*Vol. 6, No. 1, 2014, 25-35*

*ISSN: 2279-087X (P), 2279-0888(online) *
*Published on 5 May 2014 *

www.researchmathsci.org

25

Annals of

** **

**Adjacent Edge Graceful Graphs **

* T. Tharmaraj1 and P.B.Sarasija2*
1

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

2

Department of Mathematics, Noorul Islam University Kumaracoil, Tamil Nadu, India. E-mail: sijavk@gmail.com

*Received 6 April 2014; accepted 20 April 2014 *

* Abstract. Let G*(

*V*,

*E*)be a graph with

*p*vertices and

*q*edges. A (

*p*,

*q*) graph

*G*(

*V*,

*E*)

is said to be an adjacent edge graceful graph if there exists a bijection *f E G*: ( )
{1, 2,3,..., }*q*

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

( )*i*
*i*

*f e*

=

## ∑

over all edges*ei*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 path *

)
5
(*n*≥

*P _{n}* , cycle

*C*(

_{n}*n*≥5), fan

*f*(

_{n}*n*≥3)

*, Helm*

*H*(

_{n}*n*≥3), triangle snake

)
3
(*n*≥

*T _{n}* and alternate triangle snake

*A*(

*T*)(

_{n}*n*=4,6,8,...)are the adjacent edge graceful graphs and we also prove

*n-bistar Bn*,

*n*, complete bipartite graph

*Km*,

*n*, star

*graph K*

_{1}

_{,}

*and graph*

_{n}*g*are not the adjacent edge graceful graphs.

_{n}**Keywords: adjacent edge graceful graph, adjacent edge graceful labeling ****AMS Mathematical Subject Classification (2010): 05C78 **

**1. Introduction **

All graphs in this paper are finite, simple and undirected graphs. Let (*p*,*q*)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 [1].Terms not defined here are used in
the sense of Harary in [3].The concept of edge graceful labeling was first introduced in
[2] and the concept of strong edge graceful labeling was introduced in [4]. Some results
on strong edge graceful labeling of graphs are discussed in [4]. In this paper, we
introduced a new edge graceful labeling. We use the following definitions in the
subsequent sections.

26
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.[1] The bistar graph ***B _{n}*,

*is the graph obtained from two copies of star*

_{n}*n*

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

**Definition 1.3.[1] The helm ***H _{n}* is the graph obtained from a wheel by attaching a
pendent edge at each vertex of the n-cycle.

**Definition 1.4.[1] The fan ** *fn*(*n*≥2) is obtained by joining all nodes of *Pn* to a further
node called the center and contains *n*+1 nodes and 2*n*−1 edges.

**Definition 1.5.[1] A graph ***gn*(*n*≥2) be a graph with *n*+2 nodes and 3*n*−1 edges

obtained by joining all nodes of *P _{n}* to two additional nodes.

**Definition 1.6.[1] A triangular snake ***Tn* is obtained from a path *u*1*u*2...*un* by
joining *u _{i}* and

*u*

_{i}_{+}

_{1}to a new vertex

*v*for i=1,2,….,n-1.

_{i}**Definition 1.7.[5] An alternate triangular snake ** *A*(*Tn*) is obtained from a path

*n*
*u*
*u*

*u*_{1} _{2}... by joining *u _{i}* and

*u*

_{i}_{+}

_{1}(alternatively) to new vertex

*v*.

_{i}**2. Main results **

**Theorem 2.1. The triangle snake ***T _{n}*(

*n*≥3)

**admits an adjacent edge graceful labeling.**

**Proof: Let ***V*(*Tn*)={*ui* :1≤*i*≤*n*; *vi* :1≤*i*≤*n*−1}

Let

− ≤ ≤ =

− ≤ ≤ =

− ≤ ≤ =

=

+ + −

+ −

+

1 1

1 1

1 1

) (

1 1

2 1

1

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

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

*n*
*i*
*if*
*u*
*u*
*e*

*T*
*E*

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

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

*i*
*i*
*i*

*n*

Define a bijection *f* :*E*(*Tn*)→{1,2,3,....,3*n*−3} by
1

1 )

(*e* =*i* *if* ≤*i*≤*n*−

*f* *i* ; *f*(*en*−1+*i*)=*n*+2(*i*−1) *if* 1≤*i*≤*n*−1;

1 1

2 2 2 )

(*e*_{2} _{−}_{1}_{+} = *n*− + *i* *if* ≤*i*≤*n*−

*f* _{n}* _{i}* .

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

If *n*=3 , *f* *(*u*_{1})=6*n*+1 ; *f* *(*u*_{2})=10*n* ; *f* *(*u*_{3})=7*n*+2 ;

1 5 ) (

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

*f* ; *f* *(*v*_{2})=6*n*+2.

27

1 9 ) (

* *u*_{4} = *n*+

*f* ; *f* *(*v*_{1})=4*n*+3 ; *f* *(*v*_{2})=8*n*+2 ; *f* *(*v*3)=85*n* .
If *n*≥5, *f* *(*u*_{1})=4*n*+7 ; *f* *(*u*_{2})=7*n*+19 ;

40 24 ) (

* *u* _{−}_{1} = *n*−

*f* * _{n}* ;

*f**(

*u*)=14

_{n}*n*−19 ;

*f**(

*v*

_{1})=3

*n*+7 ;

16 12 ) (

* *v* −1 = *n*−

*f* *n* ; *f* *(*ui*+2)=8*n*+18+20*i* *if* 1≤*i*≤*n*−4;

3 1

12 6 4 ) (

* *v*+1 = *n*+ + *i* *if* ≤*i*≤*n*−

*f* *i* **. **

**Example 2.2. An adjacent edge graceful labeling of a triangular snake ***T*_{8} is shown in
**Fig.1. **

** Figure 1: **

**Theorem 2.3. The alternate triangular snake ***A*(*Tn*)(*n*=4,6,8,...) is an adjacent edge

**graceful graph. **

**Proof: Case (i) If the triangle starts from ***u*1**. **

Let }

2 1 : ; 1

: { )) (

(*AT* *u* *i* *n* *v* *i* *n*

*V* *n* = *i* ≤ ≤ *i* ≤ ≤ **. **

Let

≤ ≤ =

≤ ≤ =

− ≤ ≤ =

=

+ −

− + −

+

2 1

2 1

1 1

)) ( (

2 2

2 2 3

1 2 1

1

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

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

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

*u*
*e*

*T*
*A*
*E*

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

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

*i*
*i*
*i*

*n*

Define a bijection *f* :*E*(*A*(*Tn*))→{1,2,3,...,2*n*−1} by

1 1

2 )

(*e* = *i* *if* ≤*i*≤*n*−

*f* * _{i}* ;

2 1

3 4 )

(*e* _{1} *i* *if* *i* *n*
*f* _{n}_{−}_{+}* _{i}* = − ≤ ≤ ;

2 1

1 4 ) (

2 2 2 3

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

*f* _{n}_{−} _{+}* _{i}* = − ≤ ≤ .

Let *f* * be the induced vertex labeling of *f* .In this case the induced vertex label are as
follows:

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

3 6 ) (

* *u*_{4} = *n*+

28

If *n*≥6 , *f* *(*u*1)=13 ; *f* *(*u*2)=22 ;
−
≤
≤
+
=
+
2
4
1
10
32
)
(
* 2 1

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

*u*

*f* *i* ;

22 14 ) (

* *u* _{−}_{1} = *n*−

*f* * _{n}* ;

*f**(

*u*)=10

_{n}*n*−13;

− ≤ ≤ + = + 2 4 1 22 32 ) ( * 2 2

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

*u*

*f* *i* ;

12
)
(
* *v*_{1} =

*f* ;

_{−}
≤
≤
+
=
+
2
4
1
12
24
)
(

* *v* _{1} *i* *if* *i* *n*

*f* * _{i}* ; *( ) 10 12

2

−
= *n*
*v*

*f* * _{n}* .

**Case (ii) If the triangle starts from ***u*_{2}

Let }

2 1 : ; 1 : { )) (

(*AT* *u* *i* *n* *v* *i* *n*

*V* * _{n}* =

*≤ ≤*

_{i}*≤ ≤*

_{i}**.**

Let
−
≤
≤
=
_{−}
≤
≤
=
−
≤
≤
=
=
+
+
−
+
−
+
2
2
1
2
2
1
1
1
))
(
(
1
2
2
2
2
3
2
1
1
*n*
*i*
*if*
*v*
*u*
*e*
*n*
*i*
*if*
*v*
*u*
*e*
*n*
*i*
*if*
*u*
*u*
*e*
*T*
*A*
*E*
*i*
*i*
*i*
*n*
*i*
*i*
*i*
*n*
*i*
*i*
*i*
*n*

Define a bijection *f* :*E*(*A*(*Tn*))→{1,2,3,...,2*n*−3} by

1 1

1 2 )

(*e* = *i*− *if* ≤*i*≤*n*−

*f* *i* ;

−
≤
≤
−
=
+
−
2
2
1
2
4
)
( 1
*n*
*i*
*if*
*i*
*e*

*f* *n* *i* ;

_{−}
≤
≤
=
+
−
2
2
1
4
)
(
2
2
4
3
*n*
*i*
*if*
*i*
*e*

*f* _{n}* _{i}* .

In this case the induced vertex label are as follows:

If *n*=4 , *f* *(*u*1)=*n*+2 ; *f* *(*u*2)=4*n*+3 ; *f* *(*u*3)=4*n*+1 ; *f* *(*u*4)=3*n*;

2 4 ) (

* *v*1 = *n*+

*f* .

If *n*≥5 , *f* *(*u*_{1})=6 ; *f* *(*u*_{2})=19 ;
_{−}
≤
≤
−
=
2
2
1
6
24
)
(

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

*f* * _{i}* ;

12 6 ) (

* *u* = *n*−

*f* *n* ;

− ≤ ≤ − = + 2 4 1 2 32 ) ( * 2 1

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

*f* *i* ;

− ≤ ≤ + = + 2 4 1 18 32 ) ( * 2 2

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

*u*

*f* *i* ; *f* *(*un*−1)=12*n*−31.

**Example 2.4. An adjacent edge graceful labeling of an alternate triangular snake ***A*(*T*_{8})

29

**Figure 2: **

**Figure 3: **

**Theorem 2.5. The Path ***P _{n}*(

*n*≥5)

**is an adjacent edge graceful graph.**

**Proof: Let ***V*(*Pn*)={*vi* :1≤*i*≤*n*}.Let *E*(*Pn*)={*ei* =*vivi*+1 :1≤*i*≤*n*−1}**. **
Define *f* :*E*(*Pn*)→{1,2,3,...,*n*−1} by

**Case (i) n is odd , ***f*(*e _{i}*)=

*i*

*if*1≤

*i*≤

*n*−1.

**Case (ii) n is even, **

2 1

1 2 )

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

*f* *i* = − ≤ ≤ ;

2 2 1

2 )

(

2 2

− ≤ ≤ −

=

+

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

*f* *n* *i* .

Let *f* * be the induced vertex labeling of *f* .The induced vertex labels are as follows:
*If n is odd , f* *(*v*_{1})=3 ; *f* *(*v*_{2})=6; *f* *(*v _{n}*

_{−}

_{1})=3(

*n*−2);

3 2 ) (

* *v* = *n*−

*f* *n* ; *f* *(*vi*+2)=4*i*+6 *if* 1≤*i*≤*n*−4.

If *n*=6, *f* *(*v*1)=*n*−2 ; *f* *(*v*2)=*n*+3 ; *f* *(*v*3)=2*n*+1 ;
*n*

*v*

*f* *( _{4})=2 ; *f* *(*v*_{5})=2*n*−5 ; *f* *(*v*_{6})=6.

If *n*=8, *f* *(*v*1)=*n*−4 ; *f* *(*v*2)=*n*+1 ; *f* *(*v*3)=2*n* ; *f* *(*v*4)=2*n*+5 ;
*n*

*v*

30

2 6 1

) 1 ( 8 ) (

* *v*_{+}_{2} = *i*+ *if* ≤*i*≤*n*−

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

2

−
= *n*
*v*

*f* * _{n}* ; *( ) 4 10

2

2 = −

+ *n*

*v*

*f* * _{n}* ;

13 4 ) ( *

2

4 = −

+ *n*

*v*

*f* *n* ;

2 8 1

8 12 4 ) (

*

2 2 4

− ≤ ≤ −

− =

+ +

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

*v*

*f* *n* *i* .

**Example 2.6. The adjacent edge graceful labelings of ***P*_{9} and *P*_{10} are shown in the Fig.
**4 and Fig. 5 respectively. **

**Figure 4: **

**Figure 5: **

**Theorem 2.7. The cycle is an adjacent edge graceful graph. **
**Proof: Let ***V*(*Cn*)={*vi* :1≤*i*≤*n*}. Let

} ;

1 1

: {

)

(*Cn* *ei* *vivi* 1 *i* *n* *en* *v*1*vn*

*E* = = _{+} ≤ ≤ − = **. **

Define *f* :*E*(*C _{n}*)→{1,2,3,...,

*n*} by

**Case (i) n is odd ,***f*(*ei*)=*i* *if* 1≤*i*≤*n*.

**Case (ii) n is even , **

2 1

1 2 )

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

*f* *i* = − ≤ ≤ ;

2 1

) 1 ( 2 )

(

2 2

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

*f* _{n}_{+} * _{i}* = − − ≤ ≤ .

Let *f* * be the induced vertex labeling of *f* .The induced vertex labels are as follows:
*If n is odd ,f* *(*v*1)=2*n*+2 ; *f* *(*v*2)=*n*+6 ;

2 3 ) (

* *v* = *n*−

*f* * _{n}* ;

*f**(

*v*

_{i}_{+}

_{2})=4

*i*+6

*if*1≤

*i*≤

*n*−3 ;

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

*v*

*f* *( 4)=3 ; *f* *(*v*5)=2*n*+5 ; *f* *(*v*6)=2*n*+1.
If *n*≥8 , *f* *(*v*_{1})=10 ; *f* *(*v*_{2})=11 ;

2 6 1

) 1 ( 8 ) (

* *v*_{+}_{2} = *i*+ *if* ≤*i*≤ *n*−

*f* * _{i}* ;

9 4 ) ( *

2

−
= *n*
*v*

*f* * _{n}* ; *( ) 4 6

2

2 = −

+ *n*

*v*

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

2

4 = −

+ *n*

*v*

*f* * _{n}* ;

2 6 1

) 3 2 ( 4 ) (

* *v* _{−} = *i*+ *if* ≤*i*≤*n*−

31

**Example 2.8. The adjacent edge graceful labelings of ***C*_{14} and *C*_{11} are shown in the
**Fig. 5 and Fig. 6 respectively. **

** Figure 5: ** ** Figure 6: **

**Theorem 2.9. The graph ***g _{n} is not an adjacent edge graceful graph. *

**Proof: Let ***V*(*g _{n}*)={

*v*:1≤

_{i}*i*≤

*n*;

*v*

_{n}_{+}

_{1};

*v*

_{n}_{+}

_{2}}

**.**

Let

≤ ≤ =

≤ ≤ =

− ≤ ≤ =

=

+ +

−

+ +

− +

*n*
*i*
*if*
*v*
*v*
*e*

*n*
*i*
*if*
*v*
*v*
*e*

*n*
*i*
*if*
*v*
*v*
*e*
*g*
*E*

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

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

*i*
*i*
*i*

*n*

1 1

1 1

) (

2 1

2

1 1

1

Let *f* * be a bijection from E*(*g _{n}*) to {1,2,3,...,3

*n*−1}.

Let *l*1,*l*2,....,*ln*,*ln*+1,....,*l*3*n*−1 be the label of edges *e*1,*e*2,....,*en*,*en*+1,....,*e*3*n*−1
by *f* respectively. And *f* induces that *f**:*V*(*gn*)→{1,2,3,...} by

) ( )

(

* =

## ∑

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

*f* over all edges *ei* incident to adjacent vertices of *u*.

The vertices *vn*+1 and *vn*+2 have the same adjacent vertices {*v*1,*v*2,....,*vn*}.
Then *f* *(*vn*+1)=2(*l*1+*l*2+...+*ln*−1)+(*ln* +*ln*+1+...+*l*3*n*−1)

=

## ∑

## ∑

−

= −

=

+3 1

1 1

1
*n*

*i*
*i*
*n*

*i*

*i* *l*

*l* =

## ∑

## ∑

−

= −

=

+3 1

1 1

1
*n*

*i*
*n*

*i*

*i* *i*

*l* =

2 ) 1 3 ( 3 −

+ *n* *n*

*m* , where

## ∑

−

=

= 1

1
*n*

*i*
*i*
*l*
*m*

Since *vn*+1 and *vn*+2 have the same adjacent vertices,

2 ) 1 3 ( 3 )

(

* *v* _{+}_{2} =*m*+ *n* *n*−

*f* * _{n}* . That is

*f**

*is not injective. Hence gn is not an adjacent*

edge graceful graph.

**Theorem 2.10. The graph **n-bistar B_{n}_{,}_{n}** is not an adjacent edge graceful graph. **

32
Let
=
≤
≤
=
≤
≤
=
=
+
+
+
+
+
+
1
1
1
2
1
1
, 1
1
)
(
*n*
*n*
*n*
*n*
*i*
*i*
*n*
*n*
*i*
*i*
*n*
*n*
*v*
*u*
*e*
*n*
*i*
*if*
*v*
*v*
*e*
*n*
*i*
*if*
*u*
*u*
*e*
*B*
*E*

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

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

## ∑

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

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

*u*.

**Step 1: **

The adjacent vertices of *un*+1 are *u*1,*u*2,...,*un*,*vn*+1.

) ( ) ( ) ( * 1 2 1 1

1

## ∑

## ∑

+
+
=
=
+ = +
*n*
*n*
*i*
*i*
*n*
*i*
*i*

*n* *f* *e* *f* *e*

*u*

*f* ** =**

## ∑

+
=
1
2
1
)
(
*n*
*i*
*i*
*e*
*f*

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

**Step 2: **

The adjacent vertices of *vn*+1 are *v*1,*v*2,...,*vn*,*un*+1.

) ( ) ( ) ( ) (

* 2 1

2 1 1 1 + + = =

+ =

## ∑

+## ∑

+*n*

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

*n* *f* *e* *f* *e* *f* *e*

*v*

*f* ** =**

## ∑

+
=
1
2
1
)
(
*n*
*i*
*i*
*e*
*f*

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

From step(1) and step(2), the vertex labels of *un*+1 and *vn*+1 are same.
That is *f* * is not injective.Hence *B _{n}*

_{,}

_{n}is not an adjacent edge graceful graph.* Theorem 2.11. The Helm H_{n}*(

*n*≥3)

**is an adjacent edge graceful graph.**

**Proof: Let ***V*(*Hn*)={*ui* :1≤*i*≤*n*+1; *vi* :1≤*i*≤*n*}

Let
≤
≤
=
≤
≤
=
=
−
≤
≤
=
=
+
+
+
+
*n*
*i*
*if*
*u*
*u*
*e*
*n*
*i*
*if*
*v*
*v*
*e*
*u*
*u*
*e*
*n*
*i*
*if*
*u*
*u*
*e*
*H*
*E*
*n*
*i*
*i*
*n*
*i*
*i*
*i*
*n*
*n*
*n*
*i*
*i*
*i*
*n*
1
1
1
1
)
(
1
2
1
1

Define a bijection *f* :*E*(*H _{n}*)→{1,2,3,...,3

*n*} by

*f*(

*e*)=2

_{i}*i*

*if*1≤

*i*≤

*n*;

*n*

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

*f*( _{n}_{+}* _{i}*)=2 −1 1≤ ≤ ;

*f*(

*e*

_{2}

_{n}_{+}

*)=2*

_{i}*n*+

*i*

*if*1≤

*i*≤

*n*. Let

*f** be the induced vertex labeling of

*f*.

The induced vertex label are as follows:

If *n*=3 , *f* *(*u*_{1})=22*n* ; *f* *(*u*_{2})=22*n*+1 ; *f* *(*u*_{3})=20*n*+2 ;
*n*

*u*

*f* *( 4)=19 ; *f* *(*v*1)=5*n*+1 ; *f* *(*v*2)=5*n*+2 ; *f* *(*v*3)=8*n*.
If *n*≥4 ,

2
18
23
5
)
(
*
2
1
+
+
= *n* *n*

*u*

*f* ;

2
50
13
5
)
(
*
2
2
+
+
= *n* *n*

*u*

33

2 14 31 5

) ( *

2 + −

= *n* *n*

*u*

*f* *n* ;

2 5 11 ) ( *

2 1

*n*
*n*
*u*

*f* *n*

+ =

+

; 1 3

2

32 50 9 5 ) ( *

2

2 ≤ ≤ −

+ + + =

+ *if* *i* *n*

*i*
*n*

*n*
*u*

*f* *i* ;

4 4 ) (

* *v*1 = *n*+

*f* ; *f* *(*v _{i}*

_{+}

_{1})=2

*n*+4+7

*i*

*if*1≤

*i*≤

*n*−1.

**Theorem 2.12. Complete bipartite graph ***Km*,*n is not an adjacent edge graceful graph. *

**Proof: Let ***V*(*Km*,*n*)={*ui* : 1≤*i*≤*m* ;*vi* :1≤*i*≤*n*}**. **

Let *E*(*K _{m}*

_{,}

*) {*

_{n}*e*

_{i}*u*:1

_{i}v_{j}*i*

*m*,1

*i*

*n*}

*j*= ≤ ≤ ≤ ≤

= .

*Define a bijection f from E*(*K _{m}_{,n}*) to {1,2,3,...,

*n*}.

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

## ∑

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

*f* over all

edges *ei* incident to adjacent vertices of *u*..

**Case (i) ***m*<*n*

) ( )

( *

1 1

## ∑∑

= =

= *m*

*i*
*n*

*j*
*i*

*i* *f* *e _{j}*

*v*

*f* =

2 ) 1 ( ...

3 2

1+ + + +*mn*=*mn* *mn*+ for 1≤*i*≤*m* .

The label of each vertex *v _{i}* is

2
)
1
(*mn*+
*mn*

for 1≤*i*≤*m* .

**Case (ii) ***m*=*n*

)
(
* *u _{i}*

*f* = *( ) ( )

1 1

## ∑∑

= =

= *n*

*i*
*n*

*j*
*i*

*i* *f* *e _{j}*

*v*

*f* =

2 ) 1 ( ...

2 1

2 2

2= +

+ +

+ *n* *n* *n* for 1≤*i*≤*n* .

The label of each vertex *ui* and *vi* are

2 ) 1 ( 2

2 +

*n*
*n*

for 1≤*i*≤*n*.

**Case (iii) ***m*>*n*

) ( )

( *

1 1

## ∑∑

= =

= *m*

*i*
*n*

*j*
*i*

*i* *f* *e _{j}*

*u*

*f* =

2 ) 1 ( ...

3 2

1+ + + +*mn*=*mn* *mn*+ for 1≤*i*≤*n* .

The label of each vertex *u _{i}* is

2
)
1
(*mn*+
*mn*

for 1≤*i*≤*n* .

In the above three cases , *f* * is not injective.

Hence the complete bipartite graph *Km*,*n is not an adjacent edge graceful graph. *

**Corollary 2.13. The Star graph **K_{1}_{,}_{n}** is not an adjacent edge graceful graph. **

**Proof: By putting ***m*=1 in the result of the above theorem 2.12 , we can get the
**result of this corollary. **

34

**Proof: Let ***V*(*fn*)={*vi* :1≤*i*≤*n*+1}**. **

Let *E*(*f _{n}*)={

*e*=

_{i}*v*

_{i}v_{i}_{+}

_{1}:1≤

*i*≤

*n*−1;

*e*

_{n}_{−}

_{1}

_{+}

*=*

_{i}*v*

_{i}v_{n}_{+}

_{1}:1≤

*i*≤

*n*}. Define a bijection

*f*:

*E*(

*fn*)→{1,2,3,...,2

*n*−1} by

**Case (i) n is odd , ***f*(*ei*)=*i* *if* 1≤*i*≤*n*−1 and

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

*f*( _{n}_{−}_{1}_{+}* _{i}*)= −1+ 1≤ ≤ .

**Case (ii) n is even , ***f*(*e _{i}*)=2

*i*−1

*if*1≤

*i*≤

*n*−1 ;

1 1

2 )

(*e* −1+ = *i* *if* ≤*i*≤*n*−

*f* *n* *i* ;*f*(*e*2*n*−1)=2*n*−1 .
Let *f* * be the induced vertex labeling of *f* .

The induced vertex labels are as follows:

If *n* is odd ,

2 8 3

) ( *

2 1

+
+
= *n* *n*

*v*

*f* ;

2 20 13 3 ) ( *

2 1

− + =

−

*n*
*n*
*v*

*f* * _{n}* ;

2 10 7 3 ) ( *

2 _{+} _{−}

= *n* *n*

*v*

*f* * _{n}* ;

2 ) 3 5 ( ) (

* *v* _{+}_{1} =*n* *n*−

*f* * _{n}* ;

3 1

) 1 ( 6 2

16 3 3 ) ( *

2

1 + − ≤ ≤ −

+ + =

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

*n*
*n*
*v*

*f* *i* .

If *n*=4 , *f* *(*v*_{1})=6*n*+3 ; *f* *(*v*_{2})=9*n* ; *f* *(*v*_{3})=9*n*+3 ;

1 8 ) (

* *v*4 = *n*+

*f* ; *f* *(*v*5)=9*n*+1 .

If *n*≥6 , *( ) 2 7

1 =*n* +*n*+
*v*

*f* ; *( ) 2 16

2 =*n* +*n*+
*v*

*f* ;

21 11 )

(

* *v* −1 =*n*2 + *n*−

*f* *n* ; *( ) 7 11

2 + −

=*n* *n*
*v*

*f* *n* ;

1 3 3 ) (

* 2

1 = − +

+ *n* *n*

*v*

*f* * _{n}* ; *( ) 2 15 12 1 4

2 = + + + ≤ ≤ −

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

*v*

*f* * _{i}* .

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