INTERVAL VALUED INTUITIONISTIC MULTI FUZZY GRAPHjoics.org/gallery/ics-1259.pdf · Email:...

INTERVAL VALUED INTUITIONISTIC MULTI FUZZY GRAPH

A. MARICHAMY, K. ARJUNAN & K. L. MURUGANANTHA PRASAD

Department of Mathematics, Pandian Saraswathi yadav Engineering college,Sivagangai-630561,Tamilnadu,

India.Email: [email protected]

Department of Mathematics, Alagappa Government Arts College, Karaikudi-630003. Tamilnadu, India.

Email: [email protected]

Department of Mathematics, H.H. The Raja’s College, Pudukkottai-622001.

Tamilnadu, India. Email: [email protected]

ABSTRACT: In this paper, some properties of interval valued intuitionistic multi

fuzzy graph (IVIMFG) are studied and proved. Fuzzy graph is the generalization of

the crisp graph, I- fuzzy graph is the generalization of the fuzzy graph and I- multi

fuzzy graph is the generalization of multi fuzzy graph. A new structure of an interval

valued intuitionistic multi fuzzy graph is introduced.

2010Mathematics subject classification : 03E72, 03F55, 05C72

KEY WORDS: Fuzzy subset, I-fuzzy subset, intuitionistic multi fuzzy subset,

interval valued intuitionistic multi fuzzy relation, Strong interval valued intuitionistic

multi fuzzy relation, interval valued intuitionistic multi fuzzy graph, interval valued

intuitionistic multi fuzzy loop, interval valued intuitionistic multi fuzzy pseudo graph,

interval valued intuitionistic multi fuzzy spanning subgraph, Degree of interval

valued intuitionistic multi fuzzy vertex, order of the interval valued intuitionistic multi

fuzzy graph, size of the interval valued intuitionistic multi fuzzy graph, interval

valued intuitionistic multi fuzzy regular graph, interval valued intuitionistic multi

fuzzy strong graph, interval valued intuitionistic multi fuzzy complete graph.

INTRODUCTION: In 1965, Zadeh [16] introduced the notion of fuzzy set as a

method of presenting uncertainty. Since complete information in science and

technology is not always available. Thus we need mathematical models to handle

various types of systems containing elements of uncertainty. After that Rosenfeld[14]

introduced fuzzy graphs. Yeh and Bang[15] also introduced fuzzy graphs

independently. Fuzzy graphs are useful to represent relationships which deal with

uncertainty and it differs greatly from classical graph. It has numerous applications to

problems in computer science, electrical engineering system analysis, operations

research, economics, networking routing, transportation, etc. Nagoor Gani. A [11, 12]

introduced a fuzzy graph and regular fuzzy graph. Intuitionistic fuzzy set was

introduced by Atanassov K.T [6, 7]. After that intuitionistic fuzzy graphs have been

introduced by Akram. M [1]. Arjunan. K & Subramani.C [4, 5] introduced a new

structure of fuzzy graph and I-Fuzzy graph. In this paper we introduce the new structure

of a interval valued intuitionistic fuzzy graph.

1.PRELIMINARIES:

Definition 1.1[16]. Let X be any nonempty set. A mapping A: X [0, 1] is called a

fuzzy subset of X.

Definition 1.2[16]. Let X be any nonempty set. A mapping [A] : X D[0,1] is called

a I-fuzzy subset (Interval valued fuzzy subset) of X, where D[0,1] denotes the

family of all closed subintervals of [0,1] and [A](x) = [A(x), A+(x)] for all x in X,

where A and A+ are fuzzy subsets of X such that A(x) ≤ A+(x) for all x in X. Thus

Journal of Information and Computational Science

Volume 9 Issue 8 - 2019

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[A](x) is an interval (a closed subset of [0,1] ) and not a number from the interval

[0, 1] as in the case of fuzzy subset.

Definition 1.3[6]. An interval valued intuitionistic fuzzy subset (IVIFS) [A] in X

is defined as an object of the form [A] = { ( x, [A](x), [A](x) ) = ( x, [[A](x),

[A]+(x)], [[A]

(x), [A]+(x)] ) /x in X }, where [A]: XD[0, 1] and [A]: XD[0, 1]

define the degree of membership and the degree of non-membership of the element

xX respectively and for every xX satisfying 0 [A](x) + [A](x) 1.

Definition 1.4. An intuitionistic multi fuzzy subset A of a set X is defined as an

object of the form A = { ( x, µA1(x), µA2(x), .., µAn(x), A1(x), A2(x),.., An(x) )/

xX}, where µAi : X[0, 1] and Ai : X[0, 1] for all i, define the degrees of

membership and the degrees of non-membership of the element xX respectively

and every x in X satisfying 0 ≤ µAi(x) + Ai(x) ≤ 1 for all i. It is denoted as

A = ( µA , A ) where µA = ( µA1, µA2, …, µAn ) and A = (A1, A2,…, An ).

Definition 1.5. An interval valued intuitionistic multi fuzzy subset (IVIMFS) [A]

in X is defined as an object of the form [Ai](x) = {x, [Ai](x), [Ai](x) = x, [[Ai](x),

[Ai]+(x)], [[Ai]

(x), [Ai]+(x)] /x in X }, where [Ai]: XD[0, 1] and [Ai]: XD[0, 1]

for all i, define the degree of membership and the degree of non-membership of the

element xX respectively and for every xX satisfying 0 [Ai](x) + [Ai](x) 1 for

all i. It is denoted as [A] = µ[A] , [A] where µ[A] = µ[A1], µ[A2], …, µ[An] and

[A] = [A1], [A2],…, [An] .

Definition 1.6. Let [A] and [B] be any two interval valued intuitionistic multi fuzzy

subsets of X. We define the following relations and operations:

(i) [A] [B] if and only if [Ai] (x) ≤ [Bi]

(x) and [Ai]+(x) ≥ [Bi]

+(x) for all x in X

and for all i.

(ii) [A] = [B] if and only if [Ai] (x) = [Bi]

(x) and [Ai] +(x) = [Bi]

+(x) for all x in X

and for all i.

(iii) [A][B] = { ( x, rmin { [Ai](x), [Bi](x) }, rmax { [Ai](x), [Bi](x) } ) / xX }

where rmin {[Ai](x), [Bi](x)} = [ min {[Ai](x), [Bi]

(x)}, min{ [Ai]+(x), [Bi]

+(x)}]

and rmax{[Ai](x), [Bi](x)} = [ max{ [Ai](x), [Bi]

(x) }, max{ [Ai]+(x), [Bi]

+(x) } ]

for all i

(iv) [A][B] = { ( x, rmax { [Ai](x), [Bi](x) }, rmin { [Ai](x), [Bi](x) } ) / xX }

where rmax{ [Ai](x), [Bi](x) } = [max{[Ai](x), [Bi]

(x) }, max{[Ai]+(x), [Bi]

+(x) }]

and rmin { [Ai](x), [Bi](x) } = [ min { [Ai](x), [Bi]

(x) }, min { [Ai]+(x), [Bi]

+(x) } ]

for all i.

(v) [A] C = { ( x, [Ai](x), [Ai](x) ) / xX } for all i.

Definition 1.7. Let [A] = (µ[A] , [A]) be an interval valued intuitionistic multi fuzzy

subset in a set [S], the strongest interval valued intuitionistic multi fuzzy relation

on [S], that is an interval valued intuitionistic multi fuzzy relation [V] =( (µ[V1], [V1] ),

(µ[V2], [V2]),…, (µ[Vn], [Vn]) ) with respect to [A] given by µ[Vi](x,y) = rmin { µ[Ai](x),

µ[Ai](y) } and [Vi](x,y) = rmax{ [Ai](x), [Ai](y)} for all x and y in [S] and for all i.

Definition 1.8. Let V be any nonempty set, E be any set and f: EVV be any

function. Then [A] = ( µ[A], [A] ) = ( ( µ[A1], [A1] ), ( µ[A2], [A2] ),…, ( µ[An], [An] ) )

is an interval-valued intuitionistic multi fuzzy subset of V, [S] = ( µ[S], [S] ) =

((µ[S1], [S1] ), ( µ[S2], [S2] ),…,( µ[Sn], [Sn] ) ) is an interval valued intuitionistic multi

fuzzy relation on V with respect to [A] and [B] = (µ[B], [B]) = ((µ[B1], [B1]), (µ[B2],

[B2]),…, (µ[Bn], [Bn] ) ) is an interval valued intuitionistic multi fuzzy subset of E



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such that µ[Bi](e) ≤ ),(][

),(1

yxiS

f yxe

and [Bi](e) ≥ ),(][

),(1

yxiS

f

vyxe

for all i. Then the ordered

triple [F] = ( [A], [B], f ) is called an interval valued intuitionistic multi fuzzy graph

(IVIMFG), where the elements of [A] are called interval valued intuitionistic multi

fuzzy points or interval valued intuitionistic multi fuzzy vertices and the elements

of [B] are called interval valued intuitionistic multi fuzzy lines or interval valued

intuitionistic multi fuzzy edges of the interval valued Intuitionistic multi fuzzy

graph [F]. If f(e) = (x, y), then the interval valued Intuitionistic multi fuzzy points

( x, µ[A](x), [A](x) ), ( y, µ[A](y), [A](y) ) are called interval valued intuitionistic

multi fuzzy adjacent points and interval valued intuitionistic multi fuzzy points

( x, µ[A](x), [A](x) ), interval valued intuitionistic multi fuzzy line (e, µ[B](e), [B](e) )

are called incident with each other. If two district interval valued intuitionistic multi

fuzzy lines (e1, µ[B](e1), [B](e1) ) and (e2, µ[B](e2), [B](e2) ) are incident with a

common interval valued intuitionistic multi fuzzy point, then they are called interval

valued intuitionistic multi fuzzy adjacent lines.

Note : [A](x) = ((µ[A1], [A1])(x), (µ[A2], [A2])(x),…, (µ[An], [An])(x)), [B](x) = ((µ[B1],

[B1])(x), (µ[B2], [B2] )(x),…,(µ[Bn], [Bn])(x) ), [C](x) = ((µ[C1], [C1] )(x), (µ[C2],

[C2])(x),…, (µ[Cn], [Cn])(x) ) and [D](x) =( (µ[D1], [D1])(x), (µ[D2], [D2])(x),…, (µ[Dn],

[Dn])(x) ).

Definition 1.9. An interval valued intuitionistic multi fuzzy line joining an interval

valued intuitionistic multi fuzzy point to itself is called an interval valued

intuitionistic multi fuzzy loop.

Definition 1.10. Let [F] = ( [A], [B], f ) be an IVIMFG. If more than one interval

valued intuitionistic multi fuzzy line joining two interval valued intuitionistic multi

fuzzy vertices is allowed, then the IVIMFG [F] is called an interval valued

intuitionistic multi fuzzy pseudo graph.

Definition 1.11. [F] = ( [A], [B], f ) is called an interval valued intuitionistic multi

fuzzy simple graph if it has neither interval valued intuitionistic multi fuzzy multiple

lines nor interval valued intuitionistic multi fuzzy loops.

Example 1.12. F = ( [A], [B], f ), where V = { v1, v2, v3, v4, v5 }, E = { a, b, c, d, e, h,

g } and f : EVV is defined by f(a) = (v1, v2) , f(b) = (v2, v2), f(c) = (v2, v3),

f(d) = (v3, v4), f(e) = (v3, v4), f(h) = (v4, v5), f(g) = (v1, v5). An interval valued

intuitionistic multi fuzzy subset [A] = { (v1, ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]),

([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ), (v2, ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]), ([0.2, 0.3],

[0.1,0.2], [0.1, 0.2]) ), (v3, ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2],

[0.1, 0.2]) ), (v4, ([0.2, 0.3], [0.2, 0.3], [0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ),

(v5, ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ) } of V. An

interval valued intuitionistic multi fuzzy relation [S] = { ( (v1, v1), ([0.2, 0.3],

[0.1, 0.2], [0.4, 0.5]),([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ), ( (v1, v2), ([0.1, 0.2], [0.1, 0.2],

[0.3, 0.4]), ([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ), ( (v1, v3), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ( (v1, v4), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]),([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ), ( (v1, v5), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ( (v2, v1), ([0.1, 0.2], [0.1, 0.2],

[0.3, 0.4]), ([0.2, 0.3], [0.1,0.2], [0.1, 0.2])), ( (v2, v2), ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]),

([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ), ((v2, v3), ([0.1, 0.2], [0.1, 0.2],[0.3, 0.4]),

([0.2, 0.3],[0.1,0.2],[0.1, 0.2]) ), ( (v2, v4), ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]), ([0.2, 0.3],



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[0.1,0.2], [0.1, 0.2]) ), ( (v2, v5), ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]), ([0.2, 0.3], [0.1,0.2],

[0.1, 0.2]) ), ( (v3, v1), ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2],

[0.1, 0.2]) ), ((v3, v2), ([0.1, 0.2], [0.1, 0.2],[0.3, 0.4]), ([0.2, 0.3],[0.1,0.2],[0.1, 0.2]) ),

((v3, v3), ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v3,v4),

([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v3, v5),

([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v4, v1),

([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]),([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ), ((v4, v2), ([0.1, 0.2],

[0.1, 0.2], [0.3, 0.4]),([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v4, v3), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ( (v4, v4), (v4, ([0.2, 0.3], [0.2, 0.3],

[0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ), ((v4, v5), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v5, v1), ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v5, v2), ([0.1, 0.2], [0.1, 0.2],

[0.3, 0.4]), ([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ), ((v5, v3), ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]),

([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), ((v5, v4), ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]),

([0.2, 0.3], [0.1, 0.2], [0.1, 0.2])), ((v5, v5), ([0.2, 0.3], [0.1, 0.2],[0.4, 0.5]),

([0.2,0.3],[0.1, 0.2],[0.1, 0.2])) } on V with respect to [A] and an IVIMFS

[B] = { (a, ([0.1, 0.2], [0.1, 0.2], [0.2, 0.3]), ([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]) ),

(b, ([0.1, 0.2], [0.1, 0.2], [0.2, 0.3]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ), (c, ([0.1, 0.2],

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

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

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

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

[0.3, 0.4]) ) } of E.

Fig 1.1

In figure 1.1, (i) (v1, ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2],[0.2, 0.3]) )

is an interval valued intuitionistic multi fuzzy point. (ii) (a, ([0.1, 0.2], [0.1, 0.2],

[0.2, 0.3]), ([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]) ) is an interval valued intuitionistic multi

fuzzy edge. (iii) (v1, ([0.2, 0.3], [0.1, 0.2], [0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2],[0.2, 0.3]) )

and (v2, ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]), ([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ) are interval

valued intuitionistic multi fuzzy adjacent points. (iv) (a, ([0.1, 0.2], [0.1, 0.2],

[0.2, 0.3]), ([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]) ) join with (v1, ([0.2, 0.3], [0.1, 0.2],

[0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ) and (v2, ([0.1, 0.2], [0.1, 0.2], [0.3, 0.4]),

([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ) and therefore it is incident with (v1, ([0.2, 0.3],

[0.1, 0.2], [0.4, 0.5]), ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ) and (v2, ([0.1, 0.2], [0.1, 0.2],

(v2, ([0.1,0.2],[0.1,0.2],[0.3,0.4]),

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

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

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

(v3,([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v4,([0.2,0.3],[0.2,0.3],[0.4,0.5]),

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

(v5, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v1, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

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

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

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

0.3))

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

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

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

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

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

([0.4,0.5],[0.2,0.3],[0.5,0.6]))

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

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



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[0.3, 0.4]), ([0.2, 0.3], [0.1,0.2], [0.1, 0.2]) ). (v) (a, ([0.1, 0.2], [0.1, 0.2], [0.2, 0.3]),

([0.4, 0.5], [0.2, 0.3], [0.5, 0.6]) ) and (g, ([0.2, 0.3], [0.1, 0.2], [0.2, 0.3]), ([0.3, 0.4],

[0.2, 0.3], [0.3, 0.4]) ) are interval valued intuitionistic multi fuzzy adjacent lines.

(vi) (b, ([0.1, 0.2], [0.1, 0.2], [0.2, 0.3]), ([0.2, 0.3], [0.1, 0.2], [0.1, 0.2]) ) is an

interval valued intuitionistic multi fuzzy loop. (vii) (d, ([0.2, 0.3], [0.1, 0.2],[0.2, 0.3]),

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

[0.2, 0.3], [0.2, 0.3]) ) are interval valued intuitionistic multi fuzzy multiple edges.

(viii) It is not an interval valued intuitionistic multi fuzzy simple graph. (ix) It is an

interval valued intuitionistic multi fuzzy pseudo graph.

Definition 1.13. The IVIMFG [H]=([C], [D], f) where [C] = ( µ[C] , [C] ) and

[D] = ( µ[D] , [D] ) is called an interval valued intuitionistic multi fuzzy subgraph

of [F] = ([A], [B], f ) if [C] [A] and [D] [B].

Definition 1.14. The interval valued intuitionistic multi fuzzy subgraph

[H] = ( [C], [D], f ) is said to be an interval valued intuitionistic multi fuzzy

spanning subgraph of [F] = ( [A], [B], f ) if [C] = [A].

Example 1.15.

Fig.1.2 An interval valued intuitionistic multi fuzzy pseudo graph [F]=([A], [B],f)

Fig. 1.3 An interval valued intuitionistic multi fuzzy subgraph of [F]

(v2, ([0.1,0.2],[0.1,0.2],[0.3,0.4]),

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

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

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

(v3,([0.2,0.3],[0.2,0.3],[0.3,0.4]),

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

(v4,([0.2,0.3],[0.2,0.3],[0.2,0.4]),

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

(v5, ([0.1,0.2],[0.1,0.2],[0.3,0.4]),

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

(v1, ([0.2,0.3],[0.1,0.2],[0.3,0.4]),

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

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

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

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

0.3))

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

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

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

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

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

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

(v2, ([0.1,0.2],[0.1,0.2],[0.3,0.4]),

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

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

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

(v3,([0.2,0.3],[0.2,0.3],[0.4,0.5]),

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

(v4,([0.2,0.3],[0.2,0.3],[0.4,0.5]),

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

(v5, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v1, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

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

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

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

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

0.3))

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

([0.3,0.4],[0.2,0.3],[0.4,0.5]))

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

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

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

([0.4,0.5],[0.2,0.3],[0.5,0.6]))

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

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

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

([0.4,0.5],[0.2,0.3],[0.3,0.4]))

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

([0.4,0.5],[0.2,0.3],[0.5,0.6]))



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Fig 1.4 An interval valued intuitionistic multi fuzzy spanning subgraph of [F]

Definition 1.16. Let [F] = ( [A], [B], f ) be an interval valued intuitionistic multi fuzzy

graph. Then the degree of an interval valued intuitionistic multi fuzzy vertex is

defined by d(v) = ( 𝑑µ(v), 𝑑(v) ) where 𝑑µ(v) = )(

),(

][1

e

f vue

B

+ )(2

),(

][1

e

f vve

B

and 𝑑(v) = )(

),(

][1

e

f vue

Bv

+ )(2

),(

][1

e

f vve

Bv

. Here d(v) is not a unique value but it

is a structure of values.

Definition 1.17. The minimum degree of the IVIMFG [F] = ( [A], [B], f ) is

δ([F]) = (𝛿µ([F]), 𝛿([F])) where 𝛿µ([F]) ={𝑑µ(v) / vV} and 𝛿([F]) ={ 𝑑(v)

/vV } and the maximum degree of [F] is ([F]) = (∆μ([F]), ∆([F])), where

∆μ([F]) ={𝑑µ(v) /vV } and ∆([F]) = {𝑑(v) /vV}. Here δ([F]) and ([F]) is not

a unique value but they are a

Structure of values.

Definition 1.18. Let [F] = ( [A], [B], f ) be an interval valued intuitionistic multi fuzzy

graph. Then the order of IVIMFG [F] is defined to be O([F]) = (𝑂µ([F]), 𝑂([F]) )

where 𝑂µ([F]) = )(][

vVv

A

and 𝑂([F]) = )(][

vVv

Av

. Here O([F]) is not a unique

value but it is a structure of values.

Definition 1.19. Let [F] = ( [A], [B], f ) be an IVIMFG. Then the size of the

IVIMFG [F] is defined to be S([F]) = (𝑆µ([F]), 𝑆([F])) where

𝑆µ([F]) = )(

),(

][1

e

f yxe

B

and 𝑆([F]) = ),(

][1

)(yxfe

B e . Here S([F]) is not a unique

value but it is a structure of values.

(v2, ([0.1,0.2],[0.1,0.2],[0.3,0.4]),

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

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

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

(v3,([0.2,0.3],[0.2,0.3],[0.4,0.5]),

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

(v4,([0.2,0.3],[0.2,0.3],[0.4,0.5]),

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

(v5, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v1, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

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

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

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

([0.3,0.4],[0.2,0.3],[0.4,0.5]))

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

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

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

([0.4,0.5],[0.2,0.3],[0.5,0.6])) (g, ([0.2,0.3],[0.1,0.2],[0.2,0.3]),

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



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Example 1.20.

Fig 1.5 Interval valued intuitionistic multi fuzzy graph [F]

Here d(v1) = (([0.2, 0.4], [0.3, 0.5], [0.3, 0.5]), ([0.6, 0.8], [0.7, 0.9], [0.4, 0.6]) ),

d(v2) = (([0.4, 0.8], [0.7, 1.1], [0.5, 0.9]), ([1.2, 1.6], [1.3, 1.7], [0.8, 1.2]) ),

d(v3) = (([0.4, 0.7], [0.3, 0.6], [0.4, 0.7]), ([0.9, 1.2], [1.1, 1.4], [0.6, 0.9]) ),

d(v4) = (([0.4, 0.7], [0.3, 0.6], [0.4, 0.7]), ([0.9, 1.2], [1.3, 1.6], [0.6, 0.9]) ),

([F]) = (([0.2, 0.4], [0.3, 0.5], [0.3, 0.5]), ([0.6, 0.8], [0.7, 0.9], [0.4, 0.6]) ),

([F]) = (([0.4, 0.8], [0.7, 1.1], [0.5, 0.9]), ([1.2, 1.6], [1.3, 1.7], [0.8, 1.2]) ),

O([F]) = (([0.6, 1.0], [0.6, 1.0], [1.4, 1.8]), ([1.0, 1.4], [0.9, 1.3], [0.7, 1.1]) ),

S[ F] = (([0.7, 1.3], [0.8, 1.4], [0.8, 1.4]), ([1.8, 2.4], [2.2, 2.8], [1.2, 1.8) ).

Theorem 1.21. i) The sum of the degree of membership all interval valued

intuitionistic multi fuzzy vertices in an IVIMFG is equal to twice the sum of the

membership value of all IVIMFG.

i.e., )(VVvd

= 2Sμ([F]).

ii) The sum of the degree of non membership all interval valued intuitionistic multi

fuzzy vertices in an IVIMFG is equal to twice the sum of the non membership value

of all interval valued intuitionistic multi fuzzy edges. i.e., )(VVv

vd

= 2𝑆([F]).

iii) The sum of the degree of all interval valued intuitionistic multi fuzzy vertices in

an IVIMFG is equal to twice the sum of the all interval valued intuitionistic multi

fuzzy edges.

i.e., V v

d(v) = 2S([F]).

Proof. i) Let [F] = ( [A], [B], f ) be an IVIMFG with respect to the set V and E. Since

degree of an interval valued intuitionistic multi fuzzy vertex denote sum of the

membership values of all interval valued intuitionistic multi fuzzy edges incident on

it. Each interval valued intuitionistic multi fuzzy edge of [F] is incident with two

interval valued intuitionistic multi fuzzy vertices. Hence membership value of each

interval valued intuitionistic multi fuzzy edge contributes two to the sum of degrees of

interval valued intuitionistic multi fuzzy vertices. Hence the sum of the degree of all

interval valued intuitionistic multi fuzzy vertices in an IVIMFG is equal to twice the

sum of the membership value of all interval valued intuitionistic multi fuzzy edges.

i.e., )(VVvd

= 2Sμ([F]).

(v2, ([0.1,0.2],[0.2,0.3],[0.3,0.4]),

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

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

(v3,([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v4,([0.2,0.3],[0.1,0.2],[0.4,0.5]),

([0.2,0.3],[0.4,0.5],[0.2,0.3]))

(v1, ([0.1,0.2],[0.2,0.3],[0.3,0.4]),

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

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

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

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

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

(e, ([0.1,0.2],[0.1,0.2],[0.1,0.2]),

([0.3,0.4],[0.5,0.6],[0.2,0.3]))



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ii) Let [F] = ( [A], [B], f ) be an IVIMFG with respect to the set V and E. Since degree

of an interval valued intuitionistic multi fuzzy vertex denote sum of the non

membership values of all interval valued intuitionistic multi fuzzy edges incident on

it. Each interval valued intuitionistic multi fuzzy edge of [F] is incident with two

interval valued intuitionistic multi fuzzy vertices. Hence non membership value of

each interval valued intuitionistic multi fuzzy edge contributes two to the sum of

degrees of interval valued intuitionistic multi fuzzy vertices. Hence the sum of the

degree of all interval valued intuitionistic multi fuzzy vertices in an IVIMFG is equal

to twice the sum of the non membership value of all interval valued intuitionistic

multi fuzzy edges.

i.e., )(VVv

vd

= 2𝑆([F]).

iii) From i) and ii) The sum of the degree of all interval valued intuitionistic multi

fuzzy vertices in an IVIMFG is equal to twice the sum of the all interval valued

intuitionistic multi fuzzy edges. i.e., V v

d(v) = 2S([F]).

Theorem 1.22. Let [F] = ( [A], [B], f ) be an IVIMFG with number of interval valued

intuitionistic multi fuzzy vertices n, all of whose interval valued intuitionistic multi

fuzzy vertices have degree [𝑠] = ([𝑠]μ , [𝑠] ) or [𝑡] = ( [𝑡]μ , [𝑡] ). If [F] has p-

interval valued intuitionistic multi fuzzy vertices of degree [s] and (np) interval

valued intuitionistic multi fuzzy vertices of degree [t] then 2S([F]) = p[s] + (n – p)[t].

Proof. Let V1 be the set of all fuzzy vertices with degree [s]. Let V2 be the set of all

fuzzy vertices with degree [t]. Thenv V

d(v)

= 1v V

d(v)

+2v V

d(v)

which implies that

2S([F]) = ( V

vv

d1

)(

, V

vv

vd1

)( ) + ( V

vv

d2

)(

, V

vv

vd2

)( ) which implies that

2S([F]) = (p([𝑠]μ, [𝑠]) +(n–p) ([𝑡]μ, [𝑡])) which implies that 2S[(F]) = p[s]+(n– p)[t].

2. INTERVAL VALUED INTUITIONISTIC MULTI FUZZY REGULAR

GRAPH:

Definition 2.1. An IVIMFG [F] = ([A], [B], f) is called interval valued intuitionistic

multi fuzzy regular graph if d(v) = ( [s], [k] ) for all v in V. It is called interval

valued intuitionistic multi fuzzy ( [s], [k] )-regular graph, where ( [s], [k] ) =

( ( [s1], [k1]), ( [s2], [k2] ),…,( [sn], [kn] ) ).

Remark 2.2. [F] is an interval valued intuitionistic multi fuzzy ( [s], [k] )-regular

graph if and only if ([F]) = ([F]) = ( [s], [k] ).

Example 2.3.

Fig 2.1

(v3, ([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

(e, ([0.1,0.2],[0.1,0.2],[0.1,0.2]),

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

(v2,([0.3,0.4],[0.2,0.3],[0.3,0.4]),

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

(v1, ([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

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

([0.4,0.5],[0.2,0.4],[0.3,0.4]))

(d, ([0.1,0.2],[0.1,0.2],[0.1,0.2]),

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

(a ,([0.1,0.2],[0.1,0.2],[0.1,0.2]),

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

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

([0.2,0.4],[0.3,0.4],[0.3,0.5]))



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Here d(vi) = (([0.4, 0.8], [0.4, 0.8], [0.4, 0.8]), ([0.8, 1.2], [0.8, 1.2], [0.8, 1.2]) ) for

all i, ([F]) = (([0.4, 0.8], [0.4, 0.8], [0.4, 0.8]), ([0.8, 1.2], [0.8, 1.2], [0.8, 1.2]) ),

([F]) = (([0.4, 0.8], [0.4, 0.8], [0.4, 0.8]), ([0.8, 1.2], [0.8, 1.2], [0.8, 1.2]) ). Clearly it

is an interval valued intuitionistic multi fuzzy (([0.4, 0.8], [0.4, 0.8], [0.4, 0.8]),

([0.8, 1.2], [0.8, 1.2], [0.8, 1.2]) ) -regular graph.

Definition 2.4. An interval valued intuitionistic multi fuzzy graph [F] = ( [A], [B], f )

is called an interval valued intuitionistic multi fuzzy complete graph if every pair

of distinct interval valued intuitionistic multi fuzzy vertices are fuzzy adjacent and

µ[Bi](e) = ),(][

),(1

yxiS

f yxe

and [Bi](e) = ),(][

),(1

yxiS

f

vyxe

for all x, y in V and for all i.

Example 2.5.

Fig.2.2 An interval valued intuitionistic multi fuzzy complete graph

Definition 2.6. An interval valued intuitionistic multi fuzzy graph [F] = ([A], [B], f) is

an interval valued intuitionistic multi fuzzy strong graph if µ[Bi](e) = ),(][

),(1

yxiS

f yxe

and [Bi](e) = ),(][

),(1

yxiS

f

vyxe

for all e in E and for all i.

Example 2.7.

Fig. 2.3 An interval valued intuitionistic multi fuzzy strong graph

Remark 2.8. Every interval valued intuitionistic multi fuzzy complete graph is an

interval valued intuitionistic multi fuzzy strong graph. An interval valued

intuitionistic multi fuzzy strong graph need not be interval valued intuitionistic multi

fuzzy complete graph from the fig. 2.3.

(v3, ([0.2,0.3],[0.3,0.4],[0.4,0.5]),

([0.2,0.3],[0.4,0.5],[0.2,0.3]))

(v2,([0.1,0.2],[0.3,0.4],[0.3,0.4]),

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

(v1, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

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

([0.2,0.3],[0.4,0.5],[0.2,0.3]))

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

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

(v2, ([0.1,0.2],[0.2,0.3],[0.3,0.4]),

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

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

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

(v3,([0.2,0.3],[0.1,0.2],[0.4,0.5]),

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

(v4,([0.2,0.3],[0.1,0.2],[0.4,0.5]),

([0.2,0.3],[0.4,0.5],[0.2,0.3]))

(v1, ([0.1,0.2],[0.2,0.3],[0.3,0.4]),

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

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

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

(d, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

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

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

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



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Theorem 2.9. If [F] is an interval valued intuitionistic multi fuzzy ( [s], [k] )-regular

graph with p- interval valued intuitionistic multi fuzzy vertices. Then

2S([F]) = ( p[s], p[k] ) .

Proof. Given that the interval valued intuitionistic multi fuzzy graph is an interval

valued intuitionistic multi fuzzy ( [s], [k] )-regular graph, so d(v) = ( [s], [k] ) for all v

in V. Here there are p- interval valued intuitionistic multi fuzzy vertices, so

)][],[()(

VvVv Vv

ksvd = ( p[s], p[k] ) which implies that 2S([F]) = ( p[s], p[k] ).

Theorem 2.10. Let [F] = ( [A], [B], f ) be interval valued intuitionistic multi fuzzy

complete graph and [A] = ( μ[A], [A] ) is constant function. Then [F] is an interval

valued intuitionistic multi fuzzy regular graph. Here “constant” is not a unique value

but it is a structure of constant values.

Proof. Since [A] is a constant function, so [A](v) = ( [s], [k] ) (say) for all v in V and

[F] is an interval valued intuitionistic multi fuzzy complete graph, so

µ[Bi](e) = ),(][

),(1

yxiS

f yxe

and [Bi](e) = ),(][

),(1

yxiS

f

vyxe

for all x and y in V and xy and

for all i. Therefore membership and non membership value of all interval valued

intuitionistic multi fuzzy edges are [s], [k] respectively. Hence d(v) = ( (p 1)[s],

(p-1)[k] ) for all v in V.

Theorem 2.11. If [F] = ( [A], [B], f ) is interval valued intuitionistic multi fuzzy

complete graph with p- interval valued intuitionistic multi fuzzy vertices and [A] is

constant function then sum of the membership values of all fuzzy edges is 𝑝(𝑝−1)

2

μ[A](v) for all v in V and sum of the non membership values of all fuzzy edges is

𝑝(𝑝−1)

2 [A](v) for all v in V. i.e., S([F]) = ( pC2 μ[A](v), pC2 [A](v) ) for all v in V.

Proof. Suppose [F] is an interval valued intuitionistic multi fuzzy complete graph and

[A] = ( μ[A], [A] ) is a constant function. Let [A](v) = ( [s], [k] ) for all v in V and

d(v) = ( (p 1)[s], (p-1)[k]) for all v in V. Then )])[1(],)[1(()(

Vv VvVv

kpspvd

= ( p(p 1)[s], p(p-1)[k] ) which implies that 2S([F]) = ( p(p 1)[s], p(p 1)[k] ).

Hence S([F]) = ( pC2 [s], pC2 [k] ). i.e., S([F]) = ( pC2 μ[A](v), pC2[A](v) ) for all v in V.

Definition 2.12. Let [F] = ( [A], [B], f) be an interval valued intuitionistic multi fuzzy

graph. The total degree of interval valued intuitionistic multi fuzzy vertex v is

defined by dT(v) = ( 𝑑𝑇μ(v), 𝑑𝑇(v)) where 𝑑𝑇μ

(v) = )(

),(

][1

e

f vue

B

+ )(2

),(

][1

e

f vve

B

+ μ[A](v) = dμ(v) + μ[A](v) and 𝑑𝑇(v) = )(

),(

][1

e

f vue

Bv

+ )(2

),(

][1

e

f vve

Bv

+ [A](v) =

𝑑(v) + [A](v) for al v in V.

Definition 2.13. An interval valued intuitionistic multi fuzzy graph [F] is interval

valued intuitionistic multi fuzzy ( [s], [k])-totally regular graph if each vertex of

[F] has the same total degree ([s], [k]). Here ([s], [k]) is not a unique value but it is a

structure of values.



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Example 2.14.

Fig 2.4. Here dT(vi) = (([1.0, 1.5], [1.0, 1.5], [1.0, 1.5]), ([1.0, 1.5], [1.0, 1.5],

[1.0, 1.5]) ) for all i, it is interval valued intuitionistic multi fuzzy (([1.0, 1.5],

[1.0, 1.5], [1.0, 1.5]), ([1.0, 1.5], [1.0, 1.5], [1.0, 1.5]) )-totally regular graph.

Example 2.15. Fig 2.1 it is an interval valued intuitionistic multi fuzzy regular graph,

but it is not an interval valued intuitionistic multi fuzzy totally regular graph since

dT(v1) = (([0.6, 1.1], [0.7, 1.2], [0.8, 1.3]), ([1.0, 1.5], [0.9, 1.4], [0.9, 1.4]) ),

dT(v2)= (([0.7, 1.2], [0.6, 1.1], [0.7, 1.2]), ([1.0, 1.5],[0.9, 1.4], [0.9, 1.4])) and

d[T](v1) d[T](v2).

Example 2.16. Fig 2.4, it is an interval valued intuitionistic multi fuzzy totally regular

graph but it is not an interval valued intuitionistic multi fuzzy regular graph since

d(v1) = (([0.7, 1.1], [0.8, 1.2], [0.7, 1.1]), ([0.8, 1.2], [0.9, 1.3], [0.8, 1.2]) ),

d(v3) =(([0.6,1.0],[0.7,1.1],[0.6, 0.9]),([0.8,1.2],[0.9, 1.3],[0.8, 1.2])) and d(v1) d(v3).

Example 2.17.

Fig.2.5

Here d(vi) = (([0.6, 1.0], [0.6, 1.0], [0.6, 1.0]), ([0.8, 1.2], [0.8, 1.2], [0.8, 1.2]) ) for

all i, dT(vi) = (([0.8, 1.4], [0.9, 1.4], [1.0, 1.5]), ([1.0, 1.5], [0.9, 1.4], [0.9, 1.4]) ) for

all i. It is both interval valued intuitionistic multi fuzzy regular graph and interval

valued intuitionistic multi fuzzy totally regular graph.

Example 2.18.

Fig 2.6

(v3, ([0.2,0.4],[0.3,0.4],[0.4,0.5]),

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

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

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

(v2,([0.2,0.4],[0.3,0.4],[0.4,0.5]),

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

(v1, ([0.2,0.4],[0.3,0.4],[0.4,0.5]),

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

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

([0.3,0.5],[0.3,0.4],[0.3,0.4]))

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

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

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

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

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

([0.3,0.4],[0.2,0.4],[0.3,0.5]))

(v3, ([0.4,0.5],[0.3,0.4],[0.4,0.6]),

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

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

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

(v2,([0.3,0.4],[0.3,0.4],[0.3,0.5]),

([0.2,0.3],[0.2,0.3],[0.2,0.3]))

(v1, ([0.3,0.4],[0.2,0.3],[0.3,0.4]),

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

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

([0.4,0.5],[0.2,0.4],[0.3,0.4]))

(d, ([0.2,0.3],[0.2,0.3],[0.2,0.3]),

([0.2,0.3],[0.4,0.5],[0.2,0.3]))

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

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

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

([0.2,0.4],[0.3,0.4],[0.3,0.5]))

(v2, ([0.2,0.3],[0.2,0.3],[0.3,0.4]),

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

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

(v3,([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

(v1, ([0.1,0.2],[0.2,0.3],[0.3,0.4]),

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

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

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

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

([0.3,0.4],[0.5,0.6],[0.2,0.3]))



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Here d(v1) = (([0.4, 0.8], [0.8, 1.2], [0.7, 1.1]), ([1.2, 1.6], [1.5, 1.9], [0.8, 1.2]) ),

d(v2) = (([0.4, 0.7], [0.4, 0.7], [0.8, 1.1]), ([0.9, 1.2], [1.3, 1.6], [0.6, 0.9]) ) ,

d(v3) = (([0.4, 0.7], [0.4, 0.7], [0.9, 1.2]), ([0.9, 1.2], [1.2, 1.5], [0.6, 0.9]) ),

dT(v1) = (([0.5, 1.0], [1.0, 1.5], [1.0, 1.5]), ([1.4, 1.9], [1.8, 2.3], [0.9, 1.4]) ),

dT(v2) = (([0.6, 1.0], [0.6, 1.0], [1.1, 1.5]), ([1.2, 1.6], [1.4, 1.8], [0.8, 1.2]) ),

dT(v3) = (([0.6, 1.0], [0.7, 1.1], [1.3, 1.7]), ([1.2, 1.6], [1.3, 1.7], [0.8, 1.2]) ), it is

neither interval valued intuitionistic multi fuzzy regular graph nor interval valued

intuitionistic multi fuzzy totally regular graph.

Theorem 2.19. Let [F] = ( [A], [B], f ) be interval valued intuitionistic multi fuzzy

complete graph and [A] = ( [s], [k] ) is constant function. Then [F] is an interval

valued intuitionistic multi fuzzy totally regular graph. Here “constant” is not a unique

value but it is a structure of constant values.

Proof. By theorem 2.11, clearly [F] is interval valued intuitionistic multi fuzzy

regular graph. i.e., d(v) = ( (p1)[s], (p1)[k] ) for all v in V. Also given [A] is

constant function. i.e., [A](v) = ( [s], [k] ) for all v in V. Then dT(v) = ( dμ(v) + μ[A](v),

𝑑(v) + [A](v) ) = ( (p1)[s] + [s], (p1)[k] + [k] ) = ( p[s], p[k] ) for all v in V. Hence

[F] is interval valued intuitionistic multi fuzzy totally regular graph.

Theorem 2.20. Let [F] = ([A], [B], f) be an interval valued intuitionistic multi fuzzy

regular graph. Then [H] = ([C], [B], f) is an interval valued intuitionistic multi fuzzy

totally regular graph if [C](v) = ))(,)((1

][

1

][

n

i

iA

n

i

iA vv ≤ 1 for all vi in V. Here

([k1], [k2]) and ([c1], [c2]) are not a unique value but they are a structure of values.

Proof. Assume that [F] = ([A], [B], f) is an interval valued intuitionistic multi fuzzy

( [k1], [k2] )-regular graph. i.e., d(vi) = ( [k1], [k2] ) for all vi in V. Given

[C](v) = ))(,)((1

][

1

][

n

i

iA

n

i

iA vv ≤ 1 for all vi in V. Then [C](v) = ([c1], [c2]) (say) for

all v in V and dT(H)(vi) = (𝑑𝜇(vi) + μ[C](vi) , 𝑑(vi) + [C](vi)) =( [k1] + [c1], [k2] + [c2])

for all vi in V. Hence [H] is interval valued intuitionistic multi fuzzy totally regular

graph.

Theorem 2.21. Let [F] = ([A], [B], f) be an interval valued intuitionistic multi fuzzy

graph and [A] is a constant function ( ie. [A](v) = ( [c1], [c2] ) (say) for all vV ).

Then [F] is interval valued intuitionistic multi fuzzy ( [k1], [k2] )-regular graph if and

only if [F] is interval valued intuitionistic multi fuzzy ( [k1] + [c1], [k2] + [c2] )-totally

regular graph. Here ([k1], [k2]) and ([c1], [c2]) are not a unique value but they are a

structure of values.

Proof. Assume that [F] is an interval valued intuitionistic multi fuzzy ([k1], [k2])-

regular graph and [A](v) = ( [c1], [c2] ) for all v in V, so d(v) = ( [k1], [k2] ) for all v

in V. Then dT(v) = (𝑑𝜇(v) + μ[A](vi) , 𝑑(v) + [A](vi) ) = ( [k1] + [c1], [k2] + [c2] ) for

all v in V . Hence [F] is interval valued intuitionistic multi fuzzy ( [k1] + [c1],

[k2] + [c2] )-totally regular graph. Conversely, Assume that [F] is interval valued

intuitionistic multi fuzzy ([k1] + [c1], [k2] + [c2])-totally regular graph.

ie., dT(v) = ([k1] + [c1], [k2] + [c2]) for all v in V which implies that (dμ(v) + μ[A](v),

𝑑(v)+ [A](v) ) = ( [k1] + [c1], [k2] + [c2] ) for all v in V implies that

( μ[A](v), [A](v) ) = ( [c1], [c2] ) for all v in V implies that dμ(v) + [c1] = [k1] + [c1] and

𝑑(v) + [c2] = [k2] + [c2] for all v in V. Therefore dμ(v) = [k1] and 𝑑(v) = [k2]



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for all v in V. ie., d(v) = ( [k1], [k2] ) for all v in V. Hence [F] is interval valued

intuitionistic multi fuzzy ( [k1], [k2] )-regular graph.

Theorem 2.22. If [F] = ([A], [B], f) is interval valued intuitionistic multi fuzzy

regular graph and interval valued intuitionistic multi fuzzy totally regular graph then

[A] is a constant function. Here “constant” is not a unique value but it is a structure of

constant values.

Proof. Assume that [F] is a both interval valued intuitionistic multi fuzzy regular

graph and interval valued intuitionistic multi fuzzy totally regular graph. Suppose that

[A] is not constant function. Then μ[A](u) μ[A](v) or [A](u) [A](v) for some u, v in

V. Since [F] is an interval valued intuitionistic multi fuzzy ( [k1], [k2] )-regular graph.

Then d(u) = d(v) = ( [k1], [k2] ). Then dT(u) dT(v) which is a contradiction to our

assumption. Hence [A] is a constant function.

Remark 2.23. Converse of the above theorem need not be true.

Fig 2.7

Here [A](vi) = (([0.2, 0.3], [0.3, 0.4], [0.4, 0.5]), ([0.2, 0.3], [0.1, 0.2], [0.2, 0.3]) ) for

all i , d(v1) = (([0.3, 0.5], [0.4, 0.6], [0.6, 0.8]), ([0.6, 0.8], [0.6, 0.8], [0.5, 0.7]) ),

d(v2) = (([0.5, 0.8], [0.5, 0.8], [0.9, 1.2]), ([0.9, 1.2], [1.0, 1.3], [0.8, 1.1]) ) ,

d(v3) = (([0.4, 0.7], [0.5, 0.8], [0.9, 1.2]), ([0.9, 1.2], [1.2, 1.5], [0.7, 1.0]) ),

dT(v1) = (([0.7, 1.1], [0.7, 1.0], [1.0, 1.3]), ([0.8, 1.1], [0.7, 1.0], [0.7, 1.0]) ),

dT(v2) = (([0.7, 1.1], [0.8, 1.2], [1.3, 1.7]), ([1.1, 1.5], [1.1, 1.5], [1.0, 1.4]) ),

dT(v3) = (([0.6, 1.0], [0.8, 1.2], [1.3, 1.7]), ([1.1, 1.5], [1.3, 1.7], [0.9, 1.3]) ). Hence

[F] is neither interval valued intuitionistic multi fuzzy regular graph nor interval

valued intuitionistic multi fuzzy totally regular graph.

Theorem 2.24. If [F] = ( [A], [B], f ) is an interval valued intuitionistic multi fuzzy

( [c1], [c2] )-totally regular graph with p-interval valued intuitionistic multi fuzzy

vertices. Then 2S[F] + o[F] = p[c]. Here ( [c1], [c2] ) is not a unique value but it is a

structure of constant values.

Proof. Assume that [F] is an interval valued intuitionistic multi fuzzy ([c1], [c2])-

totally regular graph with p- interval valued intuitionistic multi fuzzy vertices.

Then dT(v) = ( [c1], [c2] ) for all v in V implies that ( dμ(v) + μ[A](v), 𝑑(v) + [A](v) ) =

( [c1], [c2] ) for all v in V which implies that ( ∑dμ(v) + ∑μ[A](v), ∑𝑑(v) + ∑[A](v) )

= ( ∑ [c1], ∑[c2] ) for all v in V which implies that ( 2Sμ[F] + oμ[F], 2𝑆[F] + 𝑜[F] ) =

( p[c1], p[c2] ) for all v in V implies that ( 2Sμ[F] , 2𝑆[F] ) + ( oμ[F], 𝑜[F] ) =

( p[c1], p[c2] ). Hence 2S[F] + o[F] = p[c].

Theorem 2.25. If [F] = ([A], [B], f) is both interval valued intuitionistic multi fuzzy

[k] = ( [k1], [k2] )-regular graph and interval valued intuitionistic multi fuzzy

(v2, ([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

(v3,([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

(v1, ([0.2,0.3],[0.3,0.4],[0.4,0.5]),

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

(c, ([0.2,0.3],[0.1,0.2],[0.4,0.5]),

([0.3,0.4],[0.4,0.5],[0.3,0.4]))

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

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

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))

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

([0.3,0.4],[0.4,0.5],[0.2,0.3]))



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[c] = ( [c1], [c2] )-totally regular graph with p- interval valued intuitionistic fuzzy

vertices. Then [k] + 𝑜[𝐹]

𝑝 = [c] . Here ([k1], [k2]) and ([c1], [c2]) are not a unique value

but they are a structure of values.

Proof. Assume that [F] is interval valued intuitionistic multi fuzzy [k]-regular graph

with p- interval valued intuitionistic multi fuzzy vertices. Then 2S[F] = p[k] By

theorem 2.27, 2S[F] + o[F] = p[c] implies that [k] + 𝑜[𝐹]

𝑝 = [c] .

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