maths - ijmcar - some isomorphism - sanjay sharma

International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 2 June 2012 44-59 © TJPRC Pvt. Ltd., SOME ISOMORPHISM RESULTS ON PRODUCT FUZZY GRAPHS

Y. VAISHNAW 1 & S. SHARMA2

1Department of Mathematics Yugantar Institute of technology and management, Rajnandgaon (C. G.) India

2Department of Applied Mathematics , Bhilai Institute of technology, Durg ( C. G. ) India.

ABSTRACT

We have discussed notion of ring sum of product fuzzy graphs .We further give Three

Independent Theorems based on Ring Sum, Join and Isomorphism of product fuzzy graph which

are fuzzy version of classical graph theory.

AMS Subject Classification: 05C72

KEYWORDS: Fuzzy Graph, Isomorphism, Product, Complement Of Product Fuzzy Graphs,

Complement ,Isomorphism Of Fuzzy Graph Ring Sum And Join Of Product Fuzzy Graphs.

INTRODUCTION

Fuzzy logic has developed into a large and deep subject. Zadeh [7 ] addresses the

terminology and stresses that fuzzy graphs are a generalization of the calculi of crisp graphs.

Several other formulation of fuzzy graph problems have appeared in the literature. Koczy also

gives a taxonomy of fuzzy graphs.

In [10] Rosendfeld and Yeh and Beng independently developed the theory of fuzzy graph.

A fuzzy graph is a pair G:( ), where σ is a fuzzy subset of V and is fuzzy relation on V such

that, (uv) ≤ σ (u) Λ σ (v) for all u,v ε V.

A graph isomorphism[1] search is important problem of graph theory . It consists to define

of bijective correspondence existence which preserve adjacent relation between vertex sets of two

graphs [13].In a case of fuzzy graphs the notion of isomorphism is fuzzy.

Ramaswamy and [11] replaced ‘minimum’ in the definition of fuzzy graph by ‘product’

and call the resulting structure product fuzzy graph and proved several results which are analogous

to fuzzy graphs. Further they discussed a necessary an sufficient condition for a product partial

fuzzy sub graph to be the multiplication of two product partial fuzzy sub graphs.

Some Isomorphism Results on Product Fuzzy Graphs 45

We have discussed notion of ring sum of product fuzzy graphs. We further provided Three

Independent Theorems based on Ring Sum, Join and Isomorphism of product fuzzy graph which

are fuzzy version of isomorphism on classical graphs [ 12].

PRELIMINARY

Lemma 2.1 [11] Let G be a graph whose vertex set is V, be a fuzzy sub set of V and be a

fuzzy sub set of VxV then the pair ( ) is called product fuzzy graph if

Remark :- If ( ) is a product fuzzy graph and since and are less than or equal to

1, it follows that

Hence ( ) is a fuzzy graph thus every product fuzzy graph is a fuzzy graph

Remark:- If ( ) is a product fuzzy sub graph of G whose vertex set is V, we will assume that

and is symmetric.

Example 2.1Let V = {u,v,w}, be the fuzzy subset of V defined as

Let be the fuzzy subset of VxV defined as

. It is easy to see that ( ) is a product

fuzzy graph and hence a fuzzy graph.

46 Y. Vaishnaw & S. Sharma

G: ( )

Remark:- Every product fuzzy graph is fuzzy graph but converse is not true.

Lemma 2.2 [11] a product fuzzy graph ( ) is said to be complete if

Lemma 2.3 The complement of a product fuzzy graph ( ) is where

and

It follows that itself is a product fuzzy graph. Also

2/8


Lemma 2.4 Consider the product fuzzy graph and with

and are isomorphism between two product fuzzy graphs and is a bijective

mapping

and we write in this case .

Remark 1- An automorphism of G is an isomorphism of G with itself.

Proposition 2.1 [14] Let be the intersection of graph and .

Let and are the product fuzzy graph of resp. then

Is a product fuzzy graph of G.

Lemma 2.5 [14] Let and with and and

let be the union of and . Then the union of two

product fuzzy graphs is defined as where

And

Proposition 2.2 [14] Let G be the union of the graphs and then

is a product fuzzy graph.


Proposition 2.3 [14] Let G be the join of two graphs and and let and

are the product fuzzy graph of

resp.then is a product fuzzy graph of G.

Theorem 2.1 [14] Let and be the product fuzzy graphs then

i)

ii)

3. MAIN RESULTS

Definition 3.1 Let and be the product fuzzy graphs with

and res. Then the ring sum of two product fuzzy graphs

and is denoted by where

And

Proposition 3.1 Let and be the product fuzzy graphs with

and respectively. Then the ring sum of two product fuzzy graphs

and is denoted by is a product

fuzzy graph.

Proof: Since and are the product fuzzy graphs with

and res. Then we have to show that the ring sum of two product fuzzy

is a product fuzzy graph

For this we show that


in all cases.

Case-1 and then

Case-2 and then

i.e.

Case-3 and then


i.e.

Similarly we can show that if then

for all u, v.

This completes the proof of the proposition.

Example 3.1

Remark 3.1 If is a product fuzzy graph.Then

And

.33 .6

.4 .25

.2

.33

.5

.

.7

.8

.6 .9

.40 .69

.51

.4 .25

.3

.20

.40

.7

.51

.

.9

.69


i.e.

Theorem 3.1 Let and be two product fuzzy graphs with

then

Proof:- Let and be two product fuzzy graphs and

be the identity mapping, then it is sufficient to show that

(

And (

Now,

i.e. ………………………………………..(i)

And (


Next we illustrate different cases

Case-1 When

(

i.e.

Case-2 When and

(

i.e.

Case-3 When and

(

is product fuzzy graph

i.e. (


Case-4 When and

(

i.e. (

Case-5 When and then it is obvious that



Case-8 When then


)

i.e. ) for all

Thus from Case-1 to Case-8 we have

i.e. ……………………………..…………(ii)

Hence by (i) and (ii) we have

This completes the proof of the theorem.

Theorem 3.2 Let and be two product fuzzy graphs with

then

Proof:- Let and be two product fuzzy graphs with

and

be the identity mapping, then it is sufficient to show that

(

And (


Now,

i.e. ………………………….………………..(i)

Next (

Next we illustrate different cases

Case-1 When

(

i.e (

Case-2 When and

(

i.e.

Case-3 When and

(


is product fuzzy graph

i.e. (

Case-4 When and

(

i.e. (




(

Thus from Case-1 and Case-7 we have


i.e …………………………………………….(ii)

Hence from (i) and (ii)

This completes the proof of the theorem.

Theorem 3.3 If G is the Ring sum of two subgraphs and with , then every

complete product fuzzy subgraph ( ) of G is a ring sum of complete product fuzzy subgraph of

and complete product fuzzy subgraph of .

Proof:- We define the fuzzy subset of , as follows

if , if

And if , if

Then is product fuzzy graph of and ( is product fuzzy graph of .

And by definition of ring sum of product fuzzy graph.

If then by definition of ring sum of product

fuzzy graph.

If , by definition of ring sum of product fuzzy

graph.

If , by definition of ring sum of product fuzzy

graph.


If then And other

equalities are also holds because () is a complete product fuzzy graph.

CONCLUSIONS

We have discussed product fuzzy graph and its complement . The notion of a Ring sum and join

of product fuzzy graphs are discussed.In the notion of ring sum and join of product fuzzy graphs

common vertices and edges are considered. We have also discussed isomorphism property of

product fuzzy graph which are fuzzy version of classical graphs.

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