International Journal of Pure and Applied Mathematics Volume 118 No. 20 2018, 4563-4567 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue WAVE LABELING APPROACH FOR CONNECTED GRAPH. ijpam.eu

*K. Thiagarajan1 , J. Kavitha2 , V. Rajeswari3 1Associate Professor, Department of Mathematics, PSNA College of Engineering and Technology, Dindigul, TN, India, [email protected], 2Research Scholar(Ph.D-CB- DEC2013-0334), Department of Mathematics, Bharathiar University, Coimbatore, TN, India, [email protected], 3Assistant Professor, Department of Mathematics, Don Bosco College, Dharmapuri, TN, India, [email protected]

Abstract

Now a days plays a vital role in various area of Research. Graph labeling is an assignment of Integers either to the vertices or edges or both subject to certain conditions. In this Simple connected finite Graph G with p Points and q Vertices are taken for discussion. In this paper we defined Weak Average Labeling to Edges (WAVE) as if the vertices are labeled such that 푓: 푉 → {1, 3, 5, . . . , (p + q − 1)},where p is number of vertices and q is number of edges in G and the edges set be 푓∗: 퐸 → {2, 3, … , 2 푞 − 1 } such that the edges are labeled as 푓 푢 +푓(푣) 푓∗ 푢푣 = .Finally we derived some new results for , Crown graph, 2 3-Regular graph,4-Regular Graph , Self Complementary Graph and Snake tail Graph .

Key Words

Cycle ,regular, self Complementary Graph ,Snake tail graph, Weak

Definition

A graph G is said to be Weak Average Vertex Labeling to Edges (WAVE), if the vertices are labeled with Odd numbers 1, 3, 5, ... ,(p+q-1), such that 푓: 푉 → {1, 3, 5, . . . , (p + q − 1)}, where p is number of vertices and q is number of edges in G and the edges set be 푓∗: 퐸 → {2, 3, … , 2 푞 − 1 } such that the edges are labeled as 푓 푢 +푓(푣) 푓∗ 푢푣 = which are distinct. 2

Theorem

All Cycle 퐶푛 where 푛 ≥ 3 are WAVE graph

Proof

Let 퐶푛 be Cycle Graph with n vertices 푣1, 푣2, 푣3, … , 푣푛 .

The vertices are labeled as follows

Case 1

When n is even.

푓 푣1 = 1 , 푓 푣푛 = 3

푓 푣푖 = 2푖 + 1, 푖 = 2,3, … . , (푛 − 1)

Case 2

When n is odd.

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푓 푣푖 = 2푖 − 1, 푖 = 1,2,3, … . , 푛

푓 푢 +푓(푣) Now, the edges are labeled as 푓∗ 푢푣 = 2

Therefore, All Cycle 퐶푛 where 푛 ≥ 3 are WAVE graph

Example

When n is even

When n is odd

Result 1

All Crown graphs need not be a WAVE graceful labeled graphs

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Result 2

All 3 regular graph need not be WAVE graceful labeled graph

퐶3233 퐶4244

Result 3

All 4 regular graph need not be WAVE graceful labeled graph

Result 4

All Self need not be WAVE Graceful labeled Graph

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Result 5

All Snake Tail Graph need not be WAVE Graceful labeled Graph

2 퐶31

Conclusion

In this paper we studied Cycle graph is WAVE Graceful labeled Graph and all Crown graph ,3 Regular graph, 4 Regular Graph, Self Complimentary Graph and Snake Tail Graph are need not be WAVE Graph.

Acknowledgement

The authors would like to thank Dr. Ponnammal Natarajan, Former Director – Research, Anna University - Chennai, India and currently an Advisor, (Research and Development), Rajalakshmi Engineering College, Dr. E.Sampath Kumar Acharya & Dr. L. Pushpalatha, University of Mysore, Mysore for their initiative ideas and fruitful discussions with respect to the paper’s contribution.

References

1. Gallian J.A., " A Dynamic Survey of Graph Labeling ",The Electronic journal of Combinatorics, (2015). 2. J.A. Bondy and U.S.R. Murty, "Graph Theory with Applications", London Macmillan (1976) 3. Frank Harray, Graph Theory, Narosa Publishing House pvt. ltd- ISBN 978-81-85015- 55-2. 4. K. Thiagarajan, J. Kavitha " SAVE labeling approach for connected graph", in Proceedings of 6th International Conference on Contemporary Engineering and Technology (ICCET 2018), ISBN No: 978-81-923607-3-7

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