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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
ISSN: 1548-7741
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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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Volume 9 Issue 8 - 2019
ISSN: 1548-7741
www.joics.org379
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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ISSN: 1548-7741
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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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Volume 9 Issue 8 - 2019
ISSN: 1548-7741
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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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