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Abstraction: In this paper, it is shown graceful a labeling of particular type of tree obtained from a household of n stars holding figure of subdivisions of those stars form an arithmetic patterned advance with common difference K and one of the subdivisions of each of those stars merged with one different point in turn in a common way on n vertices.

Cardinal words: graceful labeling, back uping vertices, hanging vertices, free foliages, turning stars, common difference.

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Mathematicss capable categorization 2010: 05C78

Introduction

A graph G with p vertices and Q borders, vertex set V ( G ) and border set E ( G ) is said to be graceful labeling if the bijection degree Fahrenheit: V ( G ) > { 0, 1, … , Q } such that: { U, V } is an border in G = { 1,2… Q } is the border set E ( G ) . All the border values are distinguishable.

Gallian, J A [ 5 ] gives extended study on graceful labeling. C. Huang, A. Kotzig, and A. Rosa [ 2 ] gives a new category of graceful trees, G. Sethuraman and J. Jesintha [ 3 ] shows a new category of graceful rooted trees, they showed bring forthing new graceful trees [ 4 ] and B. D. Acharya, S. B. Rao, and S. Arumugan [ 1 ] given thoughts graceful labeling of graphs.

The tree with n hanging stars whose figure of subdivisions are in arithmetic patterned advance with ( step K ) common difference K holding one of the subdivision points in each of those stars is merged with a point in a common way on n vertices as support points of those stars severally.

Main consequences

Let S1, S2. . . Sn be stars with = I for I = 1, 2, 3. . . n. Let the support points of the hanging stars S1, S2, . . . Sn be s1, s2, s3, … , tin severally and denote the free foliages of each of the stars Si by f1 ( I ) , f2 ( I ) , … . fi-1 ( I ) for I = 1, 2 … N.

Let c1, c2… , cn be the cardinal vertices of the stars S1, S2, S3 … Sn severally.

A tree with turning n hanging stars as subdivisions whose cardinality are in arithmetic patterned advance with common difference 1 ( step 1 ) is denoted

It can be verified that the figure of vertices of can be recursively defined by the relation

Besides the borders of can be defined by the relation

## ,

Because of the above relation, we define the relation between two consecutive trees and as

## ,

where – denote the difference between the figure of vertices ( borders ) of and.

A general tree drawn in figure 1.

Figure 1

We besides denote the labeling of node V in the tree as cubic decimeter ( V ) . Here, for the tree, we assign the labeling as follows in six groups.

R ( 1 ) : cubic decimeter ( s1 ) = 0 ; cubic decimeter ( c1 ) = Q ; l ( c2 ) = 1 and cubic decimeter ( s2 ) = q-1.

R ( 2 ) : cubic decimeter ( s2m+1 ) = cubic decimeter ( s2m-1 ) + ( 2m + 1 ) , thousand ? 1.

R ( 3 ) : cubic decimeter ( s2m +2 ) = cubic decimeter ( s2m ) – ( 2m + 2 ) , thousand ? 1.

R ( 4 ) : cubic decimeter ( c2m + 1 ) = cubic decimeter ( c2m-1 ) – ( 2m +1 ) , thousand ? 1.

R ( 5 ) : cubic decimeter ( c2m + 2 ) = cubic decimeter ( c2m ) + ( 2m + 2 ) , thousand ? 1.

Let the free foliages of turning mth star of at samarium be f1m, f2m… .

Then for m ? 1

The labeling of free foliages of uneven stars of S2m+1 based on its back uping vertex s2m+1 as follows.

R ( 6 ) a: labeling of free foliages of S2m + 1 is the set of whole numbers { cubic decimeter ( s2m + 1 ) ± 1, cubic decimeter ( s2m +1 ) ± 2 ) … cubic decimeter ( s2m +1 ) ± m }

The labeling of free foliages of even stars of S2m based on its back uping vertex s2m as follows.

R ( 6 ) B: labeling of free foliages of S2m is the set of whole numbers { cubic decimeter ( s2m ) ± 1, cubic decimeter ( s2m ) ± 2, . . . , cubic decimeter ( s2m ) ± ( m-1 ) , l ( s2m ) – m }

From the above labeling, we observe that the relation between labeling of back uping vertices and centre vertices of hanging stars that are given below.

cubic decimeter ( s2m + 1 ) = cubic decimeter ( c2m ) + ( thousand +1 ) , thousand ? 1.

cubic decimeter ( s2m + 2 ) = cubic decimeter ( c2m + 1 ) – ( thousand + 1 ) , thousand ? 0.

From the above labeling process we observe that the extra vertices of

over are labeled in footings of labeling of as follows recursively.

Relation table 1:

Vertexs of last subdivision of present tree

Relation

Vertexs of last subdivision of old tree

Supporting vertex cubic decimeter ( s2m+1 )

## =

cubic decimeter ( s2m )

Cardinal vertex cubic decimeter ( c2m+1 )

## =

cubic decimeter ( c2m ) + ( 2m +2 )

Leafs of S2m+1

## =

The set { , … , } ? { replacement of maximal whole number value in the above set }

Table 1

Relation table 2:

Vertexs of last subdivision of present tree

Relation

Vertexs of last subdivision of old tree

Supporting vertex cubic decimeter ( s2m )

## =

cubic decimeter ( s2m-1 ) + ( 2m + 1 )

Cardinal vertex cubic decimeter ( c2m )

## =

cubic decimeter ( c2m-1 )

Leafs of S2m

## =

The set { + ( 2m +1 ) , + ( 2m + 1 ) … , + ( 2m + 1 ) } ? { cubic decimeter ( c2m-1 ) + 1 }

Table 2

The above labeling of vertices induces a bijective function gV and germanium as follows

germanium: E ( ) > { 1, 2, 3… , }

and

gV: > { 0, 1, 2… , } ,

which could be easy verified easy that it is a graceful labeling for a given tree

from the undermentioned assignment tabular arraies.

Edge assignment of by the relation { 1, 2, 3… , } is given as follows.

Assignment table 1:

Tennessee

2

3

4

5

6

N

Edge s1c1

4

8

13

19

26

Q

Edge s1s2

3

7

12

18

25

q-1

Edge s2c2

2

6

11

17

24

q-2

Staying free foliages of S2

1

5

10

16

23

q-3

Edge s2s3

4

9

15

22

q-4

Edge s3c3

## — –

3

8

14

21

q-5

Staying free foliages of S3

2,1

7,6

13,12

20,19

q-6, q-7

Edge s3s4

5

11

18

q-8

Edge s4c4

## —

4

10

17

q-9

Staying free foliages of S4

3 to 1

9 to 7

16 to 14

q-10 to q-12

Edge s4s5

6

13

q-13

Edge s5c5

## —

5

12

q-14

Staying free foliages of S5

4 to 1

11 to 8

q-15 to q-18

Edge s5s6

## —

7

Etc. ( common

difference is 1 )

Edge s6c6

## —

6

Staying free foliages of S6

## — –

5 to 1

Table 3

The vertex assignment of as follows

Assignment table 2:

Tennessee

2

3

4

5

6

n=i

s1

0

0

0

0

0

0

c1

4

8

13

19

26

Q

s2

3

7

12

18

25

q-1

c2

1

1

1

1

1

1

of S2

2

6

11

17

24

q-2

s3

3

3

3

3

3

c3

5

10

16

23

q-3

of S3

2,4

2,4

2,4

2,4

2, 4

s4

8

14

21

q-5

c4

5

5

5

5

of S4

## — –

6,7,9

12,13,15

19,20,22

{ ( q-4 ) to ( q-7 ) except ( q-5 ) }

s5

8

8

8

c5

11

18

q-8

of S5

6,7,9,10

6,7,9,10

6,7,9,10

s6

15

q-11

c6

11

11

of S6

## — —

12,13,14,16,17

{ ( q-9 ) to ( q-14 ) except ( q-10 ) }

The general signifier of as follows. ( replace q by P remains same )

We besides observe this strategy can be extended to graceful labeling of

for K = 2, 3, 4 … with following assignments to vertices.

Assuming the common difference K, the stars are k, 2k, 3k… in go uping order. The following table gives ready mention for any difference K.

vertex assignment as follows.

Tennessee

n=k

s1

0

c1

Q

s2

q-1

c2

1

of S2

{ q-2 to ( q-1-k ) }

s3

k+2

c3

q-2-k

of S3

{ 2 to ( 2k+2 ) except ( k+2 ) }

s4

q-3-2k

c4

2k+3

of S4

{ ( q-3-k ) to ( q-3-4k ) except ( q-3-2k ) }

s5

4k+4

c5

q-4k-4

of S5

{ ( 2k+4 ) to ( 6k+4 ) except ( 4k+4 ) }

s6

q-5-6k

c6

6k+5

of S6

{ ( q-5-4k ) to ( q-5-9k ) except ( q-5-5k ) } 