Generarlized operations on fuzzy graphs

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
15
Generarlized Operations on Fuzzy Graphs
D.Venugopalam, Nagamurthi Kumar, M.vijaya kumar
Abstract
To discuss the Cartesian Product Composition, union and join on Interval-valued fuzzy graphs. We also
introduce the notion of Interval-valued fuzzy complete graphs. Some properties of self complementary graph.
Key Words : Interval-valued fuzzy graph self complementary Interval valued fuzzy complete graphs
Mathematics Subject Classification 2000: 05C99
1.Introduction
It is quite well known that graphs are simply model of relations. A graphs is a convenient way of
representing information involving relationship between objects. The objects are represented by vertices and
realtions by edges. When there is vagueness in the descriptionof the objects or in its relationships or in both, it is
natural that we need to design a Fuzy Graph Model. Application of fuzzy relations are widespread and important;
especially in the field of clustering analysis, neural networks, computer networks, pattern recognition, decision
making and expert systems. In each of these the basic mathematical structure is that of a fuzzy graph.
We know that a graphs is a symmetric binry relation on a nonempty set V. Similary, a fuzzy graph is a
symmetric binary fuzzy relation on a fuzzy subset. The first definition of a fuzzy graph was by Kaufmann [18] in
1973, based on Zadeh’s fuzzy relations [46]. But it was Azriel Rosenfeld [35] who considered fuzzy relations
onf uzzy sets and developed the theory of fuzzy graphs in 1975. During the sam etime R.T.Yeh and S.Y.Bang
[44] have also introduced various connectedness concepts in fuzzy graphs.
2 Preliminaries
Definition 2.1 : Let V be a nonempty set. A fuzzy graphs is a pair of functions.
G : (σ, µ) where σ is a fuzzy subset of v and µ is a symmetric fuzzy relation on σ i.e. σ: V→[0, 1] and µ : V x V
→ [0, 1] such that µ(u, v) ≤ σ(u) Λ σ(v) for all u, v in V.
We denot the underlying (crisp) graph of G: (σ, µ) by G*:(σ*, µ*) where σ* is referred to as the
(nonempty) set V of nodes and µ* = E ⊆V x V. Note that the crisp graph (V, E) is a special case of a fuzzy
graph with each vertex and edge of (V, E) having degree of membership 1. We need not consider loops and we
assume that µ is reflexive. Also, the underlyign set V is assumed to be finite and σ can be chosen in any manner
so as to satisfy the definition of a fuzzy graphs in all the examples.
Definition 2.2 : The fuzzy graph H: (τ, v) is called a partial fuzzy subgraph of
G : (σ, µ) if τ ⊆σ and v ⊆µ. In particular, we call H: (τ, v) a fuzzy subgraph of
G : (σ, µ) if τ(u) = σ(u) ∀ u∈τ* and v(u,v) = µ(u, v) ∀ (u, v)∈v*.For any threshold t, 0 ≤ t ≤ 1, σ′ = {u∈V :
σ(u) ≥ t} and µ′ = {(u, v) ∈V x V : µ(u, v) ≥ t}. Since µ(u, v) ≤ σ(u) Λ σ(v) for all u, v in V we have µ′ ⊆ σ′, so
that (σ′, µ′) is a graph with vertex set σ′ and edge set µ′ for t∈[0, 1].
Note1.: Let G : (σ, µ) be a fuzzy graph. If 0 ≤ t1 ≤ t2 ≤ 1, then (σ′2
, µ′2
) is a subgraph of (σ′1
, µ′1
).
Note 2.: Let H : (τ, v) be a partial fuzzy subgraph of G: (σ, µ). For any threshold 0 ≤t ≤ 1, (τ′, v′) is a subgraph
of (σ′, µ′).
Definition 2.3 : For any fuzzy subset τ of V such that τ ⊆ σ, the partial fuzzy subgraph of (σ, µ) induced by τ is
the maximal partial fuzzy subgraph of (σ, µ) that has fuzzy node set τ. This is the partial fuzzy subgraph (τ, v)
where
Τ(u, v) = τ(u) Λ µ(u, v) for all u, v ∈V.
Definition 2.4 : The fuzzy graph H: (τ, v) is called a fuzzy subgraph of G: (σ, µ) induced by P if P ⊆ V, τ(u) =
σ(u)∀ u, v∈P.
Definition 2.5 : A partial fuzy subgraph (τ, v) spans the fuzzy graph (σ, µ) if σ = τ. In this case (τ, v) is
called a aprtial fuzzy spanning subgraph of (σ, µ).
Next we introduce the concept of a fuzzy spanning subgrph as a special case of partial fuzzy spanning subgraph.
Operations 2.6: Graphs g = (D, E) are simple : no multiple edges and no loops.
An unordered pair {x, y} is deonte by xy or x – y
Operations 2.7: Graphs g = (D, E) are simple: no multiple edges and no loops.
An unordered pair {x, y} is denote by xy or x – y
An operation is a permutation on the set of graphs on D :
α : g → h
Operations 2.8: Graphs g = (D, E) are simple: no multiple edges and no loops.
An unordered pair {x, y} is denote by xy or x – y

16
An operation is a permutation on the set of graphs on D :
α : g → h
Let
DnD
ααα ,..., 21=Γ
Be the subgroup of the symmetric group generated by
Γ = {α1, α2,...,αn}
Transitivity 2.8 : The problem setting : Given operations Γ = {α1, α2,...,αn}
And any two graphs g, h on DDoes there exist a composition D
Γ∈α
α = αik, αik-1,...,α1 such that α(g) = h.
Complement 2.9 : Let 





2
D
be the set of all 2-subsets {x, y}. ( ) 













= E
D
DgC
2
,
Edges ↔ nonedges
Neighbours 2.10 : Neighbours of x Ng(x) = {y | xy∈ E} ( ) ( ) { }( )xxNDxN gg U'
=
Nonneighbours of x
Subgraphs 2.11 : The symmetric difference : A + B = (A B) U (B A)
The sub graph of g induced by :DA ⊆ [ ] 













=
2
,
A
EAAg I
Complementing Subgraphs 2.12 : Denote by g














+=⊕
2
,
A
EDAg
3.Main Results
Theorem 3.1
Let 111 ,EVG = and 222 , EVG = be two Interval Valued Fuzzy Graphs. Then
(i) 2121 GGGG U≅+
(ii) 2121 GGGG +≅U
Proof
Consider the identity map I : 2121 VVVV UU → ,
To prove (i) it is enough to prove
(a) (i) ( ) ( )ii vv '
11
'
11 µµµµ UU =
(ii) ( ) ( )ii vv '
11
'
11 γγγγ U=+
(b) (i) ( ) ( )jiji vvvv ,, '
22
'
22 µµµµ UU =
(ii) ( ) ( )jiji vvvv ,, '
11
'
22 γγγγ U=+
(a) (i) ( )( ) ( )( )ii vv '
11
'
11 µµµµ +=+ , by Definition 4.1
( )
( )


∈
∈
=
21
'
1
111
Vvifv
Vvifv
i
i
µ
µ
( )
( )



∈
∈
=
21
'
1
111
Vvifv
Vvifv
i
i
µ
µ
( )( )iv'
11 µµ U=
(ii) ( )( ) ( )( )ii vv '
11
'
11 γγγγ +=+ ,

17
( )
( )


∈
∈
=
21
'
1
111
Vvifv
Vvifv
i
i
γ
γ
( )
( )



∈
∈
=
21
'
1
111
Vvifv
Vvifv
i
i
γ
γ
( )( )iv'
11 γγ U=
(b) (i) ( )( ) ( )( )( )( ) ( )( )jijiji vvvvvv ,., '
22
'
11
'
11
'
22 µµµµµµµµ +−++=+
( )( )( )( ) ( )( ) ( )
( )( )( )( ) ( ) ( ) ( )



∈−
∈−
=
',..
,,.
'
11
'
11
'
11
21
'
22
'
11
'
11
Evvifvvvv
EEvvifvvvv
jijiji
jijiji
µµµµµµ
µµµµµµ
UU
UUUU
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )





∈−
∈−
∈−
=
',..
,,.
,,.
'
11
'
11
2
'
2
'
1
'
1
1211
Evvifvvvv
Evvifvvvv
Evvifvvvv
jijiji
jijiji
jijiji
µµµµ
µµµ
µµµ
( ) ( )
( ) ( )
( )






∈
∈
∈
=
',0
,,
,,
2
'
2
12
Evvif
Evvifvv
Evvifvv
ji
jiji
jiji
µ
µ
( )( )ji vv ,'
22 µµ U=
(b) (ii) ( )( ) ( )( )( )( ) ( )( )jijiji vvvvvv ,., '
22
'
11
'
11
'
22 γγγγγγγγ +−++=+
( )( )( )( ) ( )( ) ( )
( )( )( )( ) ( ) ( ) ( )



∈−
∈−
=
',..
,,.
'
11
'
11
'
11
21
'
22
'
11
'
11
Evvifvvvv
EEvvifvvvv
jijiji
jijiji
γγγγγγ
γγγγγγ
UU
UUUU
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )





∈−
∈−
∈−
=
',..
,,.
,,.
'
11
'
11
2
'
2
'
1
'
1
1211
Evvifvvvv
Evvifvvvv
Evvifvvvv
jijiji
jijiji
jijiji
γγγγ
γγγ
γγγ
( ) ( )
( ) ( )
( )






∈
∈
∈
=
',0
,,
,,
2
'
2
12
Evvif
Evvifvv
Evvifvv
ji
jiji
jiji
γ
γ
( )( )ji vv ,'
22 γγ U=
To prove (ii) it is enough to prove
(a) (i) ( )( ) ( )( )ii vv '
11
'
11 µµµµ UU =
(ii) ( )( ) ( )( )ii vv '
11
'
11 γγγγ +=U
(b) (i) ( )( ) ( )( )jiji vvvv ,, '
22
'
22 µµµµ +=U
(ii) ( )( ) ( )( )jiji vvvv ,, '
22
'
22 γγγγ UU =
Consider the identity map 2121: VVVVI UU →

18
(a) (i) ( )( ) ( )( )ii vv '
11
'
11 µµµµ UU =
( )
( )
( )
( )



∈
∈
=



∈
∈
=
2
'
1
11
2
'
1
11
Vvifv
Vvifv
Vvifv
Vvifv
ii
ii
ii
ii
µ
µ
µ
µ
( )( ) ( )( )ii vv '
11
'
11 µµµµ UU ==
(a) (ii) ( )( ) ( )( )ii vv '
11
'
11 γγγγ UU =
( )
( )


∈
∈
=
2
'
1
11
Vvifv
Vvifv
ii
ii
γ
γ
( )
( )



∈
∈
=
2
'
1
11
Vvifv
Vvifv
ii
ii
γ
γ
( )( ) ( )( )ii vv '
11
'
11 γγγγ UU ==
(b) (i) ( )( ) ( )( )( )( ) ( )( )jijiji vvvvvv ,., '
22
'
11
'
11
'
22 µµµµµµµµ UUUU −=
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )





∈∈−
∈−
∈−
=
21
'
11
2
'
2
'
1
'
1
1211
,0.
,,.
,,.
Vvvifvvv
Evvifvvvv
Evvifvvvv
jiji
jijiji
jijiji
µµ
µµµ
µµµ
( ) ( )
( ) ( )
( ) ( )






∈∈
∈
∈
=
21
'
11
2
'
2
12
,.
,,
,,
VvVvifvv
Evvifvv
Evvifvv
jiji
jiji
jiji
µµ
µ
µ
( ) ( )
( ) ( ) ( )



∈
∈
=
',.
,,
1
'
111
21
'
22
Evvifvv
EorEvvifvv
ji
jiji
µµ
µµ U
( )ji vv ,'
22 µµ +=
(b) (ii) ( )( ) ( )( )( )( ) ( )( )jijiji vvvvvv ,., '
22
'
11
'
11
'
22 γγγγγγγγ UUUU −=
( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )





∈∈−
∈−
∈−
=
21
'
11
2
'
2
'
1
'
1
1211
,0.
,,.
,,.
Vvvifvvv
Evvifvvvv
Evvifvvvv
jiji
jijiji
jijiji
γγ
γγγ
γγγ
( ) ( )
( ) ( )
( ) ( )






∈∈
∈
∈
=
21
'
11
2
'
2
12
,.
,,
,,
VvVvifvv
Evvifvv
Evvifvv
jiji
jiji
jiji
γγ
γ
γ
( ) ( )
( ) ( ) ( )



∈
∈
=
',.
,,
1
'
111
21
'
22
Evvifvv
EorEvvifvv
ji
jiji
γγ
γγ U

19
( )ji vv ,'
22 γγ +=
Theorem 3.2
Let 111 ,EVG = and 222 , EVG = be two Interval Valued Fuzzy Graphs. Then 21 GG o is a strong
Interval Valued Fuzzy Graphs
Proof
Let EVGGG ,21 ==o where V1 x V2 and
( )( ){ } ( ) ( ){ }111211222122 ,:,:,,:,, EvuVwwvwuEvuVuvuuuE ∈∈∈∈= U
( )( ){ }221112121 ,:,, vuEvuvvuu ≠∈U .
(i) ( )( ) ( ) ( )2221222 .,, vuuvuuu µµµ =
( ) ( ) ( )2
'
12
'
11 .. vuu µµµ= , since G2 is strong
( ) ( ) ( ) ( )2
'
112
'
11 ... vuuu µµµµ=
( )( )( )( )2
'
112
'
11 ,., vuuu µµµµ oo=
( )( ) ( ) ( )2221222 .,, vuuvuuu γγγ =
( ) ( ) ( )2
'
12
'
11 .. vuu γγγ= , since G2 is strong
( ) ( ) ( ) ( )2
'
112
'
11 ... vuuu γγγγ=
( )( )( )( )2
'
112
'
11 ,., vuuu γγγγ oo=
(ii) ( )( )( ) ( ) ( )112
'
1112 ,.,, vuwwvwu µµµ =
( ) ( ) ( )1111
'
1 .. vuw µµµ= , since G1 is strong
( ) ( ) ( ) ( )11
'
111
'
1 ... vwuw µµµµ=
( )( )( )( )wvwu ,., 1
'
111
'
11 µµµµ oo=
( )( )( ) ( ) ( )112
'
1112 ,.,, vuwwvwu γγγ =
( ) ( ) ( )1111
'
1 .. vuw γγγ= , since G1 is strong
( ) ( ) ( ) ( )11
'
111
'
1 ... vwvw γγγγ=
( )( )( )( )wvwu ,., 1
'
111
'
11 γγγγ oo=
(iii) ( )( ) ( ) ( ) ( )2
'
12
'
11122122 ..,,, vuvuvvuu µµµµ =
( ) ( ) ( ) ( )2
'
12
'
11111 ... vuvu µµµµ= , since G1 is strong
( ) ( ) ( ) ( )2
'
1112
'
111 ...| vvuu µµµµ=
( )( )( )( )21
'
1121
'
11 ,., vvuu µµµµ oo=
( )( ) ( ) ( ) ( )2
'
111211221212 .,.,,, vvuvuvvuu γγγγ =
( ) ( ) ( ) ( )2
'
12
'
11111 ... vuvu γγγγ= , since G1 is strong
( ) ( ) ( ) ( )2
'
1112
'
111 ... vvuu γγγγ=
( )( )( )( )21
'
1121
'
11 ,., vvuu γγγγ oo=
From (i), (ii), (iii), G is a strong Interval valued Fuzzy Graphs.
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