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.Mathematicss capable categorization 2010: 05C78IntroductionA 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 consequencesLet 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 denotedIt can be verified that the figure of vertices of can be recursively defined by the relationBesides 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 1We 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 ? 1The 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 ofover are labeled in footings of labeling of as follows recursively.Relation table 1:Vertexs of last subdivision of present treeRelationVertexs of last subdivision of old treeSupporting 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 1Relation table 2:Vertexs of last subdivision of present treeRelationVertexs of last subdivision of old treeSupporting 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 2The above labeling of vertices induces a bijective function gV and germanium as followsgermanium: E ( ) > { 1, 2, 3… , }andgV: > { 0, 1, 2… , } ,which could be easy verified easy that it is a graceful labeling for a given treefrom the undermentioned assignment tabular arraies.

Edge assignment of by the relation { 1, 2, 3… , } is given as follows.Assignment table 1:Tennessee23456NEdge s1c148131926QEdge s1s237121825q-1Edge s2c226111724q-2Staying free foliages of S215101623q-3Edge s2s3491522q-4Edge s3c3

## — –

381421q-5Staying free foliages of S3

## —

2,17,613,1220,19q-6, q-7Edge s3s4

## —

51118q-8Edge s4c4

## —

41017q-9Staying free foliages of S4

## — –

3 to 19 to 716 to 14q-10 to q-12Edge s4s5

## —

613q-13Edge s5c5

## —

512q-14Staying free foliages of S5

## —

4 to 111 to 8q-15 to q-18Edge s5s6

## —

7Etc. ( commondifference is 1 )Edge s6c6

## —

6Staying free foliages of S6

## — –

5 to 1Table 3The vertex assignment of as followsAssignment table 2:Tennessee23456n=is1000000c148131926Qs237121825q-1c2111111of S226111724q-2s3

## —

33333c3

## — –

5101623q-3of S3

## —

2,42,42,42,42, 4s4

## —

## —

81421q-5c4

## —

## — –

5555of S4

## — –

## — –

6,7,912,13,1519,20,22{ ( q-4 ) to ( q-7 ) except ( q-5 ) }s5

## —

## — –

## — –

888c5

## —

## — –

## —

1118q-8of S5

## —

## — –

## — –

6,7,9,106,7,9,106,7,9,10s6

## —

## — –

## — –

## — —

15q-11c6

## —

## — –

## — –

## — –

1111of 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 offor 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.Tennesseen=ks10c1Qs2q-1c21of S2{ q-2 to ( q-1-k ) }s3k+2c3q-2-kof S3{ 2 to ( 2k+2 ) except ( k+2 ) }s4q-3-2kc42k+3of S4{ ( q-3-k ) to ( q-3-4k ) except ( q-3-2k ) }s54k+4c5q-4k-4of S5{ ( 2k+4 ) to ( 6k+4 ) except ( 4k+4 ) }s6q-5-6kc66k+5of S6{ ( q-5-4k ) to ( q-5-9k ) except ( q-5-5k ) }