# Radio D-Distance Number of Some Basic Graph DOI : 10.17577/IJERTV12IS070014 Text Only Version

#### Radio D-Distance Number of Some Basic Graph

K. John Bosco

Assistant Professor, Department of Mathematics,

St. Judes college Thoothoor, Tamil Nadu Affiliated to Manonmaniam Sundaranar University,

Abishekapatti, Tirunelveli

Research Scholar (Reg. no:23113232092004), Department of Mathematics,

St. Judes college Thoothoor, Tamil Nadu Affiliated to Manonmaniam Sundaranar University,

Abishekapatti, Tirunelveli

Abstract A Radio d-distance labeling of a connected graph G is an injective map f from the vertex set V(G) to such that for two distinct vertices u and v of

G, (, ) + |() ()| + (), where (, ) denotes the d-distance between u and v and () denotes the d-diameter of G. The Radio d-distance number of f, ()

is the maximum label assigned to any vertex of G. The Radio d-

distance number of G, () is the minimum value f of G. In

this paper we find the radio d-distance number of some basic

graphs.

1. INTRODUCTION

By a graph = (V(), ()) we mean a finite undirected graph without loops or multiple edges. Let () and () denotes the vertex set and edge set of . The order and size of are denoted by and respectively.T. Jackuline,

J. Golden Ebenezer Jebamani and D. Premalatha introduced

the concept of dd-distance by considering the degrees of various vertices presented in the path, in addition to the length of the path.

Let u, v be two vertices of a connected graph G. Then the d-length of a u-v path defined as dd(u,v) = d(u, v) + deg(u) + deg(v) + deg(u) deg(v), where d(u, v) is the shortest distance between the vertices u and v

In this paper, we introduced the concept of radio d-distance labeling of a graph G. Radio d-distance labeling

is a function from V(G) to satisfying the condition

(, ) + |() ()| 1 + (), where

() is the d-distance diameter of G. A

G is defined as the minimum span of a radio coloring of G and is denoted as rn(G).

Radio labeling can be regarded as an extension of distance-two labeling which is motivated by the channel assignment problem introduced by W. K. Hale . G. Chartrand et al. introduced the concept of radio labeling of graph. Also G. Chartrand et al. gave the upper bound for the radio number of path. The exact value for the radio number of path and cycle was given by Liu and Zhu . However G. Chartrand et al. obtained different values for them. They found the lower and upper bound for the radio number of cycle. Liu  gave the lower bound for the radio number of Tree. The exact value for the radio number of Hypercube was given by R. Khennoufa and O. Togni . In  C. Fernandez et al. found the radio number for complete graph, Star graph, Complete Bipartite graph, Wheel graph and Gear graph. In this paper, we fined the radio d-distance labeling of some basic graphs.

2. MAIN RESULTS

Theorem 2.1

The radio d-distance number of the complete

graph, () =

Proof

Let, V() = {1,2, , } be the vertex set

, ( , ) = 2 1 ,

d-distance radio labeling number of G is the maximum label

assigned to any vertex of G. It is denoted by (). Let G

be a connected graph of diameter d and let k an integer such

that 1 . A radio k-coloring of G is an assignment

of colors (positive integers) to the vertices of G such that

(, ) + |() ()| 1 + for every two distinct vertices u, v of G. The radio k-coloring number () of a

It is obvious that the () = 2

(, ) + |() ()| 1 + () = 2 + 1

Now, fix (1) = 1

radio k-coloring of G is the maximum color assigned to any

(

) + |( ) ( )| 2 + |1 ( )|

vertex of G. The radio k-chromatic number () is min{ () } over all radio k-colorings of G. A radio

k-coloring of G is a minimal radio k-coloring if

1, 2

1 2 2

2 + 1

() = (. When k = Diam(G), the resulting radio

|1 (2)| 1, which implies (2) = 2

() = , 1

Hence, () = ,

Theorem 2.2

The radio d-distance number of a path

( ) 2 4 + 10, 4

Proof.

Let V() = {1, 2,, , n} be the vertex set and E() = {i+1 ; 1 1} be the edge set

Then,(1, ) = (2, ) = + 2,

(1, 2 ) = (n1, ) = 6, (, +1 ) = 9; 2 2, (2, 1 ) = + 5

It is clear that ( ) = + 5

Without loss of generality (1) < (2) < < ()

We shall check the radio d-distance condition

(, ) + |() ()| 1 + () = + 6

Fix (1) = 1 for (1, 2)

For (1, 2)

(1, 2) + |(1) (2)| 5 + |1 (2)| 2 + 3

|1 (2)| 2 2, which implies (2) = 2 1

For (2, 3)

(2, 3) + |(2) (3)| 5 + |2 1 (3)|

2 + 3

|2 1 (3)| 2 2, which implies (3) = 4 3

() = (2 2) 2 + 3, 1 Hence, (1,) = 22 4 + 3, 3

Theore. 2.4

The radio d-distance number of bistar graph,

(,) = 23 + 82 4 + 2, 2

Proof.

Let V(,) = {1, 2,, , n, 1, 2,1,2,, , n,}

be the vertex set

and E(,) = {1 , 2,12,; 1 } be the edge set

Then, (1, ) = (2, ) = 2 + 4; 1 ,

(1, 2) + |(1) (2)| 6 + |1 (2)| + 6

(

) = ( + 2)2, ( , ) = 6; 1 , ,

|1 (2)| , which implies (2) = + 1

For (2, 3)

1, 2

, (i, ) = (, ) = 5; 1 ,

(2, 3) + |(2) (3)| 9 + | + 1 (3)| + 6

| + 1 (3)| 3, which implies (3) = 2 2

It is clear that (,

) = ( + 2)2 = 2 + 4 + 4

() = ( 1) 3 + 7, 2 1

Hence, () 2 4 + 10, 4

Note. ( )= n if n = 2,3

Without loss of generality, (1) < < () <

(2) < (1) < (1) < < ()

We shall check the radio d-distance condition

(, ) + |() ()| 1 + () = 2 + 4 + 5

Fix, ( ) = 1, For ( )

1 1, 2

Theorem 2.3

The radio d-distance number of a star graph,

(1, ) = 22 4 + 3, n 3

Proof.

Let V(1, ) = {0, 1,2, , } be the vertex set, where 0 be the central vertex and

E(1,) = {0 ; 1 } be the edge set

Then, (0, ) = 2 + 2; 1 , (, ) = 5; 1 , ;

So, (1,) = 2 + 2

Without loss of generality,

(1) < (0) < (2) < < ()

We shall check the radio Gd-distance condition

(, ) + |() ()| 1 + () = 2 + 3

Fix (1) = 1, for (1, 0)

(1, 0) + |(1) (2)| 2n + 2 + |1 (0)|

2 + 3

(1,2) + |(1) (2)| 5 + |1 (2)|

2 + 4 + 5

|1 (2)| 2 + 4, which implies

(2) = 2 + 4 + 1

For (2,3)

(2,3) + |(2) (3)|

5 + |2 + 4 + 1 (3)|

2 + 4 + 5

|2 + 4 + 1 (3)| 2 + 4 which implies

(3) = 22 + 8 + 1

() = 2( 1) + (4 4)+ 1, 1

Therefore, () = 3 + 32 4 + 1

For (n, 2)

(n,2) + |() (2)|

|1 (0)| 1, which implies (0) = 2

2n + 4 + | 3

+ 32

4 + 1 (2)|

2 + 4 + 5

| 3 + 32 4 + 1 (2)| 2 + 2 + 1, which

For (, +1), 1 1

( , ) + |( ) ( )| 10 + |1 ( )|

1 2 1 2 2

implies (2) = 3 + 42 2 + 2

For (2, 1)

3 + 4

|2 (2)| 3 6, which implies (2) = 3 5

(2,1) + |(2) (1)|

For ( , ), 1

2 3

2 + 4n + 4 + | 3 + 42 2 + 2 (1)|

2 + 4 + 5

| 3 + 42 2 + 2 (1)| 1, which implies

(1) = 3 + 42 2 + 1

For (1, 1)

(2, 3) + |(2) (3)| 10 + |3 5 (3)|

3 + 4

|3 5 (3)| 3 6, which implies

(3) = 6 11

() = (3 3) 6 + 7, 1

2

(1,1) + |(1) (1)|

2n + 4 + | 3 + 42 2 + 1 (1)| 2 + 4 + 5

| 3 + 42 2 + 1 (1)| 2 + 2 + 1, which

Therefore, () = 3

For (, ), 1

9 + 7

implies (1) = 3 + 52 + 2

(, 1) + |() (1)|

8 + |32 9 + 7 (1)| 3 + 4

For (1, 2)

(1, 2) + |(1) (2)|

|32 9 + 7 (1)| 3 4 which implies

( ) = 32 6 + 3

5 + |3 + 52 + 2 (2)| 2 + 4 + 5

|3 + 52 + 2 (2)| 2 + 4, which implies

1

For ( )

(2) = 3 + 62 + 4 + 2

, +1

, 1 1

(1, 2) + |(1) (2)|

2

For (2, 3)

7 + |3

6 + 3 (2)| 3 + 4

(2, 3) + |(2) (3)|

|32 6 + 3 (2)| 3 3, which implies

(2) = 32 3

5 + |3 + 62 + 4 + 2 (3)| 2 + 4 + 5

|3 + 62 + 4 + 2 (2)| 2 + 4, which implies

(2) = 3 + 72 + 8 + 2

For (

2, 3)

() = 3 + ( + 4)2 + (4 4) + 2, 1

Hence, (,) = 23 + 82 4 + 2, 2

Theorem 2.5

(2, 3) + |(2) (3)| 7 + |32 3n (3)|

3 + 4

|32 3n (3)| 3 3, which implies

(3) = 32 3

2

The radio d-distance number of a subdivision of a

() = 3

+ (3 9) 3 + 6, 1

star, S(1, ) = 62 12 + 6, 3

Proof

Let V(S(1,)) = {0, 1, 2,, , n, 1,2,, , n,} be the vertex set, where 0 is the central vertex and E(S(1,)) = {0 , ; 1 } be the edge set

Hence, S(1,

) 62 12 + 6, 3

REFERENCES

Then, (0, ) = 3 + 3; 1 ,

(i, ) = 7, (, ) = 10; 1 ,

(, ) = 6; 1

It is clear that (S(1, )) = 3 + 3

Without loss of generality (1) < (0) < (2) < <

() < (1) < < ()

We shall check the radio d-distance condition

(, ) + |() ()| 1 + () = 3 + 4

Fix (1) = 1, for (1, 0) 1

(1, 0) + |(1) (0)| 3n + 3 + |1 (0)|

3 + 4

|1 (0)| 1 which implies (0) = 2

 F. Buckley and F. Harary, Distance in Graphs,Addition- Wesley,

Redwood City, CA, 1990.

 G. Chartrand, D. Erwinn, F. Harary,and P. Zhang, Radio labeling of graphs, Bulletin of the Institute of Combinatories and Its Applications, vol. 33,pp. 77-85, 2001.

 G. Chartrand, D. Erwinn,and P. Zhang, Graph labeling problem suggested by FM channel restrictions, Bull. Inst. Combin. Appl., 43, 43-57(2005).

 C. Fernandaz, A.Flores, M.Tomova, and C.Wyels, The Radio Number of Gear graphs, arXiv:0809. 2623, September 15, (2008).

 J.A. Gallian, A dynamic survey of graph labeling, Electron.

J.Combin. 19(2012)Â£Ds6.

 W.K. Hale, Frequency assignment: Theory and applications, Proc.

IEEE 68 (1980), pp. 1497- 1514.

 F.Harary, Graph Theory, Addition Wesley,New Delhi(1969).

 T. Jackuline, J. Golden Ebenezer Jebamani and D. Premalatha, d-

Distance in Graphs, Vol 18, Issue 8, August 2022 ISSN 1673-064X  R. Khennoufa and O. Togni, The Radio Antipodal and Radio Numbers of the Hypercube, accepted in 2008 publication in

ArsCombinatoria.

 D. Liu, X. Zhu, Radio number for trees, Discrete Math.308(7)(2008) 1153-1164.

 D. Liu,. X.Zhu, Multilevel distance labeling for paths and cycles, SIAM J. Discrete Math. 19(3)(2005) 610-621.

 T. Nicholas, K. John Bosco, Radio D-distance Number of some graphs, IJESR , vol.5 Issue 2, Feb.2017.

 T. Nicholas, K. John Bosco, M. Antony, V. Viola, Radio mean D- distance Number of Banana Tree, Thorn Star and Cone Graph

,IJARIIT, Vol.5 Issue 6, Feb.2017, ISSN:2456 132X

 T. Nicholas,V. Viola, On Radio D-distance Number of some basic graphs IJIRCT Vol 4 Issue 3 ISSN 2454-5988