Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1975-1985

© Research India Publications

http://www.ripublication.com

Fuzzy Bi-Magic labeling on Cycle Graph and Star

Graph

K.Ameenal Bibi1 and M.Devi2

1Associate Professor, D.K.M College for Women (Autonomous), Vellore-632001 2 Research Scholar, D.K.M College for Women (Autonomous), Vellore-632001

Abstract

In this paper, we introduce the concept of fuzzy Bi-Magic labeling in graphs.

We defined Fuzzy Bi-Magic labeling for cycle and star graph. Further we

investigated the Properties of such labeling on these graphs.

A fuzzy graph ,:G is known as fuzzy Bi-Magic graph if there exists

two bijective functions ]1,0[: V and ]1,0[: VV such that

)()(),( vuvu with the property that the sum of the labels on the

vertices and the labels of their incident edges is one of the constants either 1k

or 2k , independent of the choice of vertex. We investigated that fuzzy Cycle

graphs and fuzzy Star graphs are fuzzy Bi-Magic graphs. Further, some

properties related to fuzzy bridge and fuzzy cut node have been discussed.

Keywords: Fuzzy labeling, Fuzzy Bi-magic labeling, Fuzzy Bi-Magic cycle,

Fuzzy Bi- Magic Star, Fuzzy bridge, Fuzzy Cut node.

AMS Mathematical Subject Classification: 03E72, 05C72, 05C78.

1. INTRODUCTION

Fuzzy is a newly emerging mathematical framework to exhibit the phenomenon of

uncertainty in real life tribulations. It was introduced by Zadeh in 1965, and the

concepts were pioneered by various independent researchers, namely Rosenfeld,

Kauffmann, etc.

1976 Author name

A Fuzzy set is defined mathematically by assigning a value to each possible

individual in the universe of discourse, representing its grade of membership which

corresponds to the degree, to which that individual is similar or compatible with the

concept represented the fuzzy set. Based on Zadeh’s Fuzzy relation the first definition

of a fuzzy graph was introduced by Kauffmann in 1973. Baskar Babujee has

introduced the notion of Bi-Magic labeling in which there exists two constants k1 and

k2 such that the sums involved in a specified type of magic labeling is k1 and k2.

In this paper, a new concept of fuzzy Bi-magic labeling has been introduced. Fuzzy

Bi-magic labeling for the cycle graph and star graph are defined. Also, some

properties of these graphs with fuzzy bridge and fuzzy cut nodes are discussed. The

graphs which are considered in this paper are simple, finite, connected and undirected.

All basic definitions and Symbols are followed as in [2,4,5,6].

2. PRELIMINARIES AND OBSERVATIONS

Definition 2.1

A fuzzy labeling graph admits Bi-magic labeling if the sum of membership values of

vertices and edges incident at the vertices are k1 and k2 where k1 and k2 are constants

and denoted by )(~ GmB o

A fuzzy labeling graph which admits a Bi-magic labeling is called a Fuzzy Bi-magic

labeling graph.

Example: 2.2

Figure 1: Fuzzy Bi-magic path graph for n=5 )(~

5PmB o

K1=0.16 and K2=0.18

In Figure 1,

16.008.001.007.0)(),()( 2211 vvvv

16.006.002.008.0)(),()( 3322 vvvv

Fuzzy Bi-Magic labeling on Cycle Graph and Star Graph 1977

18.009.003.006.0)(),()( 4433 vvvv

18.005.004.009.0)(),()( 5544 vvvv

Note: A Fuzzy Bi-magic labeling on Path graph is satisfied only if n=5.

Definition 2.3

A Cycle (or) Circular graph is a graph that consists of a single cycle (or) in

otherwords, finite number of vertices connected in a closed chain. The cycle graph

with n vertices is denoted by Cn. The number of vertices in Cn equals the number of

edges and every vertex has degree 2.

A cycle graph which admits fuzzy labeling is called a fuzzy labeling cycle graph and

if Bi-magic labeling exists then it is called a fuzzy Bi-magic labeling cycle graph and

denoted by )(~

no CmB .

Example 2.7

Figure 2: Fuzzy Bi-magic cycle for n=5 )(~

5CmB o

Definition 2.4

A fuzzy Star graph consists of two vertex sets V and U with 1V and 1U such

that 0),( iuv and 0),( 1 ii uu for ni 1 .

1978 Author name

In a fuzzy Star graph, if a Bi-magic labeling exists then it is called a fuzzy Bi-magic

labeling Star graph and it is denoted by )(~

,1 no SmB .

Example 2.5

Figure 3: Fuzzy Bi-magic Star for n=4 )(~

4,1SmB o

Definition 2.6

Let ,:G be a fuzzy graph. The strong degree of a vertex v is defined as the sum

of membership values of all strong neighbours of v, then

)(

).,()(vNu

ss

vuvd

Definition 2.7

An edge uv is called a fuzzy bridge of G, if its removal reduces the strength of

connectedness between some pair of vertices in G. Equivalently (u,v) is a fuzzy

bridge iff there are nodes x,y such that (u,v) is a arc of every strongest x-y path.

Definition 2.8

A node is a fuzzy cutnode of ,:G if removal of it reduces the strength of

connectedness between some pair of nodes in G.

Fuzzy Bi-Magic labeling on Cycle Graph and Star Graph 1979

3. MAIN RESULTS

Proposition 3.1

If n is odd, then the cycle Cn admits a fuzzy Bi-magic labeling.

Proof:

Let Cn be any cycle with odd number of vertices and v1,v2,…,vn and v1v2, v2v3,

…,vnv1 be the vertices and edges of Cn respectively.

Let ]1,0[z such that one can choose z=0.01 if 3n .The fuzzy labeling is defined

as follows:

zinv i )42()( 2 for 2

31

ni

zinv i )42(2)( 2 for 2

1

ni

)(2

11/)()( 212 zinivMaxv ii

for 2

11

ni

nivMaxvv in 1/)(2

1),(

)(),(),( 1 zivvvv ninin for 11 ni .

Here, we investigated the results for fuzzy Bi-magic cycle )(~

7CmB o

Case (i): i is even

Then i=2x for any positive integer x

For each edge 1, ii vv , the fuzzy Bi-magic labelings are,

)(~

70 CmB = )(),()( 11 iiii vvvv

= )(),()( 121222 xxxx vvvv

=

zxnivMaxzxn

nivMaxnizxn

i

i

)1(2

11/)()2(

1/)(2

1

2

31/)42(

2

1980 Author name

=

2

11/)(1/)(

2

1)5( 2

nivMaxnivMaxzn ii

If i=2x for any positive integer x,

2

1nx

For each edge 1, ii vv , the fuzzy Bi-magic labelings are,

)(~

70 CmB = )(),()( 121222 xxxx vvvv

=

zxnivMaxzxn

nivMaxnizxn

i

i

)1(2

11/)()2(

1/)(2

1

2

1/)42(2

2

=

2

11/)(1/)(

2

1)123( 2

nivMaxnivMaxzn ii

(or)

=

2

11/)(1/)(

2

1)52( 2

nivMaxnivMaxzn ii

Case (ii): i is odd

Then i=2x+1 for any positive integer x

For each edge 1, ii vv , the fuzzy Bi-magic labelings are,

)(~

70 CmB = )(),()( 11 iiii vvvv

= )(),()( 22221212 xxxx vvvv

=

2

31/)2()42(

1/)(2

1)1(

2

11/)( 2

nizxnzxn

nivMaxzxnivMax ii

=

2

11/)(1/)(

2

1)5( 2

nivMaxnivMaxzn ii

Fuzzy Bi-Magic labeling on Cycle Graph and Star Graph 1981

If i=2x+1 for any positive integer x,

2

1nx

For each edge 1, ii vv , the fuzzy Bi-magic labelings are,

)(~

70 CmB = )(),()( 22221212 xxxx vvvv

=

zxnivMaxzxn

nivMaxnizxn

i

i

)1(2

11/)()82(

1/)(2

1

2

1/)2(2

2

=

2

11/)(1/)(

2

1)123( 2

nivMaxnivMaxzn ii

(or)

=

2

11/)(1/)(

2

1)52( 2

nivMaxnivMaxzn ii

In General, if n is odd

)(~

0 nCmB =

2

11/)(1/)(

2

12 2

nivMaxnivMaxz ii

)(~

0 nCmB =

2

11/)(1/)(

2

1)2( 2

nivMaxnivMaxzn ii

Therefore, from the above cases, we verified that G is a fuzzy Bi-magic graph if G has

odd number of vertices.

Note: If we take,

(i) The Number of vertices n=5, then

)(~

50 CmB =

2

11/)(1/)(

2

1)3( 2

nivMaxnivMaxzn ii

)(~

50 CmB =

2

11/)(1/)(

2

1)83( 2

nivMaxnivMaxzn ii

(or) =

2

11/)(1/)(

2

1)32( 2

nivMaxnivMaxzn ii

(ii) The Number of vertices n=9, then

1982 Author name

)(~

50 CmB =

2

11/)(1/)(

2

1)7( 2

nivMaxnivMaxzn ii

)(~

50 CmB =

2

11/)(1/)(

2

1)163( 2

nivMaxnivMaxzn ii

(or) =

2

11/)(1/)(

2

1)72( 2

nivMaxnivMaxzn ii

and so on.

Proposition: 3.2

For any ,4n the Star graph S1,n admits a fuzzy Bi-magic labeling.

Proof:

Let S1,n be the Star graph with v,u1,u2,…,un as vertices and vu1, vu2, …,vun as edges.

Let ]1,0[z such that one can choose z=0.01 if 4n .The fuzzy labeling is defined

as follows:

zinui ])1(2[)( for i=1,2,3

zinui }1])1(2{[)( for ni 4

nu

u ii

)()(

for ni 1

zuvMinuvMaxuv 2)}(),({)}(),({),( 111 , for i=1

)}(),({)}(),({),( 222 uvMinuvMaxuv , for i=2

zuvMinuvMaxuv 2)}(),({)}(),({),( 333 , for i=3

zuvMinuvMaxuv iii 3)}(),({)}(),({),( , for ni 4

Then the constants k1 and k2 of the fuzzy Bi-magic labeling are defined as follows:

Fuzzy Bi-Magic labeling on Cycle Graph and Star Graph 1983

To find k1:

Case (i): for i=1

)(~

,10 nSmB = )(),()( 11 uuvv

=

znziuvMin

iuvMaxninui

}1)1(2{21/)(),(

1/)(),(1/)(

1

1

=

zniuvMin

iuvMaxninui

)12(1/)(),(

1/)(),(1/)(

1

1

Case (ii): for i=2

)(~

,10 nSmB = )(),()( 22 uuvv

=

zniuvMin

iuvMaxninui

}2)1(2{2/)(),(

2/)(),(1/)(

2

2

=

nziuvMin

iuvMaxninui

22/)(),(

2/)(),(1/)(

2

2

Case (iii): for i=3

)(~

,10 nSmB = )(),()( 33 uuvv

=

znziuvMin

iuvMaxninui

}3)1(2{23/)(),(

3/)(),(1/)(

3

3

=

zniuvMin

iuvMaxninui

)12(3/)(),(

3/)(),(1/)(

3

3

1984 Author name

To find k2:

Case (iv): for ni 4

)(~

,10 nSmB = )(),()( ii uuvv

=

}4/)4(2{4/)(),(

4/)(),(1/)(

3 niinniuvMin

niuvMaxninu

ii

Hence the Star graph S1,n is fuzzy Bi-magic labeling graph )(~

,10 nSmB for 4n .

4. PROPERTIES OF FUZZY BI-MAGIC GRAPHS

Proposition 4.1

For every Bi-magic graph G, there exists atleast one fuzzy bridge.

Proof:

Let G be a fuzzy Bi-magic graph, such that there exists only one edge ),( yx with

maximum value, since is bijective.

Now, we claim that ),( yx is a fuzzy bridge, if we remove the edge (x,y) from G,

then its subgraph, we have ),(),( yxyx which implies that (x,y) is a fuzzy

bridge.

Proposition 4.2

Removal of a fuzzy cut vertex from a fuzzy Bi-magic Star graph G, the resulting

graph G* also admits a fuzzy Bi-magic labeling if 4n .

Proof:

Since G is a Star graph, there exists atleast one fuzzy cut vertex. Now if we remove

that fuzzy cut vertex from G, then it becomes a smaller Star G*. However, G* remains

to be admit fuzzy Bi-magic labeling if 4n .

Hence, we conclude that removal of a fuzzy cut vertex from a fuzzy Bi-magic Star

graph results in a fuzzy Bi-magic Star graph if 4n .

Fuzzy Bi-Magic labeling on Cycle Graph and Star Graph 1985

Observation 4.3

1. Every fuzzy Bi-magic graph is a fuzzy labelled graph, but the converse is

not true.

2. If G is a fuzzy Bi-magic graph then )()( vdud for any pair of vertices

)(, GVvu .

3. For all fuzzy Bi-magic cycle graph G, there exists a subgraph G* which is

a cycle with odd number of vertices and there exists atleast one pair of

vertices u and v such that )()( vdud ss .

5. CONCLUSION

In this paper, the concept of fuzzy Bi-magic labeling has been introduced. Fuzzy Bi-

magic labeling for Cycle and Star graphs have been discussed. Properties of fuzzy Bi-

magic graphs are investigated.

We further extend this study on some more special classes of graphs.

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1986 Author name