On the Radio Antipodal Mean Number of Some Grid Related Graphs

T. Arputha Jose; R. Prabakaran; R. Gomathy

doi:10.26713/cma.v14i2.2043

Authors

T. Arputha Jose Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Anna University, Chennai, India https://orcid.org/0000-0003-1902-1250 (unauthenticated)
R. Prabakaran Department of Mathematics, St. Joseph’s Institute of Technology, Anna University, Chennai, India https://orcid.org/0000-0002-2520-8727 (unauthenticated)
R. Gomathy Department of Education, Sri Muthukumaran College of Education, Tamiladu Teacher’s Education University, Chennai, India https://orcid.org/0000-0003-3471-9593 (unauthenticated)

DOI:

https://doi.org/10.26713/cma.v14i2.2043

Keywords:

Communication networks, Channel assignment problem, Radio labeling, Radio antipodal mean number, Triangular grid, Torus grid

Abstract

The radio antipodal mean labeling of a graph \(G\) is a function \(f\) that assigns to each vertex \(u\), a non-negative integer \(f(u)\) such that \(f(u) \neq f(v)\) if \(d(u,v) < \textrm{diam}(G)\) and \(d(u,v)+ \Big\lceil \frac{f(u)+f(v)}{2} \Big\rceil \geq \textrm{diam}(G)\), where \(d(u,v)\) represents the shortest distance between any pair of vertices \(u\) and \(v\) of \(G\) and \(\textrm{diam}(G)\) denotes the diameter of \(G\). The radio antipodal mean number of \(f\), denoted by \(r_{\textit{amn}}(f)\) is the maximum number assigned to any vertex of \(G\). The radio antipodal mean number of \(G\), denoted by \(r_{\textit{amn}}(G)\) is the minimum value of \(r_{\textit{amn}}(f)\) taken over all antipodal mean labeling \(f\) of \(G\). In this paper, the exact values of radio antipodal mean number of some grid related graphs have been obtained.

Views: 919Downloads: 437

Downloads

Download data is not yet available.

References

A. Bossard, On solving the decycling problem in a torus network, Wireless Communications and Mobile Computing 2021 (2021), Article ID 5598173, 6 pages, DOI: 10.1155/2021/5598173.

G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Mathematica Bohemica 127 (2002), 57 – 69, DOI: 10.21136/MB.2002.133978.

G. Chartrand, D. Erwin, P. Zhang and F. Harary, Radio labelings of graphs, Bulletin of the Institute of Combinatorics and its Applications 33 (2001), 77 – 85.

S. N. Daoud, Edge odd graceful labeling of cylinder and torus grid graphs, IEEE Access 7 (2019), 10568 – 10592, DOI: 10.1109/ACCESS.2018.2889293.

J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics DS6(Version 25) (2022), 623 pages, URL: https://www.combinatorics.org/files/Surveys/ds6/ds6v25-2022.pdf.

M. Giridaran, T. A. Jose and E. A. Jeony, On the radio antipodal geometric mean number of ladder related graphs, Malaya Journal of Matematik 10(1) (2022), 98 – 109, DOI: 10.26637/mjm1001/009.

V. S. Gordon, Y. L. Orlovich and F. Werner, Hamiltonian properties of triangular grid graphs, Discrete Mathematics 308(24) (2008), 6166 – 6188, DOI: 10.1016/j.disc.2007.11.040.

J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance 2, SIAM Journal on Discrete Mathematics 5(4) (1992), 586 – 595, DOI: 10.1137/0405048.

W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE 68(12) (1980), 1497 – 1514, DOI: 10.1109/PROC.1980.11899.

F. Havet, Channel assignment and multicolouring of the induced subgraphs of the triangular lattice, Discrete Mathematics 233(1-3) (2001), 219 – 231, DOI: 10.1016/S0012-365X(00)00241-7.

J. C. M. Janssen, Channel assignment and graph labeling, Chapter 5, in: Handbook of Wireless Networks and Mobile Computing, I. Stojmenovic (editor), John Wiley & Sons, Inc., pp. 95 – 117 (2002), DOI: 10.1002/0471224561.

T. A. Jose and M. Giridaran, Radial radio mean labeling of Mongolian tent and diamond graphs, International Journal for Research in Applied Science & Engineering Technology 8(7) (2020), 2078 -– 2083, DOI: 10.22214/ijraset.2020.30759.

S. M. Kang, S. Nazeer, W. Nazeer, I. Kousar and C. Y. Jung, Radio labeling and radio number of caterpillar related graphs, Mitteilungen Klosterneuburg 65(5) (2015), 149 – 159.

M. Kchikech, R. Khennoufa and O. Togni, Linear and cyclic radio k-labelings of trees, Discussiones Mathematicae Graph Theory 27(1) (2007), 105 – 123, DOI: 10.7151/dmgt.1348.

A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (International Symposium, Rome, July 1966), Dunod Gordon & Breach Science Publishers, Inc., New York and Dunod, Paris, pp. 349 – 355 (1967).

L. Saha and P. Panigrahi, A new graph radio k-coloring algorithm, Discrete Mathematics, Algorithms and Applications 11(1) (2019), 1950005, DOI: 10.1142/S1793830919500058.

S. K. Vaidya and P. L. Vihol, Radio labeling for some cycle related graphs, International Journal of Mathematics and Soft Computing 2(2) (2012), 11 – 24.

D. B. West, Introduction to Graph Theory, 2nd edition, Pearson Education, Inc., xix + 512 pages (2001).

D. A. Xavier and R. C. Thivyarathi, Radio antipodal mean number of certain graphs, International Journal of Mathematics Trends and Technology 54(6) (2018), 467 – 470.