Bitopological Harmonious Labeling of Some Star Related Graphs

Bitopological Harmonious Labeling of Some Star Related Graphs

M. Subbulakshmi¹, S. Chandrakala², G. Siva Prijith^3*

¹Associate Professor, PG and Research Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti

²Associate Professor, PG & Research Department of Mathematics, Tirunelveli Dakshina Mara Nadar Sangam College, T.Kallikulam

³Research Scholar, Reg. No. 19222052092004, G. Venkataswamy Naidu College, Kovilpatti, Affiliated to Manonmaniam Sundaranar University, Tirunelveli

DOI: https://doi.org/10.51584/IJRIAS.2025.100600147

Received: 30 June 2025; Accepted: 03 July 2025; Published: 25 July 2025

ABSTRACT

Bitopological harmonious labeling for a graph  with vertices, is an injective function , where X is any non – empty set such that  and  forms a topology on , that induces an injective function , defined  for every  such that  forms a topology on where . A graph that admits bitopological harmonious labeling is called a bitopological harmonious graph. In this paper, we discuss bitopological harmonious labeling of some star related graphs.

Keywords: Bitopological harmonious graph, bistar graph, spider graph, lilly graph, firecracker graph.

INTRODUCTION

In this paper we consider only simple, finite and undirected graphs. The graph G has a vertex set V = V(G) and edge set E = E(G). For notations and terminology we refer to Bondy and Murthy[2]. Acharya [1] established another link between graph theory and point set topology. Selestin Lina S and Asha S defined bitopological star labeling for a graph  as X be any non-empty set if there exists an injective function  which induces the function  as   for every , if    are topolologies on X then G is said to be bitopological star graph. In this paper we proved some star related graphs are bitopological harmonious graph.

Basic Definitions

Definition 2.1

Bitopological harmonious labeling of a graph  with vertices is an injective function , where X is any non – empty set such that  and  forms a topology on , that induces an injective function , defined  for every  such that  forms a topology on where . A graph that admits bitopological harmonious labeling is called a bitopological harmonious graph.              

Definition 2.2

Bistar graph   is obtained from  by attaching  pendent edges to one end of  and  pendent edges to the other end of .

Definition 2.3

A spider graph   is a star graph  such that each of which  vertices is joined to new vertex.

Definition 2.4

Lilly graph is obtained from 2 stars , by joining 2 paths  with sharing a common vertex.

Definition 2.5

Fire cracker graph  is the graph obtained by concatenation of stars by linking one leaf from each.

MAIN RESULTS

Theorem 3.1

The bistar graph  is a bitopological harmonious graph.

Proof:

Let .

Let 

Let 

Let 

Define a function  as follows:

for 

 for 

Here all the vertex labels are distinct and they form a topology on X.

Then the induced function  is given as follows:

for all 

  for 

  for 

Since  is 1-1 and so Also  forms a topology on .

Hence  is a bitopological harmonious labeling and  is a bitopological harmonious graph.

Example 3.2

Fig 3.1 Bitopological harmonious labeling of 

Theorem 3.3

The Spider graph ,  is a bitopological harmonious graph.

Proof:

Let .

Let where  be the centre vertex.

Let 

Let 

Define a function  as follows:

 for 

 for 

 for 

Here all the vertex labels are distinct and they form a topology on X.

Then the induced function  is given as follows:

 for all  

Here  for 

 for 

 for 

Since  is 1-1 and so  Also  forms a topology on .

Hence  is a bitopological harmonious labeling and  is a bitopological harmonious graph.

Example 3.4 

Fig 3.2 Bitopological harmonious labeling of 

Theorem 3.5

Lilly graph  is a bitopological harmonious graph.

Proof:

Let .

Let  

Let 

Let 

Define a function  as follows:

 for 

 for 

Here all the vertex labels are distinct and they form a topology on X.

Then the induced function  is given as follows:

 for all  

Here  for 

 for 

Since  is 1-1 and so  Also  forms a topology on .

Hence  is a bitopological harmonious labeling and  is a bitopological harmonious graph.

Example 3.6

Fig 3.3 Bitopological harmonious labeling of 

Theorem 3.7

The firecracker graph  is a bitopological harmonious graph.

Proof:

Let .

Let  

Let 

Let 

Define a function  as follows:

 for 

 for 

 for 

 for 

Here all the vertex labels are distinct and they form a topology on X.

Then the induced function  is given as follows:

 for all   

Here  for 

 for 

Since  is 1-1 and so  Also  forms a topology on .

Hence  is a bitopological harmonious labeling and  is a bitopological harmonious graph.

Example 3.8

Fig 3.4 Bitopological harmonious labeling of 

CONCLUSION

In this paper, we proved some star related graphs bistar, spider graph, lilly graph and firecracker graph are bitopological harmonious graph.

REFERENCES

1. Acharya B.D., Set valuations and their applications, MRI Lecture note in Applied Mathematics, No.2, Mehta Research Institute of Mathematics and Mathematical Physics, 1983.
2. Bondy J.A and Murthy U.S.R, “Graph Theory and Application” (North Holland). New York (1976).
3. Joseph A Gallian 2018, ‘A Dynamic Survey of Graph Labeling’, The Electronic Journal of Combinatorics.
4. Selestin Lina S, Asha S, ‘On Topological Cordial Graphs’, Journal of Science and Technology, 5(2020), 25-28.
5. Topological cordial labeling of some graphs’, Malaya Journal of Matematik, Vol. 9, No. 1, 861-863.
6. Selestin Lina, S. & Asha, S. (2022), ‘Bitopological labeling of tree related graphs’, AIP Conference Proceedings. 2385. 130016. 10.1063/5.0070851.
7. Siva Prijith, M. Subbulakshmi, S. Chandrakala, ‘Topological Cordial Labelling of Some Graphs’, Mapana – Journal of Sciences 2023, Vol. 22, Special Issue 1, 129-140 ISSN 0975-3303.