*Vol. 6, No. 2, 2014, 160-169 *

*ISSN: 2279-087X (P), 2279-0888(online) *
*Published on 26 June 2014 *

www.researchmathsci.org

160

Annals of

**Some Adjacent Edge Graceful Graphs **

**T.Tharmaraj****1**** and P.B.Sarasija****2**

1

Department of Mathematics, Udaya School of Engineering Vellamodi, Tamil Nadu, PIN-629204, India, E-mail: trajtr@gmail.com

2

Department of Mathematics, Noorul Islam University

Kumaracoil, Tamil Nadu, PIN- 629180, India, E-mail: sijavk@gmail.com

*Received 17 June 2014; accepted 22 June 2014 *

**Abstract. Let ***G*(*V*,*E*)be a graph with *p vertices and * *q* edges. A (*p*,*q*) graph
)

,
(*V* *E*

*G* is said to be an adjacent edge graceful graph if there exists a bijection
}

,..., 3 , 2 , 1 { ) (

:*E* *G* *q*

*f* → such that the induced mapping *f* * from *V(G*) by
)

( )

(

* =

## ∑

*i*
*i*
*e*
*f*
*u*

*f* over all edges *e _{i}* incident to adjacent vertices of

*u*is an injection.

*The function f is called an adjacent edge graceful labeling of G .In this paper, we prove *

the graphs *Cm*∪*Cn*(*m*≥5,*n*≥5), *Pm*∪*Cn*(*m*≥5,*n*≥5),

) 2 (

*n*

*C* ,*mK*3 and sunflower

graph *SF(n*) (*n* is odd ) are the adjacent edge graceful graphs and we also prove

*graphs B _{n}_{,n}*2 ,

*K*

_{2}+

*mK*

_{1}and

*mK*

_{2}

*are not the adjacent edge graceful graphs.*

**Keywords: adjacent edge graceful graph, adjacent edge graceful labeling. ****AMS Mathematical Subject Classification (2010): 05C78 **

**1. Introduction **

Throughout this paper, by a graph we mean a finite, undirected, simple graph. The vertex
set and the edge set of a graph *G* are denoted by *V*(*G*)and*E*(*G*) respectively. Let

)
,
(*p* *q*

*G* be a graph with*p*=|*V*(*G*)| vertices and *q*=|*E*(*G*)| edges. Graph labeling,
where the vertices and edges are assigned real values or subsets of a set are subject to
certain conditions. A detailed survey of graph labeling can be found in [2].Terms not
defined here are used in the sense of Harary in [5].The concept of adjacent edge graceful
labeling was first introduced in [18].Some results on adjacent edge graceful labeling of
graphs and some non- adjacent edge graceful graphs are discussed in [18].

In this paper, we discussed about adjacent edge graceful labeling for a few more graphs.We use the following definitions in the subsequent sections.

161

mapping *f* * from *V(G*) by *( )=

## ∑

( )*i*

*i*
*e*
*f*
*u*

*f* over all edges *e _{i}* incident to

adjacent vertices of *u is an injection. The function f is called an adjacent edge graceful *
*labeling of G. *

**Definition 1.2. [2] The bistar graph ***Bn*,*n* is the graph obtained from two copies of star

*n*

*K*1, by joining the vertices of maximum degree by an edge.

**Definition 1.3. [2] A sunflower graph SF(n) as the graph obtained by starting with an **

n-cycle with consecutive vertices *v*1,*v*2,...,*vn* and creating new vertices *w*1,*w*2,...,*wn*

with *w _{i}* connected to

*v*and

_{i}*v*

_{i}_{+}

_{1}(

*v*

_{n+}_{1}

*isv*

_{1})

**.**

**Definition 1.4. [15] For a simple connected graph ***G* the square of graph *G*is denoted
by *G*2and defined as the graph with the same vertex set as of *G* and two vertices are
adjacent in *G*2 if they are at a distance 1 or 2 apart in *G*.

**2. Main results **

**Theorem 2.1. The graph ***Cm*∪*Cn*(*m*≥5,*n*≥5) is an adjacent edge graceful graph.

**Proof: Let ***Cm*=*u*1*u*2...*umu*1 and *Cn*=*v*1*v*2*... vvn* 1 be two cycles.

Let

= −

≤ ≤ =

= −

≤ ≤ =

= ∪

+ +

+ +

*n*
*n*
*m*
*i*

*i*
*i*
*m*

*m*
*m*
*i*

*i*
*i*
*n*
*m*

*v*
*v*
*e*
*n*

*i*
*v*

*v*
*e*

*u*
*u*
*e*
*m*
*i*
*u*

*u*
*e*
*C*
*C*
*E*

1 1

1 1

; 1 1

:

; 1 1

: )

(

Define *f* :*E*(*C _{m}*∪

*C*)→{1,2,3,...,

_{n}*m*+

*n*} by

**Case (i) (***m* and *n* are both odd )
*n*

*m*
*i*
*if*
*i*
*e*

*f*( * _{i}*)= 1≤ ≤ + . Let

*f** be the induced vertex labeling of

*f*. In this case the induced vertex labels are as follows:

*f**(

*u*

_{1})=2

*m*+2 ;

*f**(

*u*

_{2})=

*m*+6;

3 1

6 4 ) (

* *u*_{+}_{2} = *i*+ *if* ≤*i*≤*m*−

*f* * _{i}* ;

*f**(

*u*)=3

_{m}*m*−2 ;

*f**(

*v*

_{1})=4

*m*+2

*n*+2 ; 6

4 ) (

* *v*_{2} = *m*+*n*+

*f* ; *f* *(*v _{n}*)=4

*m*+3

*n*−2; 3 1

6 4 4 ) (

* *v*+2 = *m*+ *i*+ *if* ≤*i*≤*n*−

*f* *i* .

**Case (ii) (***m* is odd and *n* is even )

*m*
*i*
*if*
*i*
*e*

*f*( * _{i}*)= 1≤ ≤ ;

2 1

1 2 )

(*e* *m* *i* *if* *i* *n*

*f* _{m}_{+}* _{i}* = + − ≤ ≤ ;

2 1 2 2 )

(

2 2 2

*n*
*i*
*if*
*i*
*n*

*m*
*e*

*f* *m*+*n*+ *i* = + + − ≤ ≤ .

In this case the induced vertex labels are as follows: 2

2 ) (

* *u*_{1} = *m*+

*f* ; *f* *(*u*_{2})=*m*+6; *f* *(*u _{m}*)=3

*m*−2; 3

1 6 4 ) (

* *u*_{+}_{2} = *i*+ *if* ≤*i*≤*m*−

*f* * _{i}* ;

If *n*=6, *f* *(*v*1)=4*m*+*n*+4; *f* *(*v*2)=4*m*+*n*+5;

3 2 4 ) (

* *v*3 = *m*+ *n*+

162 5

2 4 ) (

* *v*5 = *m*+ *n*+

*f* ; *f* *(*v*6)=4*m*+2*n*+1;

If *n*≥8 , *f* *(*v*_{1})=4*m*+10 ; *f* *(*v*_{2})=4*m*+11 ; *f* *(*v _{n}*)=4

*m*+13;

2 6 1

) 1 ( 8 4 ) (

* 2

− ≤ ≤ +

+ =

+

*n*
*i*
*if*
*i*
*m*
*v*

*f* *i* ; *( ) 4 4 9

2

−
+
= *m* *n*
*v*

*f* *n* ;

6 4 4 ) ( *

2

2 = + −

+ *m* *n*

*v*

*f* *n* ; *( ) 4 4 7

2

4 = + −

+ *m* *n*

*v*

*f* *n* ;

2 6 1

) 3 2 ( 4 4 ) (

* *v* _{−} = *m*+ *i*+ *if* ≤*i*≤ *n*−

*f* *n* *i* .

**Case (iii) (***m* is even and *n* is odd )

2 1 1 2 )

(*e* *i* *if* *i* *m*

*f* *i* = − ≤ ≤ ;

2 1 2 2 )

(

2 2

*m*
*i*
*if*
*i*
*m*

*e*

*f* _{m}_{+} * _{i}* = + − ≤ ≤ ;

*n*
*i*
*if*
*i*
*m*
*e*

*f*( _{m}_{+}* _{i}*)= + 1≤ ≤ .

In this case the induced vertex labels are as follows:

If *m*=6 , *f* *(*u*_{1})=2*m*−2; *f* *(*u*_{2})=2*m*−1; *f**(*u*_{3})=3*m*−3;
*m*

*u*

*f**( _{4})=3 ; *f**(*u*_{5})=3*m*−1; *f* *(*u*_{6})=2*m*+1;

If *m*≥8, *f* *(*u*1)=10; *f**(*u*2)=11;

2 6 1

) 1 ( 8 ) (

* 2

− ≤ ≤ +

=

+

*m*
*i*
*if*
*i*
*u*

*f* *i* ;

13
)
(
* *um* =

*f* ; *( ) 4 9

2

−
= *m*
*u*

*f* *m* ; *( ) 4 6

2

2 = −

+ *m*

*u*

*f* *m* ; *( ) 4 7

2

4 = −

+ *m*

*u*

*f* *m* ;

2 6 1

) 3 2 ( 4 ) (

* *u* _{−} = *i*+ *if* ≤*i*≤*m*−

*f* *m* *i* ; *f* *(*v*1)=4*m*+2*n*+2;

6 4

) (

* *v*_{2} = *m*+*n*+

*f* ; *f* *(*v _{n}*)=4

*m*+3

*n*−2; 3 1

6 4 4 ) (

* *v*_{+}_{2} = *m*+ *i*+ *if* ≤*i*≤*n*−

*f* * _{i}* .

**Case (iv) (***m* and *n* are both even)

2 1 1 2 )

(*e* *i* *if* *i* *m*

*f* * _{i}* = − ≤ ≤ ;

2 1 2 2 )

(

2 2

*m*
*i*
*if*
*i*
*m*

*e*

*f* _{m}_{+} * _{i}* = + − ≤ ≤

2 1

1 2 )

(*e* *m* *i* *if* *i* *n*

*f* *m*+*i* = + − ≤ ≤ ;

2 1

2 2 )

(

2 2 2

*n*
*i*
*if*
*i*
*n*

*m*
*e*

*f* _{m}_{+}_{n}_{+} * _{i}* = + + − ≤ ≤ .

In this case the induced vertex labels are as follows:

If *m*=6 and *n*=6, *f* *(*u*1)=2*m*−2; *f**(*u*2)=2*m*−1; *f**(*u*3)=3*m*−3;

*m*
*u*

*f**( 4)=3 ; *f**(*u*5)=3*m*−1; *f* *(*u*6)=2*m*+1; *f**(*v*1)=5*n*+4;

5 5 ) (

* *v*_{2} = *n*+

*f* ; *f* *(*v*_{3})=6*n*+3; *f* *(*v*_{4})=7*n* ; *f**(*v*_{5})=7*n*+1;
1

6 ) (

* *v*_{6} = *n*+

*f* .

If *m*=6 and *n*≥8, *f* *(*u*1)=2*m*−2 ; *f**(*u*2)=2*m*−1;

3 3 ) (

* *u*3 = *m*−

163 1

2 ) (

* *u*6 = *m*+

*f* ; *f* *(*v*_{1})=4*m*+10 ; *f* *(*v*_{2})=4*m*+11;

13 4 ) (

* *v* = *m*+

*f* *n* ;

2 6 1

) 1 ( 8 4 ) (

* 2

− ≤ ≤ +

+ =

+

*n*
*i*
*if*
*i*
*m*
*v*

*f* *i* ;

9 4 4 ) ( *

2

−
+
= *m* *n*
*v*

*f* *n* *( ) 4 4 6

2

2 = + −

+ *m* *n*

*v*

*f* *n* ; *( ) 4 4 7

2

4 = + −

+ *m* *n*

*v*

*f* *n* ;

2 6 1

) 3 2 ( 4 4 ) (

* *v* _{−} = *m*+ *i*+ *if* ≤*i*≤ *n*−

*f* *n* *i* .

If *m*≥8 and *n*≥8, *f**(*u*1)=10 ; *f* *(*u*2)=11 ;

2 6 1

) 1 ( 8 ) (

* 2

− ≤ ≤ +

=

+

*m*
*i*
*if*
*i*
*u*

*f* *i* ; *f**(*um*)=13 ; *( ) 4 9

2

−
= *m*
*u*

*f* *m* ;

6 4 ) ( *

2

2 = −

+ *m*

*u*

*f* *m* ; *( ) 4 7

2

4 = −

+ *m*

*u*

*f* *m* ;

2 6 1

) 3 2 ( 4 ) (

* *u* _{−} = *i*+ *if* ≤*i*≤*m*−

*f* *m* *i* ;

10 4 ) (

* *v*1 = *m*+

*f* ; *f**(*v*2)=4*m*+11 ; *f**(*vn*)=4*m*+13 ;

2 6 1

8 8 4 ) (

* *v*_{+}_{2} = *m*+ + *i* *if* ≤*i*≤*n*−

*f* * _{i}* ;

2 6 1

8 12 4 ) (

* *v* _{−} = *m*+ + *i* *if* ≤*i*≤*n*−

*f* *n* *i* ; *( ) 4 4 9

2

−
+
= *m* *n*
*v*

*f* *n* ;

6 4 4 ) ( *

2

2 = + −

+ *m* *n*

*v*

*f* *n* ; *( ) 4 4 7

2

4 = + −

+ *m* *n*

*v*

*f* *n* .

If *m*≥8 and *n*=6, *f**(*u*_{1})=10 ; *f* *(*u*_{2})=11 ;

2 6 1

) 1 ( 8 ) (

* *u*_{+}_{2} = *i*+ *if* ≤*i*≤*m*−

*f* * _{i}* ;

*f**(

*u*)=13 ; *( ) 4 9

_{m}2

−
= *m*
*u*

*f* * _{m}* ;

6 4 ) ( *

2

2 = −

+ *m*

*u*

*f* * _{m}* ; *( ) 4 7

2

4 = −

+ *m*

*u*

*f* * _{m}* ;

2 6 1

) 3 2 ( 4 ) (

* *u* _{−} = *i*+ *if* ≤*i*≤*m*−

*f* *m* *i* ;

4 4

) (

* *v*_{1} = *m*+*n*+

*f* ; *f* *(*v*_{2})=4*m*+*n*+5; *f* *(*v*_{3})=4*m*+2*n*+3;
*n*

*m*
*v*

*f**( 4)=4 +3 ; *f**(*v*5)=4*m*+2*n*+5 ; *f* *(*v*6)=4*m*+2*n*+1.

**Example 2.2. An adjacent edge graceful labeling of ***C*_{12}∪*C*_{10} is shown in the Fig.1.

164

**Theorem 2.3. The graph ***Pm*∪*Cn*(*m*≥5,*n*≥5) is an adjacent edge graceful graph.

**Proof: Let ***P _{m}*=

*u*

_{1}

*u*

_{2}...

*u*be a path and

_{m}*C*=

_{n}*v*

_{1}

*v*

_{2}

*... vv*

_{n}_{1}be a cycle .

Let
=
−
≤
≤
=
−
≤
≤
=
=
∪
−
+
+
+
−
+
*n*
*n*
*m*
*i*
*i*
*i*
*m*
*i*
*i*
*i*
*n*
*m*
*v*
*v*
*e*
*n*
*i*
*v*
*v*
*e*
*m*
*i*
*u*
*u*
*e*
*C*
*P*
*E*
1
1
1
1
1
;
1
1
:
;
1
1
:
)
(

Define *f* :*E*(*P _{m}*∪

*C*)→{1,2,3,...,

_{n}*m*+

*n*−1} by

**Case (i) (***m* and *n* are both odd )
1
1

)

(*e* =*i* *if* ≤*i*≤*m*+*n*−

*f* *i* **. **

In this case the induced vertex labels are as follows: 3

)
(
* *u*1 =

*f* ; *f* *(*u*2)=6 ; *f* *(*ui*+2)=4*i*+6 *if* 1≤*i*≤*m*−4; *f**(*um*−1)=3*m*−6;

3 2 ) (

* *u* = *m*−

*f* *m* ; *f**(*v*1)=4*m*+2*n*−2 ; *f* *(*v*2)=4*m*+*n*+2 ;

3 3 4 ) (

* *v* = *m*+ *n*−

*f* * _{n}* ;

*f**(

*v*

_{i}_{+}

_{2})=4

*m*+4

*i*+2

*if*1≤

*i*≤

*n*−3 .

**Case (ii) (***m* is odd and *n* is even )

1 1

)

(*e* =*i* *if* ≤*i*≤*m*−

*f* *i* ;

2
1
2
2
)
( 1
*n*
*i*
*if*
*i*
*m*
*e*

*f* *m*−+*i* = + − ≤ ≤ ;

2
1
2
1
)
(
2
2
2
2
*n*
*i*
*if*
*i*
*n*
*m*
*e*

*f* _{m}_{+}_{n}_{−} _{+} * _{i}* = + + − ≤ ≤ .

In this case the induced vertex labels are as follows: 3

)
(
* *u*_{1} =

*f* ; *f* *(*u*_{2})=6 ; *f* *(*u _{m}*

_{−}

_{1})=3

*m*−6 ; 4 1 6 4 ) (

* *u*+2 = *i*+ *if* ≤*i*≤*m*−

*f* *i* ; *f* *(*um*)=2*m*−3.

If *n*=6 , *f* *(*v*1)=4*m*+*n* ; *f* *(*v*2)=4*m*+*n*+1; *f* *(*v*3)=4*m*+*n*+5;

2 2 4 ) (

* *v*_{4} = *m*+ *n*+

*f* ; *f**(*v*_{5})=4*m*+2*n*+1 ; *f**(*v*_{6})=4*m*+*n*+3.
If *n*≥8 , *f* *(*v*1)=4*m*+6 ; *f**(*v*2)=4*m*+7;

2
6
1
8
4
4
)
(
* 2
−
≤
≤
+
+
=
+
*n*
*i*
*if*
*i*
*m*
*v*

*f* *i* ; *( ) 4 4 13

2

−
+
= *m* *n*
*v*

*f* *n* ;

10 4 4 ) ( * 2

2 = + −

+ *m* *n*

*v*

*f* * _{n}* ; *( ) 4 4 11

2

4 = + −

+ *m* *n*

*v*

*f* * _{n}* ;

2
6
1
8
8
4
4
)
(
*
2
2
4
−
≤
≤
−
−
+
=
+
+
*n*
*i*
*if*
*i*
*n*
*m*
*v*

*f* *n* *i* ; *f* *(*vn*)=4*m*+9 .

**Case (iii) (***m* is even and *n* is odd )

2 1 1 2 )

(*e* *i* *if* *i* *m*

*f* * _{i}* = − ≤ ≤ ;

2
2
1
2
)
(
2
2
−
≤
≤
−
=
+
*m*
*i*
*if*
*i*
*m*
*e*

*f* _{m}* _{i}* ;

*n*
*i*
*if*
*i*
*m*
*e*

*f*( *m*−1+*i*)= −1+ 1≤ ≤ .

In this case the induced vertex labels are as follows:

If *m*=6 , *f* *(*u*1)=*m*−2 ; *f* *(*u*2)=*m*+3; *f* *(*u*3)=2*m*+1;

*m*
*u*

165

If *m*=8 , *f* *(*u*_{1})=*m*−4 ; *f**(*u*_{2})=*m*+1; *f**(*u*3)=2*m*; *f* *(*u*4)=2*m*+5 ;

*m*
*u*

*f**( _{5})=3 ; *f**(*u*_{6})=2*m*+3; *f* *(*u*_{7})=2*m*+2 *f* *(*u*_{8})=*m*−2.

If *m*≥10, *f**(*u*1)=4 ; *f* *(*u*2)=9 ;

2
6
1
8
8
)
(
* 2
−
≤
≤
+
=
+
*m*
*i*
*if*
*i*
*u*

*f* *i* ;

11
4
)
(
*
2
−
= *m*
*u*

*f* *m* ; *( ) 4 10

2

2 = −

+ *m*

*u*

*f* *m* ; *( ) 4 13

2

4 = −

+ *m*

*u*

*f* *m* ;

2
8
1
8
12
4
)
(
*
2
2
4
−
≤
≤
−
−
=
+
+
*m*
*i*
*if*
*i*
*m*
*u*

*f* *m* *i* ; *f* *(*um*−1)=12 ; *f**(*um*)=6

2 2 4 ) (

* *v*_{1} = *m*+ *n*−

*f* ; *f* *(*v*_{2})=4*m*+*n*+2 ; *f* *(*v _{n}*)=4

*m*+3

*n*−6 ; 3 1 2 4 4 ) (

* *v*_{+}_{2} = *m*+ *i*+ *if* ≤*i*≤*n*−

*f* * _{i}* .

**Case (iv) (***m* and *n* are both even )

2 1 1 2 )

(*e* *i* *if* *i* *m*

*f* * _{i}* = − ≤ ≤ ;

2
2
1
2
)
(
2
2
−
≤
≤
−
=
+
*m*
*i*
*if*
*i*
*m*
*e*

*f* _{m}* _{i}* ;

2 1 2 2 )

(*e* _{1} *m* *i* *if* *i* *n*

*f* _{m}_{−}_{+}* _{i}* = + − ≤ ≤ ;

2
1
2
1
)
(
2
2
2
2
*n*
*i*
*if*
*i*
*n*
*m*
*e*

*f* *m*+*n*− + *i* = + + − ≤ ≤ .

In this case the induced vertex labels are as follows:

If *m*=6 , *f* *(*u*1)=*m*−2 ; *f* *(*u*2)=*m*+3; *f* *(*u*3)=2*m*+1;

*m*
*u*

*f**( 4)=2 ; *f* *(*u*5)=2*m*−5; *f* *(*u*6)=6.

If *m*=8 , *f* *(*u*_{1})=*m*−4 ; *f**(*u*_{2})=*m*+1; *f**(*u*_{3})=2*m*; *f* *(*u*_{4})=2*m*+5 ;
*m*

*u*

*f**( _{5})=3 ; *f**(*u*_{6})=2*m*+3; *f* *(*u*_{7})=2*m*+2 *f* *(*u*_{8})=*m*−2.

If *m*≥10, *f**(*u*1)=4 ; *f* *(*u*2)=9 ;

2
6
1
8
8
)
(
* 2
−
≤
≤
+
=
+
*m*
*i*
*if*
*i*
*u*

*f* *i* ;

11
4
)
(
*
2
−
= *m*
*u*

*f* *m* ; *( ) 4 10

2

2 = −

+ *m*

*u*

*f* *m* ; *( ) 4 13

2

4 = −

+ *m*

*u*

*f* *m* ;

2
8
1
8
12
4
)
(
*
2
2
4
−
≤
≤
−
−
=
+
+
*m*
*i*
*if*
*i*
*m*
*u*

*f* *m* *i* ; *f* *(*um*−1)=12 ; *f**(*um*)=6

If *n*=6 , *f* *(*v*1)=4*m*+*n* ; *f* *(*v*2)=4*m*+*n*+1; *f* *(*v*3)=4*m*+*n*+5;

2 2 4 ) (

* *v*_{4} = *m*+ *n*+

*f* ; *f**(*v*_{5})=4*m*+2*n*+1 ; *f**(*v*_{6})=4*m*+*n*+3.
If *n*≥8 , *f* *(*v*_{1})=4*m*+6 ; *f**(*v*_{2})=4*m*+7;

2 6 1 8 4 4 ) (

* *v*_{+}_{2} = *m*+ + *i* *if* ≤*i*≤*n*−

*f* * _{i}* ; *( ) 4 4 13

2

−
+
= *m* *n*
*v*

*f* * _{n}* ;

10 4 4 ) ( * 2

2 = + −

+ *m* *n*

*v*

*f* * _{n}* ; *( ) 4 4 11

2

4 = + −

+ *m* *n*

*v*

*f* * _{n}* ;

2
6
1
8
8
4
4
)
(
*
2
2
4
−
≤
≤
−
−
+
=
+
+
*n*
*i*
*if*
*i*
*n*
*m*
*v*

166

**Example 2.4. An adjacent edge graceful labeling of ***C*9∪*P*7 is shown in the Fig.

2.

** **

**Figure 2: **

**Theorem 2.5. The graph ***Cn*(2) is an adjacent edge graceful graph.

**Proof: Let ' ***w* ' be the central vertex of *Cn*(2).Let {*ui* :1≤*i*≤*n*}be the vertices of

first cycle of *Cn*(2). Let {*vi* :1≤*i*≤*n*} be the vertices of second cycle of

) 2 (

*n*
*C* .

Let

= −

≤ ≤ =

= −

≤ ≤ =

=

+ +

+

*n*
*n*
*i*

*i*
*i*
*n*

*n*
*n*
*i*

*i*
*i*
*n*

*v*
*v*
*e*
*n*

*i*
*v*

*v*
*e*

*u*
*u*
*e*
*n*

*i*
*u*

*u*
*e*
*C*

*E*

1 2 1

1 1

) 2 (

, 1 1

:

, 1 1

: )

(

Take *w*=*u*_{1}=*v*_{1} .Define *f* :*E*(*Cn*(2))→{1,2,3,...,2*n*} by
*n*

*i*
*if*
*i*
*e*

*f*( *i*)= 1≤ ≤ . Let *f* * be the induced vertex labeling of *f* .

The induced vertex labels are as follows: *f* *(*u*_{1})=8*n*+4 ; *f* *(*u*_{2})=4*n*+7;
1

6 ) (

* *u* = *n*−

*f* * _{n}* ;

*f**(

*u*

_{i}_{+}

_{2})=4

*i*+6

*if*1≤

*i*≤

*n*−3;

*f**(

*v*

_{2})=6

*n*+7; 1

8 ) (

* *v* = *n*−

*f* * _{n}* ;

*f**(

*v*

_{i}_{+}

_{2})=4

*n*+4

*i*+6

*if*1≤

*i*≤

*n*−3.

**Theorem 2.6. The graph ***K*_{2} +*mK*_{1} is not an adjacent edge graceful graph.

**Proof: Let ***G*=*K*_{2} +*mK*_{1}. Let *V*(*G*)=

### {

*u*,

*v*,

*w*:1≤

_{i}*i*≤

*m*}. Let

*E*(

*G*)=

### {

*ei*=

*uwi*,

*em*+

*i*=

*vwi*:1≤

*i*≤

*m*;

*e*2

*m*+1 =

*uv*}. Let

*f*

*be a bijection from E*(

*G*) to {1,2,3,...,2

*m*+1}.

Let

### α

_{1},

### α

_{2},....,

### α

*,*

_{m}### β

_{1},

### β

_{2},....,

### β

*,*

_{m}### α

be the label of edges*e*

_{1},

*e*

_{2},....,

*e*,

_{m}*e*

_{m}_{+}

_{1},....,

*e*

_{2}

_{m}_{+}

_{1}

respectively by *f* . And *f* induces that *f**:*V*(*G*)→{1,2,3,...} by
)

( )

(

* =

## ∑

*i*
*i*
*e*
*f*
*u*

*f* over all edges *e _{i}* incident to adjacent vertices of

*u*. Then

) ( ) ( )

( ) ( )

(

* 2 1

2

1 1

2 1

+ +

= + =

+ +

+

=

## ∑

## ∑

*m*

*m*

*m*
*i*

*i*
*m*

*m*

*i*
*i*

*i* *f* *e* *f* *e* *f* *e* *f* *e*

*w*

*f* =

## ∑

### α

+## ∑

### β

+### α

+### α

= =

*m*

*i*
*i*
*m*

*i*
*i*

167

**=**1+2+3+...+(2*m*+1)+

### α

=(*m*+1)(2

*m*+1)+

### α

for 1≤*i*≤

*m*. That is every vertex

*wi*have same vertex labels by

*f**. Hence

*f** is not injective.Therefore the

graph *K*2 +*mK*1 is not an adjacent edge graceful graph.

**Theorem 2.7.** The graph *mK is not an adjacent edge graceful graph. *_{2}

**Proof: Let the vertex set of ** *mK be *_{2} *V* =*V*_{1}∪*V*_{2}∪...∪*V _{m}* where

}
,
{ * _{i}*1

*2*

_{i}*i* *v* *v*

*V* = .Let *E*(*mK*_{2})={*e _{i}* =

*v*1

_{i}*v*2 :1≤

_{i}*i*≤

*m*}

*. Let f be a bijection*from

*E*(

*mK*2) to {1,2,3,...,

*m .*} And

*f*induces that

,....} 3 , 2 , 1 { ) ( :

* *V* *mK*2 →

*f* by *( )=

## ∑

( )*i*
*i*
*e*
*f*
*u*

*f* over all edges *e incident to i*

adjacent vertices of *u .The vertices vi*1 and

2

*i*

*v* are incident with same and only

one edge *e _{i}*.Then

*f**(

*v*1)=

_{i}*f**(

*v*2)=

_{i}*f*(

*e*).That is each vertex pairs (

_{i}*v*1,

_{i}*v*2)

_{i}*have same vertex labels by f *. Hence f * is not injective.*

Therefore the graph *mK is not an adjacent edge graceful graph. *2

**Theorem 2.8. **The graph *mK*_{3} is an adjacent edge graceful graph.

**Proof: Let the vertex set of ***mK*3 be *V* =*V*1∪*V*2∪...∪*Vm* where
}

,
,
{ * _{i}*1

*2*

_{i}*3*

_{i}*i* *v* *v* *v*

*V* = .

Let *E*(*mK*_{3})={*e _{i}* =

*v*1

_{i}*v*2 ,

_{i}*e*

_{m}_{+}

*=*

_{i}*v*2

_{i}*v*3 ,

_{i}*e*

_{2}

_{m}_{+}

*=*

_{i}*v*1

_{i}*v*3 :1≤

_{i}*i*≤

*m*}. Define

*f*:

*E*(

*mK*3)→{1,2,3,...,3

*m*} by

*f*(

*ei*)=3

*i*−2

*if*1≤

*i*≤

*m*;

*m*
*i*
*if*
*i*
*e*

*f*( *m*+*i*)=3 −1 1≤ ≤ ; *f*(*e*2*m*+*i*)=3*i* *if* 1≤*i*≤*m*.
Let *f* * be the induced vertex labeling of *f* .

The induced vertex labels are as follows: *f* *(*v _{i}*1)=12

*i*−4

*if*1≤

*i*≤

*m*;

*m*
*i*
*if*
*i*
*v*

*f* *( *i* )=12 −3 1≤ ≤

2

; *f* *(*vi* )=12*i*−5 *if* 1≤*i*≤*m*

3

.

**Example 2.9. An adjacent edge graceful labeling of ***5K*_{3}** is shown in the Fig. 3. **

168

**Theorem 2.10. The graph ***B _{n}_{.n}*2 is not an adjacent edge graceful graph.

**Proof: Let ***V*(*B _{n}*

_{.}

*2)={*

_{n}*u*,

_{i}*v*: 1≤

_{i}*i*≤

*n*+1}.

Let

= ≤

≤ =

=

≤ ≤ =

= =

+ + + +

+ + +

+ +

+

1 1 1 4 1

3 1 2

1 1

2 .

; 1

: ,

1 : ,

) (

*n*
*n*
*n*
*n*

*i*
*i*
*n*
*n*
*i*
*i*
*n*

*n*
*i*
*i*
*n*
*n*
*i*
*i*
*n*

*n*

*v*
*u*
*e*
*n*
*i*
*u*

*v*
*e*
*v*
*u*
*e*

*n*
*i*
*v*

*v*
*e*
*u*
*u*
*e*
*B*

*E*

Let *f* * be a bijection from E*(*Bn,n*2) to {1,2,3,...,4*n*+1}.

Let

### α

1,### α

2,....,### α

*n*,

### α

*n*+1,....,

### α

2*n*,

### α

2*n*+1,....,

### α

3*n*,

### α

3*n*+1,...,

### α

4*n*,

### α

be the label of edges of1 4 1 3 3 1 2 2 1 2

1,*e* ,....,*en*,*en*+,....,*e* *n*,*e* *n*+ ,....,*en*,*e* *n*+ ,...,*e* *n*+

*e* respectively by *f* . And *f*

induces that *f**:*V*(*Bn*,*n*2)→{1,2,3,....} by *( )=

## ∑

( )*i*

*i*
*e*
*f*
*u*

*f* over all edges *ei*

incident to adjacent vertices of *u*. Then *( ) ( ) 2 ( 4 1)

4

1

+ =

+

=

## ∑

*n*

*n*

*i*
*i*

*i* *f* *e* *f* *e*

*u*

*f* =

### α

+ + + +

+

+2 3 ... (2 1)

1 *n* = (2*n*+1)(4*n*+1)+

### α

for 1≤*i*≤

*n*.That is every vertex

*u*have same vertex labels by

_{i}*f**. Hence

*f** is not injective. Therefore the

graph *B _{n}_{.n}*2 is not an adjacent edge graceful graph.

**Theorem 2.11. The sunflower graph ***SF*(*n*) (*nisodd*) is an adjacent edge graceful
graph.

**Proof: Let ***V*(*SF*(*n*))=

### {

*v*,

_{i}*w*:1≤

_{i}*i*≤

*n*}.Take

*v*

_{n}_{+}

_{1}=

*v*

_{1}.

Let *E*(*SF*(*n*))=

### {

*ei*=

*vivi*+1 ,

*en*+

*i*=

*viwi*,

*e*2

*n*+

*i*=

*vi*+1

*wi*:1≤

*i*≤

*n*}. Define

*f*:

*E*(

*SF*(

*n*))→{1,2,3,...,3

*n*} by

*f*(

*ei*)=

*i*

*if*1≤

*i*≤3

*n*;

Let *f* * be the induced vertex labeling of *f* . The induced vertex labels are as follows:
6

18 ) (

* *v*_{1} = *n*+

*f* ; *f* *(*v*_{2})=14*n*+18 ; *f* *(*v _{n}*)=22

*n*−6 ; 3

1 12 18 12 ) (

* *v*+2 = *n*+ + *i* *if* ≤*i*≤*n*−

*f* *i* ;*f* *(*w*1)=8*n*+8;

*n*
*w*

*f* *( * _{n}*)=12 ;

*f**(

*w*

_{i}_{+}

_{1})=6

*n*+8+8

*i*

*if*1≤

*i*≤

*n*−2.

**Example 2.12. An adjacent edge graceful labeling of ***SF*(7) is shown in the Fig. 4.

** **

169

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