 Open Access
 Total Downloads : 17
 Authors : B. Vasudevan , K. Arjunan , K. L. Muruganantha Prasad
 Paper ID : IJERTV8IS080218
 Volume & Issue : Volume 08, Issue 08 (August 2019)
 Published (First Online): 02092019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Depth of Intuitionistic IFuzzy EDGE and Height of Intuitionistic IFuzzy EDGE of Intuitionistic IFuzzy Graph
B. Vasudevan
Department of Mathematics, Yadava College, Madurai625014.
Tamilnadu, India.
K. Arjunan
Department of Mathematics,
Alagappa Government Arts College, Karaikudi630003.
Tamilnadu, India.
K. L. Muruganantha Prasad
Department of Mathematics, H.H. The Rajas College, Pudukkottai622001. Tamilnadu, India.
Abstract: In this paper, depth of intuitionistic Ifuzzy edge and height of intuitionistic Ifuzzy edge of intuitionistic Ifuzzy graph are defined and introduced. Using this concept, some more theorems and results are given.
2010Mathematics subject classification : 03E72, 03F55, 05C72
Key Words: Fuzzy subset, Ifuzzy subset, intuitionistic Ifuzzy subset, intuitionistic Ifuzzy relation, strong intuitionistic Ifuzzy relation, intuitionisticIfuzzy graph, intuitionistic Ifuzzy loop, intuitionistic Ifuzzy pseudo graph, degree of intuitionistic Ifuzzy vertex, total degree of intuitionistic Ifuzzy vertex, order of the intuitionistic Ifuzzy graph, size of the intuitionistic Ifuzzy graph, intuitionistic Ifuzzy regular graph, intuitionistic Ifuzzy totally regular graph, intuitionistic Ifuzzy complete graph, depth of intuitionistic Ifuzzy edge, height of intuitionistic Ifuzzy edge.
INTRODUCTION:
In 1965, Zadeh [14] introduced the notation 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. Intuitionistic fuzzy set was introduced by Atanassov. K.T[4]. After that Rosenfeld[8] introduced fuzzy graphs. Yeh and Bang[13] 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. Ramakrishnan P.V and Lakshmi . T [7] introduced depth of , height of and fuzzy spanning super graphs. Arjunan. K & Subramani.C [2, 3] introduced a new structure of fuzzy graph and IFuzzy graph. Ifuzzy spanning supergraphs and intuitionistic fuzzy spanning supergraphs have been defined and introduced by Vasudevan.B et al.[11, 12]. In this paper, depth of intuitionistic Ifuzzy edge and height of intuitionistic Ifuzzy edge of intuitionistic Ifuzzy graph are defined and introduced.
PRELIMINARIES:
Definition 1.1[14]. Let X be any nonempty set. A mapping M: X [0,1] is called a fuzzy subset of X.
Definition 1.2[14]. Let X be any nonempty set. A mapping [M] : X D[0, 1] is called a Ifuzzy subset ( interval valued fuzzy subset ) of X, where D[0,1] denotes the family of all closed subintervals of [0,1] and [M](x) = [M(x), M+(x)], for all x in X, where M and M+ are fuzzy subsets of X such that M(x) M+(x), for all x in X. Thus M(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[4]. An intuitionistic fuzzy subset (IFS) A in X is defined as an object of the form [A] = { < x, A(x), A(x) > / x in X } where A:X[0, 1] and A: X[0, 1] define the degree of membership and the degree of nonmembership of the element xX respectively and for every xX satisfying A(x) + A(x) 1.
Definition 1.4. An intuitionistic Ifuzzy subset (IIFS) [A] in X is defined as an object of the form [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 nonmembership of the element xX respectively and for every xX satisfying [A] (x) + [A] (x) 1.
Example 1.5[4]. [A] = { < a, [0.4, 0.7], [0.2, 0.3] >, < b, [0.1, 0.5], [0.2, 0.5] >, < c, [0.5, 0.8], [0.1, 0.2] > } is an intuitionistic I
fuzzy subset of X = { a, b, c }.
Definition 1.6. Let [A] = { x, [A](x), [A](x) / xX }, [B] = { x, [B](x), [B](x) / xX} be any two intuitionistic Ifuzzy subsets of X. We define the following relations and operations:

[A] [B] if and only if [A](x) [B](x) and [B](x) [A](x) for all x in X.
+ +
+ +
 [A] = [B] if and only if [A](x) = [B](x) and [B](x) = [A](x) for all x in X. (iii) [A][B] = { x, rmin { [A](x), [B](x) }, rmax { [A](x), [B](x) } / xX }
where rmin {[A](x), [B](x)} = [ min {[A] (x), [B] (x)}, min{ [A] (x), [B] (x)}] and rmax{[A](x), [B](x)} = [ max{ [A] (x),
+ +
+ +
[B] (x) }, max{ [A] (x), [B] (x) } ].
(iv) [A][B] = { x, rmax { [A](x), [B](x) }, rmin { [A](x), [B](x) } / xX }
+ +
+ +
where rmax{ [A](x), [B](x) } = [max{[A] (x), [B] (x) }, max{[A] (x), [B] (x) }] and rmin { [A](x), [B](x) } = [ min {
+ +
+ +
[A] (x), [B] (x) }, min { [A] (x), [B] (x) } ].
(v) [A] C = { x, [A](x), [A](x) / xX }.
Definition 1.7. Let [M] = < Âµ[M] , [M] > be an intuitionistic Ifuzzy subset in a set S, the strongest intuitionistic Ifuzzy relation on S, that is an intuitionistic Ifuzzy relation [V] = < Âµ[V] , [V] > with respect to [M] given by Âµ[V](x,y) = rmin { Âµ[M] (x), Âµ[M](y) } and [V](x,y) = rmax{ [M] (x), [M](y)} for all x and y in S.
Definition 1.8. Let V be any nonempty set, E be any set and f: E VV be any function. Then [A] = < Âµ[A] , [A] > is an Intervalvalued intuitionistic subset of V, [S] = < Âµ[S] , [S] > is an intuitionistic Ifuzzy relation on V with respect to [A]
and [B] = < Âµ[B] , [B] > is an intuitionistic Ifuzzy subset of E such that Âµ[B](e)
[S ]1
(x, y) and [B](e)
v[S ]
1
(x, y) .
e f ( x, y )
e f ( x, y )
Then the ordered triple [F] = ( [A], [B], f ) is called an intuitionistic Ifuzzy graph, where the elements of [A] are called intuitionistic Ifuzzy points or intuitionistic Ifuzzy vertices and the elements of [B] are called intuitionistic Ifuzzy lines or intuitionistic Ifuzzy edges of the intuitionistic Ifuzzy graph [F]. If f(e) = (x, y), then the intuitionistic Ifuzzy points ( x,
Âµ[A](x), [A](x) ), ( y, Âµ[A](y), [A](y) ) are called intuitionistic Ifuzzy adjacent points and intuitionistic Ifuzzy points ( x,
Âµ[A](x), [A](x) ), intuitionistic Ifuzzy line (e, Âµ[B](e), [B](e) ) are called incident with each other. If two district intuitionistic Ifuzzy lines (e1, Âµ[B](e1), [B](e1) ) and (e2, Âµ[B](e2), [B](e2) ) are incident with a common intuitionistic Ifuzzy point, then they are called intuitionistic Ifuzzy adjacent lines.
Definition 1.9. An intuitionistic Ifuzzy line joining an intuitionistic Ifuzzy point to itself is called an intuitionistic Ifuzzy loop.
Definition 1.10. Let [F] = ([A], [B], f) be an intuitionistic Ifuzzy graph. If more than one intuitionistic Ifuzzy line joining two intuitionistic Ifuzzy vertices is allowed, then the intuitionistic Ifuzzy graph [F] is called an intuitionistic Ifuzzy pseudo graph.
Definition 1.11. [F] = ([A], [B], f) is called an intuitionistic Ifuzzy simple graph if it has neither intuitionistic Ifuzzy multiple lines nor intuitionistic Ifuzzy 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 : E VV is efined 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 intuitionistic Ifuzzy subset [A]
= { (v1, [0.5, 0.7], [0.2, 0.3] ), (v2, [0.4, 0.6], [0.1, 0.3] ), (v3, [0.4, 0.8], [0.2, 0.2] ), (v4, [0.3, 0.5], [0.2, 0.3] ), (v5, [0.3, 0.7],
[0.2, 0.2] ) } of V. An intuitionistic Ifuzzy relation [S] = { ( (v1, v1), [0.5, 0.7], [0.2, 0.3] ), ( (v1, v2), [0.4, 0.6], [0.2, 0.3] ), ( (v1,v3), [0.4, 0.7], [0.2, 0.3] ), ( (v1, v4), [0.3, 0.5], [0.2, 0.3] ), ( (v1, v5), [0.3, 0.7], [0.2, 0.3] ), ( (v2, v1), [0.4, 0.6], [0.2, 0.3] ), ( (v2,
v2), [0.4, 0.6], [0.1, 0.3] ), ( (v2, v3), [0.4, 0.6], [0.2, 0.3] ), ( (v2, v4), [0.3, 0.5], [0.2, 0.3] ), ( (v2, v5), [0.3, 0.6], [0.2, 0.3] ), (
(v3, v1), [0.4, 0.7], [0.2, 0.3] ), ( (v3, v2), [0.4, 0.6], [0.2, 0.3] ), ( (v3, v3), [0.4, 0.8], [0.2, 0.2] ), ( (v3, v4), [0.3, 0.5], [0.2, 0.3] ),
( (v3, v5), [0.3, 0.7], [0.2, 0.2] ), ( (v4, v1), [0.3, 0.5], [0.2, 0.3] ), ( (v4, v2), [0.3, 0.5], [0.2, 0.3] ), ( (v4, v3), [0.3, 0.5], [0.2, 0.3]
), ( (v4, v4), [0.3, 0.5], [0.2, 0.3] ), ( (v4, v5), [0.3, 0.5], [0.2, 0.3] ), ( (v5, v1), [0.3, 0.7], [0.2, 0.3] ), ( (v5, v2), [0.3, 0.6], [0.2, 0.3]
), ( (v5, v3), [0.3, 0.7], [0.2, 0.2] ), ( (v5, v4), [0.3, 0.5], [0.2, 0.3] ), ( (v5, v5), [0.3, 0.7], [0.2, 0.3] ) } on V with respect to [A] and
an intuitionistic Ifuzzy subset [B] = {(a, [0.4, 0.5], [0.2, 0.4]), (b, [0.3, 0.5], [0.2, 0.3] ), (c, [0.4, 0.6], [0.2, 0.4] ), (d, [0.2, 0.5],
[0.3, 0.4]), (e, [0.3, 0.5], [0.2, 0.3]), (h, [0.3, 0.5], [0.3, 0.4]), (g, [0.3, 0.6], [0.2, 0.4]) } of E.(v1, [0.5,0.7],[0.2,0.3])
(g, [0.3,0.6],[0.2,0.4])
(a , [0.4,0.5],[0.2,0.4])
(b, [0.3,0.5],[0.2,0.3] )
(v5, [0.3,0.7],[0.2,0.2])
(v2, [0.4,0.6],[0.1,0.3])
(h, [0.3,0.5],[0.3,0.4])
(v4, [0.3,0.5].[0.2,0.3])
(d, [0.2,0.5],[0.3,0.4])
(e, [0.3,0.5].[0.2,0.3])
Fig 1.1
(c, [0.4,0.6],[0.2,0.4])
(v3, [0.4,0.8],[0.2,0.2])
In figure 1.1, (i) ( v1, [0.5, 0.7], [0.2, 0.3] ) is an intuitionistic Ifuzzy point. (ii) ( a, [0.4, 0.5], [0.2, 0.4] ) is an intuitionistic I
fuzzy edge. (iii) ( v1, [0.5, 0.7], [0.2, 0.3] ) and ( v2, [0.4, 0.6], [0.1, 0.3] ) are intuitionistic Ifuzzy adjacent points. (iv) ( a, [0.4,
0.5], [0.2, 0.4] ) join with ( v1, [0.5, 0.7], [0.2, 0.3] ) and ( v2, [0.4, 0.6], [0.1, 0.3] ) and therefore it is incident with ( v1, [0.5,
0.7], [0.2, 0.3] ) and ( v2, [0.4, 0.6], [0.1, 0.3] ). (v) ( a, [0.4, 0.5], [0.2, 0.4] ) and ( g, [0.3, 0.6], [0.2, 0.4] ) are intuitionistic I
fuzzy adjacent lines. (vi) ( b, [0.3, 0.5], [0.2, 0.3] ) is an intuitionistic Ifuzzy loop. (vii) ( d, [0.2, 0.5], [0.3, 0.4] ) and ( e, [0.3, 0.5]. [0.2, 0.3] ) are intuitionistic Ifuzzy multiple edges. (viii) It is not an intuitionistic Ifuzzy simple graph. (ix) It is an intuitionistic Ifuzzy pseudo graph.
Definition 1.13. The fuzzy graph [H] = ([C], [D], f) where [C] = < Âµ[C] , [C] > and [D] = < Âµ[D] , [D] > is called an
intuitionistic Ifuzzy subgraph of [F] = ([A], [B], f) if [C] [A] and [D] [B].
Definition 1.14. Let [F] = ([A], [B], f) be an intuitionistic Ifuzzy graph. Then the degree of an intuitionistic Ifuzzy vertex is
defined by d(v) = (Âµ(v),(v)) where Âµ(v) =
(e) + 2
1 [ B] 1
(e)
[ B]
and
(v) =
[ B](e) + 2
v
v
1 1
e f
v[ B](e) .
(u,v)
e f
(v,v)
e f
(u,v)
e f
(v,v)
Definition 1.15. Let [F] = ([A], [B], f) be an intuitionistic Ifuzzy graph. The total degree of intuitionistic Ifuzzy vertex v is
defined by dT(v) = ( (v), (v)) where (v) =
(e) + 2
1 [ B] 1
(e) + [A](v) = d(v) + [A](v) and
[ B]
e f
(u,v)
e f
(v,v)
(v) =
[ B](e) + 2
v
v
1 1
v[ B](e) + [A](v) = (v) + [A](v) for al v in V.
e f
(u,v)
e f
(v,v)
Definition 1.16. The minimum degree of the intuitionistic Ifuzzy graph [F] = ([A], [B], f) is [F] = ( Âµ[F] , [F] ) where
Âµ[F] = rmin{ Âµ(v) / vV } and [F] = rmax{ (v) / vV} and the maximum degree of [F] is [F] = ([F], [F]) where[F] = rmax{ Âµ(v) / vV }and[F] = rmin{ (v) / vV }.
Definition 1.17. Let [F] = ([A], [B], f) be an intuitionistic Ifuzzy graph. Then the order of intuitionistic Ifuzzy graph [F] is
defined to be O[F] = (Âµ[F] , [F] ) where Âµ[F] = (v) and [F] = v[ A](v)
vV
[ A]vV
Definition 1.18. Let [F] = ([A], [B], f) be an intuitionistic Ifuzzy graph. Then the size of the intuitionistic Ifuzzy graph [F] is
defined to be S[F] = (Âµ[F] , [F] ) where Âµ[F] =
(e) and [F] =
1 [ B]
[ B](e) .
v
v
1
e f ( x, y)
e f ( x, y )
Definition 1.19. An intuitionistic Ifuzzy graph [F] = ( [A], [B], f ) is called intuitionistic Ifuzzy regular graph if d(v) = [m, n] for all v in V. It is also called intuitionistic Ifuzzy [m, n]regular graph.
Definition 1.20. An intuitionistic Ifuzzy graph [F] is an intuitionistic Ifuzzy [m, n]totally regular graph if each intuitionistic Ifuzzy vertex of [F] has the same total degree [m, n].
Theorem 1.21. The sum of the degree of all intuitionistic Ifuzzy vertices in a intuitionistic Ifuzzy graph [F] = ( [A], [B], f ) is equal to twice the sum of the membership value of all intuitionistic Ifuzzy edges. That is d (v) 2S ([F ]) .
vV
2. DEPTH OF INTUITIONISTIC IFUZZY EDGE AND HEIGHT OF INTUITIONISTIC IFUZZY EDGE OF INTUITIONISTIC IFUZZY GRAPH
Definition 2.1. Let [] = ([], [], ) be an intuitionistic Ifuzzy graph. Then the depth of intuitionistic Ifuzzy edge [] is defined by D([B]) = ( [d]( [B] ), [d]( [B] ) ) = ( rmin { [B](e) / eE) }, rmax { [B](e) / eE) } ).
Definition 2.2. Let [] = ([], [], ) be an intuitionistic Ifuzzy graph. Then the height of intuitionistic Ifuzzy edge [B] is defined by H([B]) = ( [d]( [B] ), [d]( [B] ) ) = ( rmax { [B](e) / eE) }, rmin { [B](e) / eE) } ). .
Example 2.3.
(u, [0.4,0.5],[0.2,0.3])
(f, [0.3,0.5],[0.2,0.4])
(w, [0.3,0.5],[0.2,0.4])
(a, [0.3,0.5],[0.2,0.5])
(e, [0.2,0.4],[0.2,0.4])
(d, [0.2,0.5],[0.3,0.4])
(b, [0.1,0.4],[0.2,0.4])
(v, [0.3,0.6],[0.2,0.4])
(c, [0.2,0.5],[0.2,0.4])
(x, [0.2,0.5],[0.1,0.4])
Fig 2.1 Intuitionistic Ifuzzy graph [F]
Here D(B) = ( [0.1,0.4],[0.3,0.5] ) and H(B) = ( [0.3,0.5], [0.2,0.4] ).
Remark 2.4. Clearly([]) []() ([]), since [D]([B]) [B](e) [H]([B]) and [D]([B]) [B](e) [H]([B]).
Theorem 2.5. Let [F] = ( [A], [B], f ) be any intuitionistic Ifuzzy graph with  = and  = . Then ([]) ([])
([]).
Proof. Suppose [F] = ( [A], [B], f ) is any intuitionistic Ifuzzy graph with pintuitionistic Ifuzzy vertices.
Obviously, [D]([B]) []() ([]) D([B]) [B](e) H ([B])
eE
eE
eE
([]) ([]) ([]) ([]) ([]) ([]).
Theorem 2.6. Let[F] = ([A], [B], f) be any intuitionistic Ifuzzy simple graph with pintuitionistic Ifuzzy
vertices.Then2([]) ([]).
(1)
Proof. By Theorem 2.5,([]) ([]) ([]) ([]) 2([]) ([]).
(1)
Theorem 2.7. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices.
Then ([]) 2([]) ([]).
(1)
Proof. By Theorem 2.5,([]) ([]) ([]) ([]) ([]) ([])
Since is intuitionistic Ifuzzy complete graph,
(1) ([]) ([]) (1) (]).Which implies that
2 2
([]) 2([]) ([]).
(1)
Theorem 2.8. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and [] be [s,t]constant function.
Then ([]) = 2([]) = ([]).
(1)
Proof. Assume that [] is a intuitionistic Ifuzzy complete graph with intuitionistic
[B] [] [B] [] [B] [] [B] []
Ifuzzy vertices and []() = [, ] for all in .That is Âµ (e) = (, ) and (e) = (, ) for all x, y
1(,) 1(,)
inV. Then implies that Âµ[](e) = []() []() and [](e) = []() []() = [s,t] for all and in , so ([]) = []() = ([])
D([B]) [B](e) H ([B]) ([]) = ([]) = ([])
eE
eE
eE
which implies (1) ([]) = ([]) = (1) ([]).
2 2
Hence([]) = 2([]) = ([]).
(1)
Corollary 2.9. Let [F] = ( [A], [B], f ) be an intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and [] be a [s, t]constant function. Then
d (v) p( p 1)H ([B]) p( p 1)D([B])
vV
Theorem 2.10.If [] is an intuitionistic Ifuzzy [m,n] regular graph with

intuitionistic Ifuzzy vertices. Then ([]) [,].
1
Proof. Suppose [] is an intuitionistic Ifuzzy [m,n] regular graph with intuitionistic Ifuzzy vertices. Here () = [m, n] for
p[m, n]
all v in , d (v) [m, n] p[m, n] . We get 2([]) = [, ] implies that ([]) = 2 . By 2.6 Theorem,
vV vV
[,] (1) ([]) [,] ([]) which implies that ([]) [,].2 2 1 1
Theorem 2.11. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and [] be [s, t] constant function.
Then ([]) = [s, t] = ([]).
Proof. Assume that [] is a intuitionistic Ifuzzy complete graph with intuitionistic Ifuzzy vertices and []() = [s, t] for
[B] [] [B] []all in . That is Âµ (e) = (, )
1(,)
1(,)
1(,)
and [B](e) = [] (, ) for all x, y inV. Then implies that Âµ[](e) = []() []() and [](e) = []() []()
= [s,t] for all and in . Therefore () = ( 1)[s, t] for all in . Which implies that
d (v) ( p 1)[s, t] p( p 1)[s, t]
. By Corollary 2.9,
d (v) p( p 1)H ([B]) = ( 1)([]).
vV
vV
vV
Hence([]) = [s, t] = ([]).
Theorem 2.12. Let [F] = ( [A], [B], f ) be any intuitionistic Ifuzzy simple graph with pintuitionistic Ifuzzy vertices . Then
([]) ( 1)([]).
Proof. For any intuitionistic Ifuzzy graph, ([]) 2([]). By Theorem 2.6, 2([]) ( 1)([]) which implies that
([]) ( 1)([]).
Theorem 2.13. Let[F] = ([A], [B], f ) be an intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and [] be [s, t]constant function.
Then([]) = ([]) = ( 1)([]) = ( 1)([]).
Proof. By Theorem 2.11, () = ( 1)[, ] for all in and
() = () = [, ] also ([]) = ([]) = ( 1)[, ]
implies that ([]) = [(]) = [, ] ([]) = ([]) = ([]) = ([])
implies that ([]) = ([]) = ( 1)([]) =
1
( 1)([]).
1
1
1
Theorem 2.14. If [F] = ( [A], [B], f ) is a intuitionistic Ifuzzy [s,t] totally regular graph with pintuitionistic Ifuzzy vertices.
Then dT(v) (p1) H([B]) + ([]).
Proof. For any intuitionistic Ifuzzy graph, 2S([F]) + O([F]) = p[s, t].
By Theorem 2.6,([]) (1) ( 1)([]) + O([F]) [, ]
2
( 1)([]) + O([F]) = dT(v). Hence dT(v) (p1) H([B]) + ([]).
Theorem 2.15. If [F] = ( [A], [B], f ) is both intuitionistic Ifuzzy [m,n]regular graph and intuitionistic Ifuzzy [s,t]totally
regular graph with pintuitionistic Ifuzzy vertices. Then dT(v) (p1) H([B]) + ([]).
Proof. By Theorem 2.10, ([]) [,]. By hypothesis,[m, n] + ([]) = [s, t]
1
(p1)H([B]) +([]) [s, t] dT(v) (p1) H([B]) + ([]).
Theorem 2.16. If[F] = ( [A], [B], f ) is a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and [] is a [s, t]constant function.
Then ([]) = ([]) = ([]).
Theorem 2.17. Let [F] = ( [A], [B], f ) be any intuitionistic Ifuzzy graph with respect to set and where  = and  =
d (v)
. Then ([]) vV
2
([]).
Theorem 2.18. Let [F] = ( [A], [B], f ) be any intuitionistic Ifuzzy simple graph with pintuitionistic Ifuzzy vertices. Then
d (v) ( 1)([]).
vV
Theorem 2.19. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices.
Then( 1)([]) d (v) ( 1)([]).
vV
Theorem 2.20. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and
[] be []constant function.Then ( 1)([]) = d (v) = ( 1)([]).
vV
Proof. By Theorem 2.8, ([]) = 2([]) = ([]). Since [F] is a intuitionistic Ifuzzy complete graph with intuitionistic I
(1)
fuzzy vertices and by Theorem 2.19, so ( 1) ([]) = d (v) = ( 1)([]).
vV
Theorem 2.21. Let [F] = ( [A], [B], f ) be a intuitionistic Ifuzzy complete graph with pintuitionistic Ifuzzy vertices and []
be [s,t]constant function. Then dT (v) = p2 H([B]) = p2 D([B]).
vV
Proof. By Theorem 2.20, d (v)
vV
( 1) ([]) = ( 1) ([]).
dT (v) = d (v) + [ A](v) , since [] is [s,t]constant function,
vV
vV
vV
=( 1) ([]) + ([]) = p2 H([B])
Similarly dT (v) = p2 D([B]).
vV
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