xiyi eE,,x2y2eE2} respectively, which satisfies the following:

(Qb1 *N ) (

I ((x1, x2 )(y 1, y2 )) = S (x1 y1 ), Q (x2y2 ) > s^NA; (x1), Q1 (y1)j, S^NA? (x?), Q (y?) = S^S^Na; (xi), na? (x?^ , s^NQ (y;), Na2 (y?)N = s([NAi xNA?)(x;,x?), (NatxNA?)(y;,y?) j.

This completes the proof.

Example 3.13. Let G, = (V,, E,) and G2 = (V2, E2) be two graphs such that V, = {u,v}, V2 = {z,w}, E, = {uv}, and E2 = {zw}. Consider two interval-valued intuitionistic (S,T)-fuzzy graphs G, = (A,,B,) and G2 = (A2,B2), where

A; = ( V;,

[0.2, 0.4]' [0.3, 0.5]/ V[0, 0.2]'[0, 0.1],//'

[0.3, 0.5]' [0.4, 0.6]/ V[0.1, 0.3]' [0, 0.2]

B; = ( E;,

[0.1, 0.3]/ V[0, 0.2]//'

[0.2, 0.4]/' V[0.1, 0.3]

MA(x1, x2)= T (MAl (x1), MA? (x2))((xi, x?)2V; X V2), Na(xi, x? )=S(NAi (xi), Na? (x?)) u 1 1

PB((x1, x2)(yi, y?))= T (PB1(xiyi), PBi(x2y2)) (xiy;2El x2y22E2). Qb((x;, x?)(yi, y2))= S(Qsi (x;yi), Qb? (x?y2 )) 11 1

Proposition 3.12. Let G; = (A1,B1) and G2 = (A2,B2) be two interval valued intuitionistic (S,T)-fuzzy graphs. The direct product G; *G2 is an interval-valued intuitionistic (S,T)-fuzzy graph. Proof. Let x1y12E; and x?y22E2. Thus,

By a routine computation, it is clear that G,*G2 is an interval-valued intuitionistic (S,T)-fuzzy graph. Fig. 2.

Definition 3.14. The Lexicographic product G, • G2 of two interval valued intuitionistic (S,T)-fuzzy graphs G, = (A,,B,) and G2 = (A2,B2) of

((xi, x2)(yi, y?)) = TSPsi (x;yi), Pp2 (x?y? )

< t St Sma; (xi ), MAi (y; ) !, T SMa2 (x?), mA2 (y?) = T Ss ^MAi (x;), Q (x?)j, s(mA; (yi), MfA2 (y?) = TllMAi x Ma2)(x;,x?)JMA; xMa2)(yi,y?)

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G, = (V,, E1)andG2 = (V2, E2)rejspectively is defined as a pair (A,B), where A = (MA, NA) and B = (PB, QB) are interval-valued intuitionistic fuzzy sets on V = V, x V2 and E = {(x,x2)(x,y2)|xe V,, x2y2eE2}u{(xi,x2)(yi,y2)|xiyieEi,x2y2eE2}, respectively which satisfies the following:

MA(xi,x2) = T(MA(xi),Ma?(x2)) ((xi,x?)2V; X V?), Na(x;, x?)=S(Na; (xi), Na? (x?)r 1 1

Pb((x,x?)(x,y?)) = T(MA (x),PBi(x?y2)) (x2V xy 2E) Qb( (x,x?)(x,y?)) = S(Nai (x), Qb?(x?y?)r 2 2 2

Pb ( (x1, x2) (yi, y?))= T(PBi(xiyi), PBL(x2y2)) (xiy;2El x?y2 2E?). Qb ( (xi, x?) (yi, y2))= S( Qbj (x;yi), Qb? ^2))

Proposition 3.15. Let G; = (A1,B1) and G2 = (A2,B2) be two interval valued intuitionistic (S,T)-fuzzy graphs. The Lexicographic product G; G2 is an interval-valued intuitionistic (S,T)-fuzzy graph. Proof. Let x2V; and x2y22E2, then

fl-p ( (x,x?)(x,y? )) = t(mA; (x), P^y?^

< t(mA; (x), t(m(2 (xS ), n (y?)

= T ^ S SmA; (x), MQ (x?) j, S^Mal (x), Ma2 (y?)

= t((ma;-ma;) (x, x?), (ma;-mA,

(x,y?^,

([0.2,0.4], [0,0.2]) ") ( ([0.3,0.5], [0.1,0.3])

([0.1,0.3], [0,0.2])

([0.2,0.4], [0.1,0.3])

QB; -Qb~?J ( (x,x?)(x,S 2)) = S^Na; (x), Qb2(x?y?)

> s^NN (x), s^N! (xS ), NQ2 (y?) = S (s SsN(x), JQ(x?) j , S (nA; (x), na2 (y?) = S^JQQ (x,x?), (na Ifx;y;2E; andx2yy22E2, then

pi- Pp2 S((xi, x?)(yi, y?)) = T SPb; (x;yi), Pb2 (x?y?) < t St SmaI (x; ), MA; (y; )(, T SmA2 (x?), N (y?) = t ^ S Sq (xi ), MQ (x?^, S (mA; (yi ! MA2 (y?)

= t((mAi-MaA (xi, x?), ( ma;-Ma2] (yi, y?) 1.

(x), n (x?y?)j

al -Q (x,y??j.

bi -(N S (

Fig. 2. Interval valued intuitionistic (S,T)-fuzzy graphs Gi, G2 and Gi*G2.

NN 1 ((xi, x2)(yi,y?)) = S (x;yi), Q (x?y?) > S SS S2;(xi),Q (yi)(,S SnQ(x?),NQ(y?)N = s(s SnA; (xi), (Q (x?)j , S (nA; (yi!Q (y?)(

=s( fe -Na2 j (xi, x?), (na;-na^ j (yi, y?) j.

4. Busy vertices and free vertices in interval valued intuitionistic (S,T)-Fuzzy graphs

In this section, we introduce busy values, busy vertices and free vertices in interval valued intuitionistic (S,T)-fuzzy graphs.

Definition 4.1. The busy value of a node, v, of an interval valued intuitionistic (S,T)-fuzzy graph, G = (A,B), is defined to be

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D(v) = (Dm(v),Dn(v)), where Dm ( v) = e MA ( v)aMa ( v,), and Dn(v) = Y1 NA(v)vNA(vj). Here, vi, are neighbors of v, and the busy value ofan interval valued intuitionistic (S,T)-fUzzy graph G is defined to be the sum of the busy values of all vertices of G (i.e., D(G) = ED(vi)), where vi are vertices of G. i » Example 4.2. Consider a graph G = (V,E) such that V = (vi,v2,v3,v4) and E = (viv2,v2v3,viv4,v3v4). Let A = (MA, NA)bejm interval valued intuitionistic fuzzy subset of V and let B = (PB, QB) be an interval valued intuitionistic fuzzy subset ofE4V x V defined by

Proof. Since Gi y G2, we have

Ma1 (Х1)=М^2(g(X1)), NA1 (X1)=NA~2(g(X1)), for allx12V1.

Hence,

dM(X) = X Ma1 (y) = X Ma2 (g (y)) = dM(g(X)),

y 2 v1 y2 v1

A = ( V,

B = E,

[0.3, 0.5]' [0.4, 0.5]' [0.4, 0.6]' [0.5, 0.6]/ V[0.2, 0.4]' [0.1, 0.3]' [0.1, 0.3]' [0.2, 0.4] v1v2 v2v3 v3v4 v1v4 Л / v1v2 v2v3 v3v4 v1v4

[0.3; 0.5] [0.2; 0.4] [0.3; 0.5] [0.2; 0.4] [0.2; 0.4] [0.1; 0.3] [0.2;0.4] [0.2; 0.4]

By routine computations, we have

[0.7,1], Dm(v3) = [0.8,1.1], = [0.3,0.7], Dn(v3) = [0.3,0.7],

DMЫ = [0.6,1], DMЫ = Dm ( v4) = [0.7,1.1] Dn(v1) = [0.4,0.8], Dn(«2) Dn(v4) = [0.4, 0.8]

Subsequently, D(v1)

([0.6,1], [0.4, 0.8]), D(v2) ( [0.7,1], [0.3, 0.7]), D(v3) ([0.8,1.1], [0.3, 0.7]), D(v4) ([0.7,1.1], [0.4, 0.8])

Definition 4.3. Let G = (A,B) be an interval valued intuitionistic (S,T)-fuzzy graph. Then, the open neighborhood degree of a vertex x is defined by deg(x) = (dMx),dN(x)), where dM(x) = £ MA (y) and

dN (x)= E Na (y). y2N(x)

y2N(x)

Definition 4.4. A vertex, v, in an interval valued intuitionistic (S,T)-fuzzy graph G = (A,B) is said to be a busy vertex ifMA (v) < dM(v) and NA (v) > dN(v), otherwise it is called a free vertex.

For the graph of Fig. 3 there are not any busy vertices.

Definition 4.5. An edge, uv, ofan interval valued intuitionistic (S,T)-fuzzy graph G (A,B) is said to ibe jm effective edge if PB (uv) = MA (u)aMa (v) and QB (uv) = NA (u)vNA (v).

For the graph of Fig. 3 the edge v1v2 is an effective edge.

Lemma 4.6. Let G1 y G2 and g be an isomorphism from G1 to G2. Then, deg(x) = deg(g(x)) for all xeV.

([0.2,0.4], [0.2,0.4])

([0.2,0.4], [0.1,0.3])

Vi ([0.3,0.5], [0.2,0.4]) „3 Fig. 3. Interval valued intuitionistic (S,T)-fuzzy graph G.

dN(x) =J2 NA1 (y) =J2 NA2 (g(y)) = dN(g(x)).

y 2 V1 yev,

Additionally, we know that deg(x) = (dM(x),dN(x)) for all xeV. Thus, deg(x) = deg(g(x)) for all xeV.

Theorem 4.7. If G, = G2 and v is a busy vertex in G,, then v is a busy vertex in G2.

Proof. Let g:Vi / V2 be an isomorphism from G, to G2. Thus,

MA, (x) = mA2 (g (x)) and NA, (x)=NAl (g (x)) for all xeV,, and

PB, (xy) = Pb2 (g(x)g(y)) and Qb,(xy) = Qb2(g(x)g(y)) for all xyeE,.

Additionally, g preserves the degree of vertices, by Lemma 4.6 (i.e., dM(x) = dM(g(x)), dN(x) = dN.g(x)), for all xeV). Ifx is a busy vertex in G,, then MA, (x) < dM(x) and NA, (x) > dN(x). Then,

MA2 (g(x)) < dM(g(x)) and NA2 (g(x)) > dN(g(x)). Hence, g(x) is a busy vertex in G2.

Theorem 4.8. Let interval valued intuitionistic (S,T)-fuzzy graph G, be weak isomorphism with G2. IfueV, is a busy vertex in G, then its image under a weak isomorphism in G2 is also busy.

Proof. Let g: V, / V2 be a weak isomorphism between G, and G2. Thus, for all x,yeV,

MA1 (x) = Ma2 (g (x)), NA1 (x) = Na2 (g (x)) (4.1)

Pb1 (xy) < Pb2 (g (x)g(y)), Qb1 (xy) > Qb2(g(x)g(y)). (4.2)

Let u 2 V1 be a busy vertex. Then

Ma1 ( u) < dM(u), Nat( u) > dN(u). (4.3)

From (4.1) and (4.3) we have

ma2(g(u)) = MA1 (u) < dM(u) =J2 MA1 (v) = J2 ma2 (g(v))

v2 v1 v2 v1

= dM (g (u)).

Hence, Ma2(g(u)) < dP(g(u)). Additionally,

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NA2 (g(u)) = NAl (u) > dN(u) = £ NA1 (v) = £ Na2 (g(v))

= dN (g(u)).

Therefore, NA2 (g(u)) > dN(g(u)). Hence, g(u) is a busy vertex in

5. Regular interval valued intuitionistic (S,T)-Fuzzy graphs

Definition 5.1. Let G = (A,B) be an interval valued intuitionistic (S,T)-fuzzy graph. Then, the degree of a vertex x is defined by deg(x) = (degM(x),degN(x)), where degu(x) = E Ps(xy) and

degN (x) = E Qs(xy). IfdegM(Vi) = ki and degN(Vi) = kj, for all Vi,Vj2V

then the graph is called regular interval valued intuitionistic (S,T)-fUzzy graph of degree (ki,kj).

Definition 5.2. Let G = (A,B) be an interval valued intuitionistic (S,T)-fuzzy graph. The closed neighborhood degree of vertex x is defined by deg[x] = (dM[x],dN[x]), where

dM[x] = degM (x) + Ma(x), dN [x] = degN (x) + Na(x).

If each vertex of G has same closed neighborhood degree m = (m,, m*2), then the graph is called totally regular interval valued intuitionistic (S,T)-fuzzy graph of degree m.

Example 5.3. Consider a graph G* such that V = {v,,v2,v3} and E = {v,v2,v2v3,v3v,}. Let A be an interval valued intuitionistic fuzzy subset of V and B be an interval valued intuitionistic fuzzy subset of E defined by

(ii) if G1 is a totally regular interval valued intuitionistic (S,T)-fuzzy graph, then G2 is also.

Proof. Let G1 y G2 and G1 is an n = (n1,n2)- regular interval valued intuitionistic (S,T)-fuzzy graph. Since, deg(x) = (degM (x),

degN(x)) = ( E Pb(xy), E QB(xy)) = (ni,n2), we have

V xsy xsy /

ni = degM (x) = J2 PB(xy) = J2 PB(g(x)g(y)) = degM (g(x)),

xsy xsy

n2 = degN(x) = J2 QB(xy) = J2 QB(g(x)g(y)) = degN(g(x)).

xsy xsy

Therefore, G2 is an n-regular interval valued intuitionistic (S,T)-fuzzy graph. Now let G1 y G2, where G1 is an m = (m1,m2)- totally regular interval valued intuitionistic (S,T)-fuzzy graph. By Definition 5.2, w have, deg[x] = (dM[x],dN[x]), where dM[x] = degM(x)+Ma(x) and dN[x] = degN(x)+Na(x) Therefore,

mi = degM (x) +Ma(x) = degM (g(x)) + Ma(g(x)) = dM [g(x)],

m2 = degN (x)+NA(x) = degN (g(x)) + Na(g(x)) = dN [g(x)].

It follows that G2 is an m- totally regular interval valued ^intuitionistic (S,T)-fuzzy graph.

Theorem 5.6. Let G = (A,B) be^an interval valued intuitionistic (S,T)-fuzzy graph of graph G*. If A = (MA, NA) is a constant function, then the following are equivalent:

(i) G is a regular interval valued intuitionistic (S,T)-fuzzy graph,

(ii) Gisa totally regular interval valued intuitionistic (S,T)-fuzzy graph.

B =( E;

' V[0.3, 0.5^ [0.3, 0.5]' [0.3, 0.5]/ ' V[0.1, 0.3^ [0.1, 0.3f [0.1, 0.3]

^ V1V2 V2V3 V1V3 л ( V1V2 V2V3 V1V3 Л J0.2, 0.4], [0.2, 0.4], [0.2, 0.4^ , ^[0.1, 0.3], [0.1, 0.3], [0.1, 0.3],

Routine computations show that the interval valued intuitionistic (S,T)-fuzzy graph G is both regular and totally regular.

Proposition 5.4. The size of an n— regular interval valued intuitionistic (S,T)-fuzzy graph G is ^k, where |V| = k.

Proof. The size of G 4is S(G) = p PB(xy), p QB(xy)

xsy xsy

y x,yeV x,yeV J

Since, G is n— regular, dG(v) = n, for all veV. We have

2S(G) ^ dc(v) =

E dc(v) = 2^ E PB(xy), T, QB(xy)). So,

V2V xy2E xy2E

En = nk. Hence, S(G) =

Proposition 5.5. Let G1 y G2. Then

(i) if Gn is a regular interval valued intuitionistic (S,T)-fuzzy graph, then G2 is also.

Proof. Suppose that A = (MA,NA) is a constant function and MA(x) = c1, NA(x) = c2, for all xeV.

(1)0(11): Assume that G is an n— regular interval valued intuitionistic (S,T)-fuzzy graph, then degM(x) = nm, degN(x) = nn, for all xeV. So, dM[x] = degM(x) +Ma(x) = nm + C1, dN[x] = degN(x) + NA(x) = nn + c2, for all xeV. Hence, G is totally regular interval valued intuitionistic (S,T)-fuzzy graph.

(ii)0(i): Suppose that G is a totally regular interval valued intuitionistic (S,T)-fuzzy graph, then dM[x]= k1, dN[x] = k2 for all xeV, or degM(x) + Ma(x) = къdegN(x)+Na(x) = k2, or degM(x) + c1 = k1,degN(x) + c2 = k2 for all xeV, or degM(x) = k1 — c1,degN(x) = k2 — c2, for all xeV. Thus, G is a regular interval valued intuitionistic (S,T)-fuzzy graph.

Proposition 5.7. If the interval valued intuitionistic)S,T)-fuzzy graph G is both regular and totally regular, then A = (MA, NA) is a constant function.

Proof. Let G be a regular and totally regular interval valued intuitionistic (S,T)-fuzzy graph. Then

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degM(x) = n1,degN(x) = n2,for all x2V1,

dM[x] = k1,dN[x] = k2,for all x2V1. It follows that,

dM [x] = k1^degM(x) + ma (x) = k1

_ on1 + ma (x) = k1 <&MA (x) = k1 - n1, for all x 2 V.

Similarly, we can show that, NA (x) = k2 - n2, for all x 2 V. Hence, A = hMA, NA) is a constant function.

Remark 5.8. Let G = (A,B) be an interval valued intuitionistic (S,T)-fuzzy graph, where crisp graph G» is the cycle C:v0,v1,^,vn = v0. Then, we have the following:

(i) if n is odd, G is regular if and only ifBJs a constant/unction.

(ii) ifn is even, GJs regular if and only ifPB (vi-1vi) = PB (vi+1vi+2), QB(vi-1vi) = QB(vi+1vi+2),1 < i < n, which i + 1 and i + 2 are in module n.

6. Conclusions

Graphs are among the most ubiquitous models of both natural and human made structures. Graphs can be used to model many types of relationships and process dynamics in physical, biological and social systems. In this paper, three new types of product operations of interval valued intuitionistic (S,T)-fuzzy graphs are defined. Additionally, regular and totally regular interval valued intuitionistic (S,T)-fuzzy graphs are introduced. The concept of interval valued intuitionistic (S,T)-fuzzy graphs can be applied in various areas of engineering and computer science (e.g., database theory, expert systems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, medical diagnosis). In our future work, we will investigate balanced interval valued intuitionistic (S,T)-fuzzy graphs and investigate certain properties of irregularity in interval valued intuitionistic (S,T)-fuzzy graphs.

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