INTUITIONISTIC FUZZY LINE GRAPH OF AN
INTUITIONISTIC FUZZY HYPERGRAPH
S. Vimala
I. Pradeepa
Department of Mathematics
Arul Anandar College
Mother Teresa Women’s University
Karumathur, Madurai - 625 016
Kodaikanal - 624101
E-mail: pradeepanatarajan@gmail.com
E-mail: tvimss@gmail.com
Abstract
In this paper, the concept of Intuitionistic fuzzy line graph of an Intuitionistic
fuzzy hypergraph is introduced. Further some of the properties of Intuitionistic fuzzy
line graph of the Intuitionistic fuzzy hypergraph are also examined.
Keywords
Intuitionistic fuzzy set, line graph, line graph of hyper graph, fuzzy hyper
graph
1.Introduction
In 1983, Krassimir T. Atanassov introduced the concept of Intuitionistic Fuzzy
sets. An Intuitionistic fuzzy set is characterized by two functions expressing the
degree of membership and the degree of non-membership of elements of the universe
to the IFS. Among the various notions of higher-order fuzzy sets, IFS proposed by
Atanossov provide a flexible framework to explain uncertainty and vagueness. This
domain has recently motivated new research in several directions.
The theory of fuzzy graphs was introduced by Rosenfeld in 1975. The fuzzy
relations between fuzzy sets were also considered by Rosenfeld and he developed the
structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts.
Bhattacharya gave some remarks on fuzzy graphs, and some operations on fuzzy
graphs were introduced by Moderson and Peng.Lee-Kwang and Lee generalized and
redefined the concepts of fuzzy hypergraphs whose basic idea was given by
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Kaufmann. Recently Parvathi et al., defined intuitionistic fuzzy hypergraphs. The line
graph of the hypergraph is generalization of the line graph of simple graphs. The
name line graph comes from a paper by Harary and Normon (1960) although both
Whitney (1932) and Kraunz (1943) used the construction before this. In this paper, we
introduce the concept of Intuitionistic fuzzy line graph of an Intuitionistic fuzzy
hypergraph and examined some of their properties.
2. Basic definitions
Definition 2.1:
A fuzzy graph G   ,  ) is a pair of function  : X  [0,1] and
 : X  X  [0,1] where  ( x, y )  min(  ( x ),  ( y )) for all x, y  X . The underlying
crisp graph of G  ( ,  ) is denoted by G*  ( X , E ) where X  {x  X ;  ( x)  0}
and E  {( x, y )  X  X :  ( x, y )  0}.
Definition 2.2:
An Intuitionistic fuzzy graph with underlying set V is defined to be a pair
G  ( A, B) where (i) the functions  A : V  [0,1] and  A V  [0,1] denote the
degree of membership and non-membership of the element x V respectively such
that 0   A ( x)   A ( x)  1
 x  V . (ii) the function B  E  V V  [0,1] and
VB : E  V V  [0,1] are defined by
B ({x, y})  min ( A ( x),  A ( y)) and
 B ({x, y})  max(  A ( x),  A ( y)) such that
0  B ({x, y})   B ({x, y})  1 {x, y}  E
Definition 2.3:
A hyper graph H is an ordered pair H  ( X , E ) where
i)
X  {x1 , x2 ,..., xn } a finite set of vertices.
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ii)
E  {E1 , E2 ,...Em } a family of subsets of V .
iii)
E j   , j  1, 2, ...m and
iv)
 E j  X . The set X is called the set of vertices and E is the set of edges (or
j
hyper edges).
Definition 2.4:
Let X be a finite set and let  be a finite family of non-trivial fuzzy subsets
of X such that X   Supp( ) . The pair H  ( X ,  ) is called a fuzzy hypergraph
E
(on X ) and  is called the edge set of H which in sometimes denoted  (H ) . The
members of a  are called the fuzzy edges of H.
Definition 2.5:
The IFHG H is an ordered pair H  { X , E ) where
i)
X  {x1 , x2 ...xn } is a finite set of vertices.
ii)
E  {E1 , E2 ,...., Em } is a family on intuitionistic fuzzy subsets of X .
iii)
E j  {( xi ,  j ( xi ),  j ( xi ));  j ( xi ), V j ( xi )  0 &  j ( xi )   j ( xi )  1}
j  1, 2 ... m
iv)
E j   , j  1, 2, ..., m
v)
 Supp( E j )  X , j  1, 2, ... m
j
Here the edges E j are an IFSs of vertices  j ( xi ) and  j ( xi ) denote the degree of
membership and non-membership of vertex xi to edge E j .
Definition 2.6:
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The line graph L(G*) is by definition the intersection graph P(E ). (i.e.)
L(G*)  ( Z , W )
Z  {{x}  {ux , vx } | x  E , ux , vx V , x  ux vx }
where
and
W  {Sx Sy | Sx  Sy   , x, y  E , x  y} and Sx  {x}  {ux , vx }, x  E.
Definition 2.7:
For a hypergraph H * , the line graph of the hypergraph L (H *) is defined as
follows. (i) The vertex set of H *, VL ( H *)  EH (hyperedges of H). In accordance
with the definition of a hypergraph, VL (H *) is a set and E H is a family. In this
situation
the
above
equality
means
that
if
E H  {Ei / 1  i  m}
then
VL ( H )  {E1 , E2 ...Em } is an m-element set. In otherwords, the multiple edges of H
give rise to different vertices of L(H).
(ii)
Vertices E i and E j are adjacent in L(H ) if and only if E i  E j   .
Definition 2.8:
Let L(G*)  ( Z ,W ) be line graph of a simple graph G*  (V , E ) . Let
A1  ( A1 ,  A1 ) and B1  (  B1 ,  B1 ) be intuitionistic fuzzy subsets of V and E
respectively. Let A2  (  A2 ,  A2 ) and B2  (  B2 ,  B2 ) be intuitionistic fuzzy sets of
Z & W respectively. We define an intuitionistic fuzzy line graph L(G)  ( A2 , B2 ) of
the intuitionistic fuzzy graph G  ( A1 , B1 ) as follows
(i)
 A ( S x )   B ( x )   B ( ux v x )
(ii)
 A ( Sx )   B ( x )   B (ux vx )
(iii)
 B ( S x S y )  min(  B ( x),  B ( y ))
(iv)
 B ( S x S y )  max(  B ( x),  B ( y )) for all S x , S y  Z , S x S y  W .
2
2
2
2
1
1
1
1
1
1
1
1
3.Intuitionistic fuzzy line graph of intuitionistic fuzzy hypergraph
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Definition 3.1:
Let L( H *)  ( Z , W ) be a line graph of a simple hypergraph H *  ( X , E ) . Let
H  ( X ,  ) be IFHG of H * . We define IFLG L( H )  ( A1 , B1 ) where A1 is the
vertex set of L(H ) and B1 is the edge set of L(H ) as follows:
(i)
A1 and B1 are IFS of Z and W respectively.
(ii)
 A ( Ei )  min (E ( x))
(iii)
 A ( Ei )  max ( E ( x)), Ei  
(iv)
 B ( E j Ek )  min (min(  E ( xi ),  E ( xi ))
(v)
 B ( E j E k )  max (max(  E ( xi ),  E ( xi )) where xi  E j  E k , j, k = 1, 2, … n
1
1
1
xEi
i
xEi
i
j
i
k
j
i
k
Example 3.2:
Consider an IFHG H  ( X ,  ) such that
X  {x1 , x2 , x3 , x4 },   {E1, E2 , E3 , E4 } Here E1  {( x1 ,0.2,0.5), ( x2 ,0.3,0.6)} ,
E2  {( x2 ,0.3,0.6) ( x3 ,0.4,0.5)} , E3  {( x3 ,0.4,0.5) ( x4 ,0.6,0.2)} and
E4  {( x4 ,0.6,0.2), ( x1 ,0.2,0.5)}
Intuitionistic fuzzy hepergraph H
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The line graph of H can be obtained as L(H) = (A1, B1).
Where A1  {( E1 ,0.2,0.6) ( E2 ,0.3,0.6)( E3 ,0.4,0.5)( E4 ,0.2,0.5)} is the vertex
set and B1  {( E1 E2 ,0.3,0.6) ( E2 E3 ,0.4,0.5) ( E3 E4 ,0.6,0.2) ( E4 E1 ,0.2,0.5)} is the edge
set of the Intuitionistic fuzzy line graph.
Intuitionistic fuzzy line graph L(H)
Example 3.3:
Consider an IFHG H  ( X ,  ) such that
X  {a, b, c, d } ,   {E1, E2 ,
E3 , E4 , E5 } when E1  {( a,0.7,0.2) (b,0.7,0.2)}, E2  {( a,0.9,0) (b,0.9,0) (d ,0.4,0.3)} ,
E4  (b,0.7,0.2) (c,0.7,0.2) (d ,0.4,0.3)},
E3  {( b,0.9,0) (c,0.9,0)} ,
E5  {( a,0.4,0.3) (c,0.4,0.3) (d ,0.4,0.3)} .
The line graph of H can be obtained as
L( H )  ( A1 , B1 )
where
A1  {( E1 ,0.7,0.2) ( E2 ,0.4,0.3) ( E3 ,0.9.0) ( E4 ,0.4.03) ( E5 ,0.4,0.3)} is the vertex set
and
B1  {( E1 E2 ,0.7,0.2) ( E1 E3 ,0.7,0.2) ( E1 E4 ,0.7,0.2) ( E1 E5 ,0.4,0.3) ( E2 E3 ,0.9,0)
( E2 E4 ,0.4,0.3) ( E2 E5 ,0.4,0.3) ( E3 E4 ,0.7,0.2) ( E3 E5 ,0.4,0.3) ( E4 E5 ,0.4,0.3)}
edge set of the Intuitionistic fuzzy line graph.
Proposition3.4:
6
is
the
L( H )  ( A1 , B1 ) is an
Intuitionistic fuzzy line graph corresponding to
intutionistic fuzzy hypergraph H  ( X ,  ).
Proof
Obvious from Definition.
Proposition3.5:
If L(H ) is a Intuitionistic fuzzy line graph of the Intuitionistic fuzzy
hypergraph H then L (H *) is the line graph of H * .
Proof:
Since H  ( X ,  ) is an Intuitionistic fuzzy hypergraph and L( A)  ( A1 , B1 ) is
an Intuitionistic fuzzy line graph.
We have  A1 ( Ei )  min ( Ei ( x)), Ei  
xEi
and  A1 ( Ei )  max ( Ei ( x))
xEi
so Ei  Z  x  Ei and Ei  
Also
 B ( E j Ek )  min i (min(  E ( xi ), E ( xi )) and
1
j
k
 B ( E j Ek )  max i (max(  E ( xi ),  E ( xi ))
1
j
where xi  E j  Ek ,
j , k  1, 2, ..., n
K
for
all
E j , Ek  Z
and
so
W  {E j Ek ; E j  Ek   , E j , Ek   , j  k}. This completes the proof.
Conclusion
In this paper, Intuitionistic fuzzy line graph of a Intuitionistic fuzzy
hypergraph is defined and properties are examined.
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