Operations on Intuitionistic Fuzzy Hypergraphs

Operations on Intuitionistic Fuzzy Hypergraphs

R.Parvathi

Department of Mathematics Vellalar College for Women Erode, Tamilnadu, India.

S.Thilagavathi

Department of Mathematics Vellalar College for Women Erode, Tamilnadu, India.

M.G.Karunambigai

Department of Mathematics Sri Vasavi College, Erode

Tamilnadu, India.

ABSTRACT

Hypergraph is a graph in which an edge can connect more than two vertices. Hypergraphs can be applied to analyze architecture structures and to represent system partitions. The concept of hypergraphs was extended to fuzzy hypergraph

.

In this paper, we extend the concepts of fuzzy hypergraphs into that of intuitionistic fuzzy hypergraphs. Based on the definition of intuitionistic fuzzy graph, operations like complement, join, union, intersection, ringsum, cartesian product, composition are defined for intuitionistic fuzzy graphs. The authors further proposed to apply these operations in clustering techniques.

Keywords

Intuitionistic fuzzy hypergraph (IFHG), Complement, Union, Join, Intersection, Ringsum, Cartesian product, Composition.

1.

INTRODUCTION

Hypergraph theory, originally developed by C.Berge in 1960, is a generalization of graph theory. The concept of hypergraphs can model more general types of relations than binary relations. The notion of hypergraphs has been extended in fuzzy theory and the concept of fuzzy hypergraphs was proposed by Lee-Kwang and S.M.Chen. The concept of an intuitionistic fuzzy graph (IFG) was introduced by Atanassov [1,2,3,4]. The authors have already introduced the concept of intuitionistic fuzzy hypergraph [7]. Operations on IFGs have also been analyzed by the authors [5,6]. Akram [8] applied the concepts in [7] to a real-life problem with a numerical example.

2.

PRELIMINARIES

Definition 2.1

An intuitionistic fuzzy graph (IFG) is of the V, E

G  where (i) V v , v , ...,v



_{1 2 n}



such that

1 0 1

μ :V_ , _ and γ :V1 0 1,  denote the degree of membership and non-membership of the element v_i  V respectively and

0 μ (v ) γ (v )  1 i  1 i 1 (1)

for every v_i  V , ( i = 1, 2, …, n ), (ii)E  V  V where μ :V V2    0 1, and γ :V V2    0 1, are such that

μ (v , v ) μ (v ). μ (v )_{2 i j}  _{1 i 1 j} (2)

γ (v , v ) γ (v ). γ (v )_{2 i j}  _{1 i 1 j} (3)

and 0μ (v , v ) γ (v , v ) 2 i j  2 i j 1 (4)

for every (v , v ) E_{i j}  .

Definition 2.2

An IFHG H is an ordered pair H  V, E where (i) V 



v₁, v₂, ...,v_n



, a finite set of vertices (ii) E



E₁, E₂, ...,E_n



, a family of intuitionistic fuzzy

subsets of V. .(iii)Ej





v ,μ v ,γ vi j

   

i j i



: μ v ,γ vj

   

i j i 0 and 0j( )vi j( ) 1v i  , j = 1, 2 ,…, m (iv) Ejφ

j = 1,2,…,m (v) Ujsupp

 

Ej V, j=1, 2 ,…, m

Here the edges Ej are IFSs. j( ) andx i j( )xi denote the

degree of membership and non-membership of the vertex vi to edgeE . Thus, the elements of the incidence matrix of IFHG _j are of the form a ,μ x ,γ x



ij j

   

i j i



. The sets V and E are crisp sets.

Notations

The triple v , μ , γi 1i 1i denotes the degree of membership and non-membership of the vertexv_i. The triple e , μ , γij 2ij 2ij

denotes the degree of membership and non-membership of the edge eij

 

v ,vi j on V. That is,

1i 1 i 1i 1 i and

μ μ (v ), γ γ (v ) μ2ijμ (v ,v ), γ2 i j 2ijγ (v ,v )2 i j

.

Note 2.3

Definition 2.4

An IFHG , G V, E is said to be a semi - μ strong IFHG if μ2ijμ .μ1i 1jfor every i and j.

Definition 2.5

An IFHG , G V, E is said to be a semi - γ strong IFHG if γ2ijγ .γ1i 1j for every i and j.

Definition 2.6

An IFHGG V, E is said to be a strong IFHG if

2ij 1i 1j

μ μ .μ and γ_2ijγ .γ_{1i 1j}for all ( , )v vi j E.

Definition 2.7

An IFHG , G V, E is said to be a complete -μ strong IFHG if μ2ijμ .μ1i 1j and γ2ijγ .γ1i 1j

Definition 2.8

An IFHG , G  V, E is said to be a complete -γ strong IFHG if μ2ijμ .μ1i 1j and γ2ijγ .γ1i 1j for all i and j.

Definition 2.9

An IFHG , G V, E is said to be a complete IFHG if

2ij 1i 1j

μ μ .μ and γ2ijγ .γ1i 1j, for every (v ,v ) Vi j  .

3.

COMPLEMENT OF AN IFHG

Definition 3.1

The complement of an IFHG G  V, E is an IFHG, E

, V

G  , where



V  V

 μ μ_1i _1i and γ γ_1i _1i, for all i = 1, 2, …, n.

 μ (μ . μ ) μ_2ij _{1i 1j}  _2ijandγ (γ . γ ) γ_2ij _{1i 1j}  _2ij for all i, j = 1, 2, …, n.

Example 3.2

[image:2.595.54.542.68.601.2]

Let V



v ,v ,v ,v1 2 3 4



and E



e ,e1 2



Fig 1: An Intuitionistic fuzzy hypergraph and its complement

Note 3.3

One can easily verify thatG  G

.

Proposition 3.4

(i) The complement of a completeμ - strong IFHG is a complete IFHG.

(ii) The complement of a complete γ- strong IFHG is a complete IFHG.

Proof

(i) LetG  V, E be a complete μ - strong IFHG. Therefore,

2ij 1i 1j

μ μ .μ and γ2ijγ .γ1i 1j for all i and j.

To prove that either (a) μ₂ij0 orγ2ij0, (b) μ2ij0 or

2ij 0

γ  .

That is, μ2ij



μ .μ1i 1j



μ2ij

= 0 if μ2ij0

= μ_1i if μ₂ij0 and



1 1



2

2ij i j ij 0

γ  γ .γ γ  for every i and j, since G is a complete μ - strong IFHG. (ii) Proof of (ii) is the same as proof of (i).

Definition 3.5

Consider the two IFHGs G1 V , E1 1 andG2 V , E2 2 . An isomorphism between two IFHGs G1andG2 , denoted by G₁G₂, is a bijective map

1 2

h :V V which satisfies

1 i 1 i ;

μ (v ) μ'(h(v )) γ (v ) γ'(h(v ))_{1 i}  _{1 i} and

v

4

v

1

v

2

v

3

(0.1,0.2)

(0.3,0.5)

(0.4,0.6)

(0.2,0.2)

(0.2,0.2)

v

4

v

1

v

2

v

3

(0.1,0.2)

2 i j 2 i j ;

μ (v ,v ) μ '(h(v ),h(v )) γ (v ,v ) γ '(h(v ),h(v )2 i j  2 i j for

every( v_ι,v_j)V .

Definition 3.6

An IFHG G is self-complementary if G is isomorphic toG. Symbolically, G1G2.

Theorem 3.7

Let G V, E be a self-complementary IFHG. Then, there exists an isomorphism h:V V such that

1 i 1i

μ (v ) μ ;γ (v ) γ_{1 i}  _1i, for every v_i V and

2 i j 2i

μ (h(v ),h(v )) μ ; γ (h(v ),h(v )) γ_{2 i j}  _2i for every V

) j ,v ι

( v  .

Proof

By Definition 3.1, we have

2 i j 1 i 1 j 2ij

μ (h(v ),h(v )) ( μ (h(v )).μ (h(v ))) μ  2 i j 1i 1j 2ij

μ (v ,v ) μ .μ μ

   μ2ijμ .μ1i 1jμ2ij

2ij 1i 1j 2ij

μ (μ .μ ) μ

i j i j i j

     

   2 μ2ij (μ .μ )1i 1j

i j i j

   

 

2 1 1

1 2

ij i j

μ (μ .μ )

i j i j

   

  . and

2 i j 1 i 1 j 2ij

γ (h(v ),h(v )) ( γ (h(v )).γ (h(v ))) γ  2ij 1i 1j 2ij

γ (γ .γ ) γ

   γ2ij (γ .γ )1i 1j γ2ij

i j i j i j

     

  

2 1 1

2 γ_ij (γ .γ )_{i j}

i j i j

   

  2 1 1

1 2

ij i j

γ (γ .γ )

i j i j

   

  .

Remark 3.8

The condition in the above theorem is not sufficient.

Theorem 3.9

If G is a strong IFHG, then Gis also strong.

Proof

Let uvE. Then

2 1 1 2

μ (uv) μ (u). μ (v) μ (uv) 

=μ (u). μ (v) μ (u). μ (v)_{1 1}  _{1 1} , since G is strong

= 0 and γ (uv) γ (u).γ (v) γ (uv)2  1 1  2 =γ (u).γ (v) γ (u). γ (v)1 1  1 1 , since G is strong = 0.

Let uvE. Then μ (uv) (μ (u). μ (v)) μ (uv)2  1 1  2 =

1 1

μ (u).μ (v) and γ (uv) γ (u). γ (v) γ (uv)₂  _{1 1}  ₂ = γ (u). γ (v) _{1 1} .

4.

OPERATIONS ON INTUITIONISTIC

FUZZY HYPERGRAPHS

Definition 4.1

Let G₁ V , E_{1 1} and G₂ V , E_{2 2} be two IFHGs with

1 2

VIV φ. Then the union of G1andG2 is an IFHG

1 2

G G UG  V1UV , E2 1UE2 defined by

1 1 1

1 1

1 2 1

μ (v) if v V V

(μ μ')(v)

μ'(v) if v V V   

  _{ }

 U

1 1 2

1 1

1 2 1

γ (v) if v V V

(γ γ')(v)

γ'(v) if v V V   

  _{ }

 U

and 2 2 2 1 2

2 2 1

ij ij i j

ij ij

μ if e E E

(μ μ ')(v v )

μ ' if e E E   

  _{ }

 U

2 1 2

2 2

2 2 1

ij ij i j

ij ij

γ if e E E

(γ γ ')(v v )

γ ' if e E E   

  _{ }

 U

where (μ ,γ )_{1 1} and (μ ',γ ')_{1 1} refer the vertex membership and non-membership of G₁andG₂ respectively; (μ ,γ )_{2 2} and

2 2

(μ ',γ ')refer the edge membership and non-membership of 1and 2

G G respectively.

Definition 4.2

The join of two IFHGs G1andG2is an IFHG GG1G2 =

1 2 1 2

VUV , E UE UE' defined by

1 1 1 1 if 1 2

(μ μ')(v) (μ Uμ')(v) v V UV

1 1 1 1 if 1 2

(γ γ')(v) (γ Uγ')(v) v V UV

2 2 i j 2 2 i j

if

i j 1 2

(μ



μ ')(vv ) (μ μ ')(v v ) vv E E



U



U

1 i 1 j

if

i j

(μ (v ).μ'(v )



v v E'



and

2 2 2 2 1 2

1 1 if if

i j i j i j

i j i j

(γ γ ')(v v ) (γ γ ')(v v ) v v E E (γ (v ).γ '(v ) v v E'.

  

 

U U

Definition 4.3

Let G1 V , E1 1 and G2 V , E2 2 be two IFHGs. Then the intersection of IFHGs G1andG2 denoted by G1I G2is an

IFHG defined by 1 1 1 1 2

1 2 1

if and

if and

(v) v V V

( ')(v)

'(v) v V V

        _  I

1 1 2

1 1

1 2 1

if and

if and

γ (v) v V V (γ γ')(v)

γ'(v) v V V  

  _

and

2 1 2

2 2

2 1 2

if and

if and

ij ij i j

ij ij

μ e E E

(μ μ ')(v v )

μ ' e E E 



  _

 I

2 1 2

2 2

2 1 2

if and

if and

ij ij i j

ij ij

e E E

( ')(v v )

' e E E

  



 

  _

 I

Definition 4.4

Let G1 V , E1 1 and G2 V , E2 2 be two IFHGs. Then the ringsum of IFHGs G1andG2 is an IFHG defined by



 



1 2 1 2 1 2

G G  GUG  GI G .

Example 4.5

[image:4.595.149.471.238.738.2]

Let V₁



v ,v ,v ,v_{1 2 3 4}



and V₂



u ,u_{1 2}



such that V₁IV₂φ

Fig 2: G1andG2

Fig 3:

G

₁

U

G

₂

Fig 4: 

v

4

v

1

v

2

v

3

(0.1,0.2)

(0.3,0.5)

(0.4,0.6)

(0.2,0.2)

(0.7,0.1)

u

2

u

1

(0.3,0.2)

v

1

(0.1,0.2)

v

4

v

2

v

3

(0.3,0.5)

(0.4,0.6)

(0.2,0.2)

(0.7,0.1)

u

2

u

1

(0.3,0.2)

v

1

(0.1,0.2)

v

4

v

2

v

3

(0.3,0.5)

(0.4,0.6)

(0.2,0.2)

(0.7,0.1)

u

2

Example 4.6

For the operation intersection and ringsum we consider





1 1 2 3

V v ,v ,v and V₂



v ,v ,v_{1 2 3}



and V₁I V₂φ

Theorem 4.7

Let G1 V , E1 1 and G2 V , E2 2 be two IFHGs. Then (i)

1 2 1 2

G G GUG (ii)G1UG2G1G2

Fig 5: a) G1 b) G2 c) G1IG2 d) G1G2

Proof

Consider the identity map I :V₁UV₂V₁UV₂.To prove (i), it is enough to prove

a) (i) ( μ1μ' )(v ) ( μ1 i  1Uμ' )(v )1 i

(ii) ( γ₁γ' )(v ) ( γ_{1 i}  ₁Uγ' )(v )_{1 i}

(b) (i) ( μ2μ ' )(v ,v ) ( μ2 i j  2Uμ ' )(v ,v )2 i j

(ii) ( γ₂γ ' )(v ,v ) ( γ_{2 i j}  ₂Uγ ' )(v ,v )_{2 i j} .

(a) (i) ( μ₁μ' )(v ) (μ_{1 i}  ₁μ')(v )_{1 i} , by Definition 4.1

= 1 1

1 2

if if

i i i i

μ (v ) v V μ'(v ) v V  

 _



= 1 1

1 2

if if

i i

i i

μ (v ) v V μ'(v ) v V

 _

 



 = ( μ1Uμ' )(v )1 i .

(ii) ( γ1γ' )(v ) (γ1 i  1γ')(v )1 i , by Definition 4.1

= 1 1

1 2

if if

i i i i

γ (v ) v V γ'(v ) v V

   _  = 1 1 1 2 if if i i i i

γ (v ) v V γ'(v ) v V

 _     = ) i )( v ' γ γ

( ₁ ₁ .

(b) (i) 2 2 1 1 1 1

2 2

( )

i j i j

i j

( ' )(v ,v ) ' (v ).( ')(v )

( ')(v ,v )

     

 

    



1 1 1 1 2 2

1 2

1 1 1 1 1 1

if

i j i j

i j

i j i j

(μ μ')(v ).(μ μ')(v ) (μ μ ')(v ,v ) (v ,v ) E E (μ μ')(v ).(μ μ')(v )) μ (v ).μ'(v )



 



U U U

U

U U

if i j

(v ,v ) E'.

      _ 

1 1 2 1

1 1 2 2

1 1 1 1

if if

if

i j i j i j i j i j i j i j i j i j

μ (v ).μ (v ) μ (v ,v ) (v ,v ) E μ'(v ).μ'(v )) μ '(v ,v ) (v ,v ) E μ (v ).μ'(v ) μ (v ).μ'(v ) (v ,v ) E'

        _{ }  2 1 2 2 if if 0 if

i j i j

i j i j i j

μ (v ,v ) (v ,v ) E μ '(v ,v ) (v ,v ) E

(v ,v ) E'

 _



 

 _



= ( μ2Uμ ' )(v ,v )2 i j .

(b) (ii) 2 2 1 1 1 1

2 2

( )

i j i j

i j

( γ γ ' )(v ,v ) γ γ' (v ).(γ γ')(v ) (γ γ ')(v ,v )

    



1 1 1 1 2 2

1 2

1 1 1 1 1 1

if

i j i j

i j

i j i j

(γ γ')(v ).(γ γ')(v ) (γ γ ')(v ,v )

(v ,v ) E E (γ γ')(v ).(γ γ')(v ) γ (v ).γ'(v )



 



U U U

U

U U

if i j

(v ,v ) E'.

      _ 

1 1 2 1

1 1 2 2

1 1 1 1

if if

if i j i j i j i j i j i j i j i j i j

γ (v ).γ (v ) γ (v ,v ) (v ,v ) E γ'(v ).γ'(v ) γ '(v ,v ) (v ,v ) E γ (v ).γ'(v ) γ (v ).γ'(v ) (v ,v ) E'

    _    _{ }  2 1 2 2 if if 0 if

i j i j

i j i j i j

γ (v ,v ) (v ,v ) E γ '(v ,v ) (v ,v ) E (v ,v ) E'

 



 

 _



= ( γ2Uγ ' )(v ,v )2 i j .

v

2(0.3,0.2)

v

1(0.6,0.2)

a)

d)

v

3

b)

v

1(0.6,0.2)

(0.12,0.1)

v

2(0.3,0.2)

(0.08,0.01)

v

2(0.3,0.2)

v

1(0.6,0.2)

v

3

c)

v

3

v

2(0.3,0.2)

v

1(0.1,0.03)

(0.4,0.5)

(0.4,0.5)

(0.4,0.5)

[image:5.595.67.565.171.311.2]

To prove (ii), it is enough to prove that

(a) (i) ( μ₁Uμ' )(v ) ( μ_{1 i}  ₁μ' )(v )_{1 i}

(ii) ( γ1Uγ' )(v ) ( γ1 i  1γ' )(v )1 i

(b) (i) ( μ₂Uμ ' )(v ,v ) ( μ_{2 i j}  ₂μ ' )(v ,v )_{2 i j}

(ii) ( γ2Uγ ' )(v ,v ) ( γ2 i j  2γ ' )(v ,v )2 i j .

Consider the identity map : I :V₁UV₂V₁UV₂.

(a) (i) ( μ1Uμ' )(v )1 i = (μ1Uμ')(v )1 i

1 1 1 2 if if i i i i

μ (v ) v V μ'(v ) v V

    _  1 1 1 2 if if i i i i

μ (v ) v V μ'(v ) v V

 _

  





1 1 i

( μ μ' )(v )

 U ( μ1μ' )(v )1 i .

(a)(ii) ( γ1Uγ' )(v )1 i = (γ1Uγ')(v )1 i

1 1 1 2 if if i i i i

γ (v ) v V γ'(v ) v V     _  1 1 1 2 if if i i i i

γ (v ) v V γ'(v ) v V

 

  





1 1 i

( γ γ' )(v )

 U ( γ₁γ' )(v )_{1 i} .

(b)(i)

2 2 1 1 1 1

2 2

i j i j

i j

( μ μ ' )(v ,v ) (μ μ')(v ).(μ μ')(v ) (μ μ ')(v ,v )

 

U U U

U

1 1 2 1

1 1 2 2

1 1 1 2

if if 0 if

i j i j i j i j i j i j

i j i j

μ (v ).μ (v ) - μ (v ,v ) (v ,v ) E μ '(v ).μ '(v ) - μ '(v ,v ) (v ,v ) E μ (v ).μ '(v ) - v V , v V

      _{ }  2 1 2 2

1 1 1 2

if if

if

i j i j

i j i j

i j i j

μ (v ,v ) (v ,v ) E μ '(v ,v ) (v ,v ) E

μ (v ).μ'(v ) v V , v V

 



 

 _{ }



2 2 1 2

1 1

if or

if

i j i j

i j i j

( μ μ ' )(v ,v ) (v ,v ) E E μ (v ).μ'(v ) (v ,v ) E'

       U

2 2 i j

( μ μ ' )(v ,v )

 

(b)(ii) 2 2 1 1 1 1

2 2

i j i j

i j

( γ γ ' )(v ,v ) (γ γ')(v ).(γ γ')(v ) (γ γ ')(v ,v )

 

U U U

U

1 1 2 1

1 1 2 2

1 1 1 2

if

if

0

if

i j i j i j

i j i j i j

i j i j

γ (v ).γ (v ) γ (v ,v ) (v ,v ) E

γ'(v ).γ'(v ) γ '(v ,v ) (v ,v ) E

γ (v ).γ'(v )

v V , v

V



















_

_

_



2 1 2 2

1 1 1 2

if

if if i j i j

i j i j i j i j

γ (v ,v ) (v ,v ) E γ '(v ,v ) (v ,v ) E

γ (v ).γ'(v ) v V , v V

      _{ } 

2 2 1 2

1 1

if or

if

i j i j i j i j

( γ γ ' )(v ,v ) (v ,v ) E E γ (v ).γ'(v ) (v ,v ) E'

 _      U

2 2 i j

( γ γ ' )(v ,v )

  .

Definition 4.8

Let G G ₁G V , E'' ₂ be the Cartesian product of two graphs G₁andG₂ where V V₁ V₂ and









2 2 1 2 2 2

1 1 2 1 1 1

E (u, u )(u, v ) : u V ,u v E

(u , w)(v , w) : w V ,u v E .

   

 

U

Then, the Cartesian product of IFHGs G1andG2 is an IFHG defined by G G 1G V , E'' 2 where

(i)

1 1 1 2 1 1 1 2 1 2

(

μ



μ ' u ,u

)(

)



μ u .μ ' u

( )

(

) for every (

u ,u

)



V

and

1 1 1 2 1 1 1 2

for every

1 2

(γ



γ ')(u ,u ) γ (u ).γ '(u )



(u ,u ) V



(ii)

2 2 2 2 1 2 2 2

1 2 2 2

2 2 2 2 1 2 2 2

1 2 2 2

for every

and

for every

and

and

(μ

μ ')(u,u )(u,v ) μ (u).μ (u v )

u V ,

u v

E

(γ

γ ')(u,u )(u,v ) γ (u).γ (u v )

u V ,

u v

E

















2 2 1 1 1 2 1 1

2 1 1 1

2 2 1 1 1 2 1 1

2 1 1 1

for every

and

for every

and

(μ

μ ')(u ,w)(v ,w) μ (w).μ (u v )

w V ,

u v

E

(γ

γ ')(u ,w)(v ,w) γ (w).γ (u v )

w V ,

u v

E .

















Example 4.9

Fig 6:G1G 2

Definition 4.10

Let GG G1o 2=



V1V ,E2



be the composition of two

graphs G1andG2, where













2 2 1 2 2 2

1 1 2 1 1 1

1 2 1 2 1 1 1 2 2 .

E (u, u )(u, v ) : u V ,u v E (u , w)(v , w) : w V ,u v E (u , u )(v , v ) : u v E ,u v

  

 

 

U U

Then the composition of IFHGs G1andG2, denoted by

1 2

GG Go , is an IFHG defined by

(i)(μ₁oμ')(u ,u ) μ (u ).μ'(u_{1 1 2}  _{1 1 1 2}) for every (u ,u ) V V_{1 2}  _{1 2} and

1 1 1 2 1 1 1 2 for every 1 2 1 2

(γ oγ')(u ,u ) γ (u ).γ'(u ) (u ,u ) V V 

(ii) 2 2 2 2 1 2 2 2 1

2 2 2

for every and

(μ μ ')(u,u )(u,v ) μ (u).μ (u v ) u V ,

u v E

 

 o

2 2 2 2 1 2 2 2 1

2 2 2

for every V ,

and

(γ γ ')(u,u )(u,v ) γ (u).γ (u v ) u u v E

 

 o

and

2 2 1 1 1 2 1 1

2 1 1 1

for every

and

(

')(u ,w)(v ,w)

(w).

(u v )

w V ,

u v

E

 











o

2 2 1 1 1 2 1 1

2 1 1 1

for every

and

(γ

γ ')(u ,w)(v ,w) γ (w).γ (u v )

w V ,

u v

E







o

2 2 1 2 1 2 1 2 1 2 2 1 1 1 2 1 2

for every

and

(μ

μ ')(u ,u )(v ,v ) μ'(u ).μ'(v ).μ (u ,v )

(u ,u )(v ,v )

E

E''



 

o

2 2 1 2 1 2 1 2 1 2 2 1 1

1 2 1 2

for every

,

(γ

γ ')(u ,u )(v ,v )

γ '(u ).γ '(v ).γ (u ,v )

(u ,u ) (v ,v )

E

E''



 

o

where









2 2 1 2 2 2

1 1 2 1 1 1

for every

for every

.

E'' (u, u )(u, v ) : u V , u v E (u , w)(v , w) : w V , u v E

  

 

U

Theorem 4.9

Let G₁ V , E_{1 1} and G₂  V , E_{2 2} be two strong IFHGs. Then, G1oG2 is a strong IFHG.

Proof

Let G1oG2= G V,E where V V1 V2 and













2 2 1 2 2 2

1 1 2 1 1 1

1 2 1 2 1 1 1 2 2

E (u, u )(u, v ) : u V ,u v E (u , w)(v , w) : w V ,u v E (u , u )(v , v ) : u v E ,u v .

  

 

 

U

U

(i)μ (u,u )(u,v ) μ (u).μ (u v ) 2 2 2  1 2 2 2

1 1 2 1 2

μ (u).(μ'(u ).μ'(v )),



since G₂ is strong

u

1

v

2

(0.09,0.01)

u

1

v

3

(0.12,0.12)

u

1

v

4

(0.06,0.04)

u

2

v

1

u

2

v

2

(0.21,0.05)

u

2

v

3

(0.28,0.06)

u

2

v

4

(0.14,0.02)

(0.021,0.002)

(0.063,0.005)

(0.084,0.006)

(0.042,0.002)

(0.03,0.04)

_(0.07,0.02)

1 1 2 1 1 2

μ (u).μ'(u ).μ (u).μ'(v )



1 1 2 1 1 2

(μ μ')(u,u ).(μ μ')(u,v )

 o o .

2 2 2 1 2 2 2

γ (u,u )(u,v ) γ (u).γ (u v ) 

1 1 2 1 2

γ (u).(γ'(u ).γ'(v )),

 since G2 is strong

1 1 2 1 1 2

γ (u).γ'(u ).γ (u).γ'(v )



1 1 2 1 1 2

(γ γ')(u,u ).(γ γ')(u,v ).

 o o

(ii)μ ((u ,w)(v ,w)) μ'(w).μ (u ,v ) _{2 1 1}  _{1 2 1 1}

1 1 1 1 1

μ'(w).(μ (u ).μ (v )),

 since G₁ is strong

1 1 1 1 1 1

μ'(w).μ (v ).μ'(w). μ (v )



1 1 1 1 1 1

(μ μ')(u ,w).(μ μ')(v ,w).

 o o

1 1 1 1 2 1 1

γ ((u ,w)(v ,w)) γ'(w).γ (u ,v ) 

1 1 1 1 1

γ'(w).(γ (u ).γ (v )),



since G1 is strong

1 1 1 1 1 1

γ'(w). γ (v ).γ'(w). γ (v )





(γ

₁

o

γ')(u ,w).(γ γ')(v ,w)

_{1 1 1}

o

_{1 1}

(iii)μ (u ,u )(v ,v ) μ (u ,v ).μ'(u ).μ'(v ) _{2 1 2 1 2}  _{2 1 1 1 2 1 2}

1 1 1 1 1 2 1 2

(μ (u ).μ (v )).μ'(u ).μ'(v ),

 G₁ is strong

1 1 1 2 1 1 1 2

μ (u ).μ'(u ).μ (v ).μ'(v )



1 1 1 2 1 1 1 2

(μ μ')(u ,u ).(μ μ')(v ,v ).

 o o

2 1 2 1 2 2 1 1 1 2 1 2

γ (u ,u )(v ,v ) γ (u ,v ).γ'(u ).γ'(v ) 

1 1 1 1 1 2 1 2

(γ (u ).γ (v )).γ'(u ).γ'(v ),

 G1 is strong

2 1 1 1 2 1 1

1( u ).γ'( u )).γ ( v ).γ'( v )

(γ



1 1 1 2 1 1 1 2 .

(γ γ')(u ,u ).(γ γ')(v ,v )

 o o

From (i), (ii), (iii), G is a strong IFHG.

Example 4.11

Let Let V1



v ,v ,v ,v1 2 3 4



and V2



u ,u1 2



such that

1 2

VIV φ

[image:8.595.158.492.420.641.2]

.

Fig 7: G₁oG₂

5.

CONCLUSION

In this paper, operations like complement, join, union, intersection, ringsum, Cartesian product, composition on IFHGs are defined. Currently, the authors are working on

existing clustering techniques. Further, it is proposed to apply the properties of IFHGs to develop a new clustering algorithm and the same may be checked with a numerical dataset .

u

1

v

1

u

1

v

2

u

1

v

3

u

1

v

4

u

2

v

1

u

2

v

2

u

2

v

3

u

2

v

4

(0.021, 0.002)

(0.063, 0.005)

(0.084, 0.006)

6.

REFERENCES

[1] K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20, (1986), 87-96.

[2] K.T.Atanassov, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems 33, (1989), 37-46.

[3] K.T.Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems 61, (1994), 137-142.

[4] K.T.Atanassov, Remarks on the intuitionistic fuzzy sets-III, Fuzzy Sets and Systems 75, (1995), 401-402. [5] S.Thilagavathi, R.Parvathi and M.G.Karunambigai,

Operations on Intuitionistic fuzzy graphs II,

Developments in fuzzy sets, intuitionistic fuzzy sets, generalized nets and related topics 1, (2008), 319-331. [6] R.Parvathi, M.G.Karunambigai and K.T.Atanassov

Operations on Intuitionistic fuzzy graphs, Fuzzy Systems, 2009, Fuzz-IEEE 2009. IEEE international conference 1396-1401.

[7] S.Thilagavathi, R.Parvathi and M.G.Karunambigai, Intuitionistic fuzzy hypergraphs, J.Cybernetics and Information Technologies 9(2), (2009), 46-53, Sofia. [8] M.Akram and W.A.Dudek Intuitionistic fuzzy