Some results on (a,d)-distance antimagic labeling

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-2020-02-0022

Keywords:

Distance magic graphs, (a, d)-distance antimagic graphs, Circulant graphs, Cartesian and corona product of graphs

Abstract

Let G = (V,E) be a graph of order N and f : V ? {1, 2,...,N} be a bijection. For every vertex v of graph G, we define its weight w(v) as the sum ?u?N(v) f(u), where N(v) denotes the open neighborhood of v. If the set of all vertex weights forms an arithmetic progression {a, a + d, a + 2d, . . . , a + (N ? 1)d}, then f is called an (a, d)-distance antimagic labeling and the graph G is called (a, d)-distance antimagic graph. In this paper we prove the existence or non-existence of (a, d)- distance antimagic labeling of some well-known graphs.

Author Biographies

S. K. Patel, Government Engineering College.

Dept. of Mathematics.

Jayesh Vasava, Gujarat University.

Dept. of Mathematics.

References

S. Arumugam and N. Kamatchi, “On (a, d)-distance antimagic graphs”, Australasian journal of combinatorics, vol. 54, pp. 279-287, 2012. [On line]. Available: https://bit.ly/2yHobye

D. Froncek, P. Kovar and T. Kovarova, “Fair incomplete tournaments”, Bulletin of the institute of combinatorics and its applications, vol. 48, pp. 31-33, 2006.

D. Froncek, “Handicap distance antimagic graphs and incomplete tournaments”, AKCE international journal of graphs and combinatorics, vol. 10, no. 2, pp. 119-127, 2013.

J. Gallian, “A Dynamic Survey of Graph Labeling”, 19th ed. The electronics journal of combinatorics, vol. # DS6, pp. 1-394, 2016. [On line]. Available: https://bit.ly/2KxqG95

J. L. Gross and J. Yellen, Graph theory and its applications. Boca Raton, CA: CRC, 1999.

M. Miller, C. Rodger and R. Simanjuntak, “Distance magic labelings of graphs”, Australasian journal of combinatorics, vol. 28, pp. 305-315, 2003. [On line]. Available: https://bit.ly/2y2gumn

N. Nalliah, "Antimagic labelings of graphs and digraphs", Ph. D. Thesis, Kalasalingam University, Faculty of science and humanities, 2013. [On line]. Available: https://bit.ly/3cMl2Mj

M. F. Semeniuta, “(a, d)-Distance antimagic labeling of some types of graphs”, Cybernetics and systems analysis, vol. 52, no. 6, pp. 950–955, Nov. 2016, doi: 10.1007/s10559-016-9897-z

R. Simanjuntak and K. Wijaya, ”On distance antimagic graphs”, Dec. 2013, arxiv:1312.7405v1.

K. A. Sugeng, D. Froncek, M. Miller, J. Ryan and J. Walker, “On distance magic labeling of graphs”, Journal of combinatorial mathematics and combinatorial computing, vol. 71, pp. 39-48, 2009.

V. Vilfred, “Σ-labelled graph and circulant graphs”, Ph. D. Thesis University of Kerala, Department of mathematics, 1994. [On line]. Available: https://bit.ly/2y1wIvT

Published

2020-04-27

How to Cite

[1]
S. K. Patel and J. Vasava, “Some results on (a,d)-distance antimagic labeling”, Proyecciones (Antofagasta, On line), vol. 39, no. 2, pp. 361-381, Apr. 2020.

Issue

Section

Artículos