Integrated Intelligent Research (IIR) International Journal of Computing Algorithm Volume: 06 Issue: 01 June 2017 Page No.27-30 ISSN: 2278-2397 A Study on Graceful and -Graceful Labeling of Some Graphs

M. Radhika1, V. S. Selvi2 Research Scholar1, Professor2 1,2 Department of Mathematics, Theivanai Ammal College for Women (Autonomous) Villupuram, India. Email: [email protected], [email protected]

Abstract – In this paper, we prove that result on are distinct. The concept of graceful labeling is an graceful labeling of path and symmetrical trees are active area of research in which has graceful graphs. We also prove that result on ϕ- rigorous applications in coding theory, graceful labeling of paths, flower graph, friendship communication networks, optimal circuits layouts graph. and graph decomposition problems. For a dynamic survey of various problems along Keywords: Path, symmetrical trees, flower graph, with an extensive bibliography we refer to Gallian [1].

I. INTRODUCTION The following three types of problems are considered generally in the area of graph labeling. Graph labeling is an active area of research in graph theory. Graph labeling where the vertices are 1. How particular labeling is affected under assigned some value subject to certain condition. various graph operations; Labeling of vertices and edges play a vital role in 2. Investigation of new families of graphs which graph theory. To begin with simple, finite, admit particular graph labeling; connected and undirected graph ) )) 3. Given a graph theoretic property and with | )| and | )| . For standard characterizing the class/classes of graphs with terminology and notation we follow Gallian [1], property that admits particular graph labeling. Gross and Yellen [2]. The definitions and other information which serve as prerequisites for the It is clear that the problems of second type are present investigation. largely studied than the problems of first and third types. The present work is aimed to discuss some II. DEFINITIONS problems of the first kind in the context of graceful and -graceful labeling. Definition 2.1: A function is called graceful Definition 2.4: Let G be a graph with order ‘ ’ and labeling of a graph G if ) is size , such that exactly one node is adjacent to injective and the induced function every other nodes, this is also known as flower graph. The resulting graph is flower graph ) defined as with petal.

) | ) )| Definition 2.5: A friendship graph is a graph is bijective. The graph which admits graceful which consists of triangles with common vertices. labeling is called a graceful graph. Definition 2.2: A rooted in which every level III. RESULTS ON GRACEFUL contains vertices of the same degree is called LABELING symmetrical trees. The following result is a gracefully labeled symmetrical tree on 15 vertices. Theorem 3.1: Every path ( ) graph is graceful Definition 2.3: A function is called a -graceful graphs. labeling of graph G, if ) is injective and the Proof: Let G be a . induced function ) is defined as Label the first vertex , and label every other vertex ) ) ) , then edge labels increasing by each time. 27 Integrated Intelligent Research (IIR) International Journal of Computing Algorithm Volume: 06 Issue: 01 June 2017 Page No.27-30 ISSN: 2278-2397 Label the second vertex and label every other ) vertex decreasing by each time. This completes the induction. There are vertices, All symmetrical trees are graceful. so the first set will label it’s vertices with numbers from the set ⁄ if is even and from the Illustration: set )⁄ if q is odd. Gracefully labeled symmetrical tree on 15 vertices is The second set will label it’s vertices with numbers shown in fig.2. from the set )⁄ if is even and )⁄ if is odd. Thus, the vertices are labeled legally. With the vertices labeled in this manner, the edges attain the values in that order. Thus, this is a graceful labeling, so G is graceful. All path graphs are graceful.

Illustration: Graceful labeling of the graph is shown in fig.1.

Fig2: Graceful labeling of symmetrical tree on 15- vertex.

IV. RESULTS ON -GRACEFUL LABELING

Theorem 4.1: Path ) graphs are -graceful graphs.

Proof: Let be a path with vertex set and be the edges of the path. We define the vertex labeling ) Fig1: Graceful graph of ) Theorem 3.2: All symmetrical trees are graceful. ) : Proof: We prove the theorem by induction method. : The number of layers that all symmetrical trees are ) , vertex labeling can be done in both graceful, and there exists a graceful labeling which directions. assigns the number 1 to the root. The edge labeling function is defined as follows ) is defined by If ‘T ’ is a symmetrical tree with 0 layers, ) ) )

Then, it consists of 0 edges and just one vertex. ) ) ) And clearly, there is a graceful which assigns 1 to : the vertex. : Suppose, we have proved that for some ) such that the edge labels . are distinct. All symmetrical trees with layers are In view of above labeling pattern the paths are - graceful and each of them has a graceful which graceful labeling. assigns the number 1 to the root. Hence is a -greceful graph. 28 Integrated Intelligent Research (IIR) International Journal of Computing Algorithm Volume: 06 Issue: 01 June 2017 Page No.27-30 ISSN: 2278-2397 Illustration: ) -Graceful labeling of the graph is shown in ) fig.3. : : Theorem 4.2: Flower graphs are -graceful graphs. ) Proof: Let G be a flower graph. Then G has n The edge labeling function is defined as follows vertices and (n-1) edges. ) is defined by

Therefore and ) ) )

) ) ) = 2 ; We define the vertex labeling : f : V(G) {0, 1, 2, …, n-1} : ) then the edge labels are ) distinct. : In view of the above labeling pattern the is a - : graceful labeling. Hence is a -graceful graph. ) such that labeling of the vertices may be clockwise or anticlockwise. The edge labeling function is defined as follows ) is defined by ) ) )

) ) ) : : then the edge labels are distinct and it’s in increasing order. Hence all flower graphs are -graceful graph.

Fig4: - Graceful graph of flower graph.

Illustration: - Graceful labeling of the graph is shown in fig.5.

Fig3: - Graceful graph of .

Illustration:

Flower graph with 8 petals in fig.4.

Theorem 4.3: A Friendship graph - graceful graph. Proof: Let be the Friendship Graph. Let be the vertices of and edges are . We define the vertex labeling ) Fig5: - Graceful graph of graph. 29 Integrated Intelligent Research (IIR) International Journal of Computing Algorithm Volume: 06 Issue: 01 June 2017 Page No.27-30 ISSN: 2278-2397 V. CONCLUSION

In this paper we have shown that every path graphs are Graceful and - Graceful graph, Friendship graph is a - Graceful graph, every flower graph is a - Graceful graph. Are investigated it can also verified for some graphs.

Acknowledgment We would like to give the grateful thanks to Dr. Seenuvasan and Dr. Rishivarman who gives for their valuable suggestions and improvements to the paper of research project and work. Moreover, we want to thank Parent’s, Friends, ICMA17 – GT165 and IJCTA for their encouragement and support.

References [1] A Gillian. “A dynamic Survey of graph labeling”, the Electronics Journal of Combinatories Dec 20,(2013). [2] J. Gross and J. Yellen, “Graph theory and its applications”, CRC Press,(1999). [3] Frank Harray, “Graph Theory”, Narosa Publishing House . [4] Gray Chatrand, Ping Zhang, “Introduction to graph theory”, McGraw –Hill international Edition. [5] Murugesan N, Uma.R , “Graceful labeling of some graphs and their sub graphs” Asian Journal of Current Engineering and Maths 6 Nov-Dec. (2012). [6] V. J. Kaneria, S. K. Vaidya, G. V.Ghodasara and S. Srivastav, “Some classes of disconnected graceful graphs”, Proceedings of the First International Conference on Emerging Technologies and [7] Applications in Engineering, Technology and Sciences, (2008), pp 1050 –1056. [8] S. K. Vaidya and Lekha Bijukumar, “Some new graceful graphs”, International Journal of Mathematics and Soft Computing,1(1)(2011), pp 37-45. [9] K. Kathiresan, “Two classes of Graceful graphs”, Ars. Combin.55 (2000), pp 129-132. [10] Jacob Yunker, “Graceful Labeling Necklace Graphs”, (2012) . [11] M. M. A. A. Md Forhad Hossain and M. Kaykobad, “New classes of graceful trees”. Journal of Discrete mathematics, (2014).

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