-MULTIPLICATIVE GRAPHS

n

Z -MULTIPLICATIVE GRAPHS

Dr Ajitha V

P.G Department of Mathematics,

Mahatma Gandhi College, Iritty-670703, Kerala, India

Silja C

Research Scholar,

Mary Matha Arts & science College, Mananthavady

Kannur University, Kannur- 670 567,Kerala, India

Abstract

Let

G



( , )

V E

be a graph with

p

vertices and

q

edges. A graph G is said to admit Zq1 multiplicative labeling if its vertices can be labeled by non negative integers such that induced edge labels obtained by the products of

the labels of end vertices modulo

q



1

are the positive numbers up to q. A graph admit such a labeling called Zq1 multiplicative graphs. In this paper we give certain graphs which admit Zq1multiplicative labeling and certain

graphs not admit Zq1multiplicative labeling.

Keywords

1 q

Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856) 1. Introduction

By a graph we mean a finite, connected, simple undirected graph with. The vertex set and the edge set of

the graph G are denoted by V(G) and E(G) respectively. For various graph theoretic notations and terminology we

follow F. Harrary [1]

Definition 1.1 Let

G



( , )

V E

be a

( , )

p q

graph. The graphG is said to be a 1

q

Z _ -multiplicative labeling if there exist a one-to-one function

f V G

: ( )



N

(where N is the set of all positive integers ) that induces a function *

1

: ( )

_q

{0}

f

E G



Z

_



of the edges of G

defined by *

( ) ( ) ( ) mod( 1)

f uv  f u f v q _{for every}

e



uv



E G

( )

. The graph which admits such a labeling is called a

Z

q1-multiplicative graph.

Example 1.2 Z7-multiplicative graph.

Define

f V G

: ( )



N

by

1 2 3 4 5 6

( ) 4, ( ) 3, ( ) 2, ( ) 1, ( ) 5, ( ) 10.

f v  f v  f v  f v  f v  f v 

Now we discus about star graph.

Theorem 1.3 : Let The star graphK1n is Zn1-multiplicative graphs.

Proof : Let v0 be the apex vertex and v v1, 2,...,vn be the pendent vertices of the

star graph K1n

define f v( )0  n 2

( )_i

f v i for

i



1, 2,...,

n

.





1 0 1 0 2 0

(

_n

)

,

,...,

_n

For

i



1, 2,...,

n

f v( ) ( )0 f vi (n2) mod(i n1)



i

mod(

n



1)

* 0

( _i)

f v v



We see that the induced edge labels obtained by the product of the labels of the

vertices are the first n positive integers.

Theorem 1.4 : The cycle C3 is not Z4-multiplicative graph.

Proof : Let v v v1, 2, 3 be vertices ofC3. Define

f V G

: ( )



N

by

1 2 3

( ) , ( ) , ( ) ,

f v a f v b f v c where

a b c

, ,

are distinct integers. Let f*(v v1 2)2(mod 4)

that is f v f v( ) ( )1 2 ab2(mod 4).



a or b must be even.



ac or bc must be even.



* *

1 3 1 3 2 3 2 3

( ) ( ) ( ) or b ( ) ( ) ( )

ac f v f v  f v v c f v f v  f v v must be even.

Then either

f

*

(

v v

1 3

) nor f (

*

v v

2 3

)

congruent to 1 or 3 modulo 4. Which contradict definition of f*.



C3 is notZ4-multiplicative graph.

Theorem 1.5 : The cycle C₄ is Z₅-multiplicative graph.

Proof : Define

f V G

: ( )



N

_,by

1 2 3 4

( ) 1, ( ) 6, ( ) 3, ( ) 4

f v  f v  f v  f v 

Theorem 1.6 : The cycle C5 is not a Z6-multiplicative graph.

Proof : Let v v v v v1, 2, ,3 4, 5 be vertices of C5. Define

f V G

: ( )



N

by

1 2 3 4 5

( ) , ( ) , ( ) ,, ( ) , ( ) ,

Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856) integers.

Let f*(v v1 2)3(mod 6)

that is f v f v( ) ( )1 2 ab3(mod 6)



a or b must be a multiple of 3. Let it be a. Now,

if a is even



a



0(mod 6)



ab



0 mod 6

if b is even



ab0 mod 6)

that is f*(v v1 2) f v f v( ) ( )1 2 ab0(mod 6) . Which is not possible.

Theorem 1.7 : Complete graph K3 is not Z4multiplicative .

Proof : We know that K3 C3.

Already proved thatC3 is not Z4multiplicative graph.

Theorem 1.8 : For a Complete graph Kn with

(

p



1)

edges where

p

is a prime number then Knis not Zp1 multiplicative graph .

Proof : Let v v1, ...,2 vnbe vertices of Kn.

Consider vi, with *

1

( _{i i} ) 1mod( 1)

f v v_  p

*

1 1

( ) (_{i i} ) ( _{i i} ) 1mod( 1)

f v f v_  f v v_  p

This imply f v( ) 1mod(i  p1) and f v( i1) 1mod( p1) or

( )_i 1mod( 1)

f v   p & f v( i1) 1mod(p1)

If f v( ) 1mod(_i  p1) & f v( i1) 1mod( p1)

Considervj,

j



i i

,



1

*

(

_{i j}

)

( ) ( )

_{i j}

( ) mod(

_j

1)

f

v v



f v f v



f v

p



*

1 1

(

_{i j}

)

(

_i

) ( )

_j

( ) mod(

_j

1)

f

v v

_



f v

_

f v



f v

p



.

Which is not possible

If f v( )i  1mod(p1) & f v( i1) 1mod(p1)

Considervj,

j



i i

,



1

*

*

1 1

(

_{i j}

)

(

_i

) ( )

_j

( ) mod(

_j

1)

f

v v

_



f v

_

f v

 

f v

p



Which is not possible. There for no such labeling exist.

Theorem 1.9 : For a Complete graph Kn with 2

1

p  edges where

p

is a prime number then Knis not Z_p2

multiplicative graph .

Proof : Let v v1, ...,2 vnbe vertices of Kn. Define

f V G

: ( )



N

by f v( )i ai where aiare distinct integers , for every

i

.

Since Kn is complete then the edge set E G( ){v vi j ; 1i j, n i,  j}.

Assume Kn is Z_p2 multiplicative graph .

Let

f

*

(

v v

_{k j}

)



p

2

( ) ( )

k j

(mod

)

f v

f v

p

p





2

(mod

)

k j

a a

p

p





Then either akor

a

jis a multiple of

p

.

Let it be ak.

i k

a a

 Is a multiple of

p

, for every 1 i n.

*

( _{i k}) _{i k}

f v v a a

  is a multiple of

p

, for every 1 i n.

*

( _{i k})

f v v

 has the possibilities

p

, 2 ,..., (

p

p



1)

p

there are

p



1

possibilities . But there are n1 edges , Then

p

  

1

n

1

must satisfy

 

n

p

(

1)

(

1)

n n

 

p p



(1)

Since Kn has

(

1)

2

n n



edges.

2 2

(

1)

1

(

1)

2

2

2

n n

p

n n

p











 



(1) Imply 2p2 2 n n(  1) p p( 1)

2

2 2 2 0

p     p





p



2



2

0

p  Is a contradiction .

Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856)

Theorem 1.10 : For a Complete graph Kn with 1 m

p  edges where

p

is a prime number and m1 is any

integer then Knis not Z_pm  multiplicative graph .

Proof : For m2 the result is above theorem. Now assume m2.

Let v v1, ...,2 vnbe vertices of Kn. Define

f V G

: ( )



N

by f v( )i ai where aiare distinct integers , for every

i

.

Since Kn is complete then the edge set

E G

( )



{

v v

i j

; 1



i j

,



n i

,



j

}

. Assume Kn is Z_pm multiplicative graph .

Let

f

*

(

v v

_{k j}

)



p

( ) ( )

_{k j}

(mod

m

)

f v

f v

p

p





(mod

m

)

k j

a a

p

p





Then either akor ajis a multiple of

p

.

Let it be ak.

i k

a a

 Is a multiple of

p

, for every 1 i n.

*

( i k) i k

f v v a a

  is a multiple of

p

, for every 1 i n.

*

( _{i k})

f v v

 has the possibilities 2 2 2 1 1

, 2 ,..., ( 1) , , 2 ,..., ( 1) ,..., m ,..., ( 1) m

p p p p p p p p p  p p 

there are

(

n



1)(

p



1)

possibilities . But there are n1 edges , Then

(

m



1)(

p

  

1)

n

1

must satisfy .

(

1) [(

1)(

1)][(

1)(

1) 1]

n n

m

p

m

p



 







 

imply

2 2 2 2

( 1) ( 2 1) ( 2 5 3) 3 2

n n  p m  m  p m  m m  m (2)

Since Kn has

(

1)

2

n n



edges.

(

1)

1

(

1)

2

2

2

m m

n n

p

n n

p











 



2 2 2 2

2pm p m( 2m 1) p m(2 5m 3) (m 3m 4) 0

           (3)

Consider,

2 2 2 2

2pmp m( 2m 1) p m(2 5m 3) (m 3m4)

1 2 2 2

2m 4(m 2m 1) 2(2m 5m 3) (m 3m 4)

         





p



2



1 2 1 2

2m m m 6 2m (m 1) 3m 7

        

1 2

2m (m 1)

   .

Since m+1 2

2 1 3 2 ( 1)

m   m   m

m+1 2

2 (m1) 0

Which imply 2 2 2 2

2pm p m( 2m 1) p m(2 5m 3) (m 3m 4) 0

          

Contradiction to equation (3) .

There for Kn is not Z_pm multiplicative graph .

Example 1.11: Complete graph K4 are Z7 multiplicative.

Define

f V G

: ( )



N

by f v( ) 1, ( )1  f v2 2, ( )f v3 6, ( )f v4 4

Example 1.12: Wheel W3 are Z7multiplicative.

Define

f V G

: ( )



N

by f v( ) 1, ( )0  f v1 4, ( )f v2 2, ( )f v3 6

Example 1.13: Wheel W4 is not Z9multiplicative.

Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856)

Define

f V G

: ( )



N

by f v( )_i a_i, where aiare distinct integers , for every 0 i 4. Let *

1 2

( ) 3(mod 9)

f v v 

1 2

( ) ( ) 3(mod 9)

f v f v

 

1 2 3(mod 9)

a a

 

Then a1 or a2 is a multiple of 3.

Let it be a1.

1 i

a a

 Is a multiple of 3.

2, 4

v v are adjacent to v₁

2 1 and 4 1

a a a a

 is a multiple of 3.

* *

1 2 1 4

( ) & ( )

f v v f v v are multiple of 3

* *

1 2 1 4

(

)

6(mod 9) &

(

)

6(mod 9)

f

v v

f

v v







Which is a contradiction . So W₄ is not Z₉multiplicative.

Example 1.14: Wheel W5 are Z11multiplicative.

Define

f V G

: ( )



N

by f v( )0 4, ( ) 10, ( )f v1  f v2 7, ( )f v3 6, ( )f v4 2, ( )f v5 8.

Example 1.15: Wheel W6 are Z13multiplicative.

Next consider about paths,

Example 1.16: The path Pnis Znmultiplicative graph for n10.

For P₂, f v( ) 1, ( )1  f v2 3

For P3, f v( ) 1, ( )1  f v2 4, ( )f v3 2

For P4, f v( )1 2, ( ) 1, ( )f v2  f v3 5, ( )f v4 3

For P5, f v( )1 2, ( ) 1, ( )f v2  f v3 6, ( )f v4 4, ( )f v5 7

Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856)

For P7, f v( ) 1, ( )1  f v2 8, ( )f v3 3, ( )f v4 4, ( )f v5 5, ( )f v6 6, ( ) 10f v7 

For P8, f v( ) 1, ( )1  f v2 9, ( )f v3 2, ( )f v4 6, ( )f v5 5, ( )f v6 7, ( )f v7 3, ( ) 13f v8 

For P₉, f v( ) 1, ( ) 10, ( )₁  f v₂  f v₃ 2, ( ) 11, ( )f v₄  f v₅ 8, ( )f v₆ 3, ( )f v₇ 7, ( )f v₈ 5, ( ) 19f v₉ 

For P10, f v( ) 1, ( ) 11, ( )1  f v2  f v3 2, ( )f v4 7, ( )f v5 4, ( )f v6 9, ( )f v7 3, ( ) 13, ( )f v8  f v9 21, (f v10)5

Problem 1.17 The path Pnis Znmultiplicative graph for all n.

REFERENCES [1] F. Harary Graph Theory, Addition-Wesley, Reading, Mass, 1972

[2] David M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown

Company Publishers, 1980.

[3] J.A. Gallian, A Dynamic Survey of Graph Labeling, Electronic Journal of