Some Graceful and $\alpha$-Graceful Labeling for Cycle Related Graphs

# V. J. Kaneria^{1}, Payal Akbari^{1} and Nishaba Parmar^{1}

^{1}Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India.**Abstract:** In this paper we present graceful and $\alpha $-graceful labeling for some cycle related graphs. We have proved that the graph obtained by adding two pendent vertices at distance two and one chord between them in a cycle $C_n$ and the graph obtained by adding arbitrary pendent vertices at two different places at distance two in a cycle $C_n$, when $n$ is odd are graceful graphs, while the graph obtained by adding alternate pendent vertices in a cycle $C_n$, when $n$ is even and the graph obtained by adding arbitrary pendent vertices at two different places at distance two in a cycle $C_n$, when $n$ is even are $\alpha $-graceful graphs.

**Keywords:** Labeling, Graceful Labeling, $\alpha$-Graceful Labeling, Cycle Related Graphs.

**Cite this article as:** V. J. Kaneria, Payal Akbari and Nishaba Parmar, *Some Graceful and $\alpha$-Graceful Labeling for Cycle Related Graphs*, Int. J. Math. And Appl., vol. 9, no. 4, 2021, pp. 65-73.

**References**

- F. Harary, Graph theory, Addition Wesley, Massachusetts, (1972).
- A. Rosa, On certain valuation of graph, Theory of Graphs (Rome, July 1966), Goden and Breach, N. Y. and Paris, (1967), 349-355.
- S. W. Golomb, How to number a graph, Graph Theory and Computing (R. C. Read. Ed.) Academic Press, New York, (1972), 23-37.
- J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronics Journal of Combinatorics, 22(2020), \#DS6.
- V. J. Kaneria and H. M. Makadia, Some graceful graphs, J. of Math. Res., 4(1)(2012), 54-57.
- V. J. Kaneria, H. M. Makadia and M. M. Jariya, Graceful labeling for cycle of graphs, Int. J. of Math. Res., 6(2)(2014), 173-178.