Star graph, Bistar graph, Cordial labeling, Cordial graph.

Abstract

1,2 and 1,2. EMAIL: [e-mail] (Pariksha Gupta)

1. Introduction

graph for its cordiality. 1.1. 1.2.

Definition

having three edges is called as claw 1.3.

WCNC-2022: Workshop on Computer Networks and Communications, April 22 – 24, 2022, Chennai, India. ORCID: 0000-0003-3041-7694 (Pariksha Gupta)

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Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).

CEUR Workshop Proceedings (CEUR-WS.org)

Star of Bistar graph is formed by adding a bistar graph to each vertex of 1, . For reference, consider the below example – Star of Bistar graph 2,2.

2. Results

2.1. Theorem: Star of bistar graph , is a cordial graph when = .

Proof: Let 1, and 1, are two-star graph forming a bistar graph , by joining their centre vertex to an edge.

Let 1, 2, 3, … … … … , and 1, 2, 3, … … … … , are the nodes of 1, and 1, for internal node of star graph and and are the centre nodes of bistar graph by which 1, and 1, are joined.

Let ( 11, 12, … … 1 ), ( 21, 22, … … . . 2 ),……...,( ( 11, 12, … … 1 ), ( 21, 22, … … . . 2 ),……...,( 1, 2, … … … 1, of m – leaves of star of bistar graph. 1, 2, … … … ) and ) be the nodes of 1, and

The vertices of graph are attached to and are attached to and for the centre vertices of graph 1, and 1, of m – leaves are denoted as and .

Let is a star graph of bistar graph. To define the function : ( ) → {0,1}, below cases will arise: Case 1: When = ≡ 0,2 ( 4) then we define labeling as: Case 2: When = ≡ 1,3 ( 4) then we define labeling as: ( ) = ( ) = {10;; ≡≡01,,23(( 4)

4) ( ) = ( ) = {10;; ++ ≡≡01,,23(( 4) 4) ( ) = ( ) = ( ) = ( ) = {10;; ≡≡01,,23(( 44)) ( ) = ( ) = {10;; ≡≡01,,23(( 4)

4) ( ) = ( ) = {10;; ++ ≡≡01,,23(( 4)

4) 0; ≡ 1,3 ( 4) ( ) = ( ) = {1; ≡ 0,2 ( 4)

1; ≡ 1,3 ( 4) ( ) = ( ) = {0; ≡ 0,2 ( 4)

The above explained cases satisfy the condition for both vertices and edges, to be cordial which is as shown below in table 1. The above cases justify that the graph G is cordial graph Example 1.1. Star of Bistar graph 3,3 is a cordial graph.

Edge Labeling

(0) + 1 = (1) (1) = (0) + 1

Vertices Labeling

(0) = (1) (0) = (1) 2.2. Theorem: Star of Bistar graph , is a cordial graph when ≠ .

Proof: Let 1, and 1, are two-star graph forming a bistar graph , by joining their centre vertex to an edge.

Let 1, 2, 3,…………, and 1, 2, 3,…………, are the nodes of 1, and 1, for internal node of star graph and and are the centre nodes of bistar graph by which 1, and 1, are joined.

Let ( 11, 12,…… 1 ), ( 21, 22,…….. 2 ),……...,( 1, 2,……… ) and ( 11, 12,…… 1 ), ( 21, 22,…….. 2 ),……...,( 1, 2,……… ) be the nodes of 1, and 1, of m – leaves of star of bistar graph.

The vertices of graph are attached to and are attached to and for the centre vertices of graph 1, and 1, of m – leaves are denoted as and .

Let is a star graph of bistar graph. To define the function : ( ) → {0,1}, the below cases will arise: Case 1: When ≡ 0,2 ( 4) and ≡ 0,2 ( 4) and ≠ then we define labeling as: Case 2: When ≡ 1,3 ( 4) and ≡ 1,3 ( 4) and ≠ then we define labeling as: Case 3: When ≡ 1 ( 4) and ≡ 2 ( 4): n ≡ 1,3 (mod 4); m ≠ n when m ≡ 1 (mod 4) & n ≡ 2 (mod 4) when m ≡ 1 (mod 4) & n ≡ 0 (mod 4) when m ≡ 3(mod 4) & n ≡ 0(mod 4) when m ≡ 3 (mod 4) & n ≡ 2(mod 4) When m ≡ 2 (mod 4) & n ≡ 1 (mod 4) When m ≡ 2 (mod 4) & n ≡ 3 (mod 4) When m ≡ 0 (mod 4) & n ≡ 1 (mod 4) When m ≡ 0 (mod 4) & n ≡ 3 (mod 4)

Edge Labeling The above cases justify that the graph G is cordial graph

Example 1.2: Star of Bistar graph 4,5 is a cordial graph.

Vertices Labeling vf(0) = vf(1) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0) vf(1) = vf(0)

3. Conclusion

5. References

Graph labeling in graphs is an important area of research. We have presented the cordiality for star of bistar graph. Also, we have seen some examples which justify the above theorem. Labeling for star of other graphs is an open research area.

4. Acknowledgements

[1] Amit H. Rokad, Kalpesh M. Patadiya, (2017), Cordial labeling of some graphs, Aryabhatta

Journal of Mathematics and Informatics, 589 – 597. [2] Cahit, (1987), Cordial graphs: A weaker version of graceful and harmonious graphs, Ars

Combin. [3] F. Harary, (1972), Graph Theory, Addition - Wesley, Reading MA [4] J. A. Gallian, (2019), A dynamic survey of graph labeling, The electronic Journal of combinatorics [5] S. Sudhakar, V. Maheshwari and V. Balaji, (2018), Cordial labeling for Star Graphs,

International Journal of Mathematics and its Application, 51 – 54. [6] S.K. Vaidya, N.H. Shah, (2014), Cordial labeling for some Bistar related graphs, International

Journal of Mathematics and soft computing, 33-39. [7] S.K. Vaidya, N.A. Dani, K.K. Kanani, P.L. Vihol, (2009), Cordial and 3 – Equitable labeling for some star related graphs, International Mathematical Forum, 1543 – 1553. [8] Vaidya, S. K., G. V. Ghodasara, Sweta Srivastav, and V. J. Kaneria, (2008), Cordial and 3equitable labeling of star of a cycle, Mathematics Today, 24 (54-64).

References