Pebbling in Sierpiński type graphs

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics and Actuarial Science, B.S. Abdur Rahman Crescent Institute of Science & Technology, Chennai, India

2 Department of Mathematics, Bharathi Women’s College, Chennai, India

Abstract

Graph pebbling is a network optimization technique for the movement of resources in transit. A pebbling move in a connected graph G can be defined as a distribution of pebbles on the vertices of a graph, which involves removing two pebbles from a vertex, placing one pebble on one of its adjacent vertices, and discarding the other pebble. For a graph G, the pebbling number f(G) is the minimum number of pebbles required such that one pebble is moved to any arbitrary vertex of the graph G. Fractals are described as intricate patterns that are identical at different dimensions or identical in all dimensions. In this paper, the strategy of pebbling is applied to Sierpi'nski graphs which are well known fractals and several critical points are scrutinized and verified for Generalized Sierpi'nski graph S(G,t),t≥2, Sierpi'nski graph S(Kn,t), t≥1, n≥2 and Sierpi'nski triangle graph Sm, m≥2.

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References

[1] L. Alcón and G. Hurlbert, Pebbling in powers of paths, Discrete Math. 346 (2023), no. 5, Article ID: 113315. https://doi.org/10.1016/j.disc.2023.113315
[2] M. Chris Monica and A. Sagaya Suganya, Pebbling in flower snark graph, J. Pure Appl. Math. 13 (2017), no. 6, 1835–1843.
[3] F.R.K. Chung, Pebbling in hypercubes, SIAM J. Discrete Math. 2 (1989), no. 4, 467–472.  https://doi.org/10.1137/0402041
[4] D. Flocco, J. Pulaj, and C. Yerger, Automating weight function generation in graph pebbling, Discrete Appl. Math. 347 (2024), 155–174.  https://doi.org/10.1016/j.dam.2023.12.022
[5] S. Gravier, M. Kovše, and A. Parreau, Generalized sierpi´nski graphs, 2011. 
[6] A.M. Hinz, S. Klavžar, and S.S. Zemljič, A survey and classification of sierpiński-type graphs, Discrete Appl. Math. 217 (2017), no. 3, 565–600.  https://doi.org/10.1016/j.dam.2016.09.024
[7] M. Joice Punitha, A Sagaya Suganya, and S. Kalyan SK, Computational game theory model for adhoc networks, Solid State Technol. 63 (2020), no. 6, 1816–1829.
[8] F. Kenter, D. Skipper, and D. Wilson, Computing bounds on product graph pebbling numbers, Theoret. Comput. Sci. 803 (2020), 160–177.  https://doi.org/10.1016/j.tcs.2019.09.050
[9] F. Kenter, D. Skipper, and D. Wilson, Computing bounds on product graph pebbling numbers, Theoret. Comput.
Sci. 803 (2020), 160–177.  https://doi.org/10.1016/j.tcs.2019.09.050
[10] S. Klavžar and U. Milutinović, Graphs s(n, k) and a variant of the tower of hanoiproblem, Czechoslovak Math. J. 47 (1997), no. 1, 95–104.  https://doi.org/10.1023/A:1022444205860.
[11] S.L. Lipscomb and J.C. Perry, Lipscomb’s L(A) space fractalized in hilbert’s L2(A) space, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1157–1165.
[12] A. Lourdusamy, I. Dhivviyanandam, and S. Kither Iammal, Monophonic pebbling number and t-pebbling number of some graphs, AKCE Int. J. Graphs Comb. 19 (2022), no. 2, 108–111. https://doi.org/10.1080/09728600.2022.2072789
[13] N. Pleanmani, Graham’s pebbling conjecture holds for the product of a graph and a sufficiently large complete graph, Theory Appl. Graphs 7 (2020), no. 1, Article 1.  https://doi.org/10.20429/tag.2020.070101
[14] N. Pleanmani, N. Nupo, and S. Worawiset, Bounds for the pebbling number of product graphs, Trans. Comb. 11 (2022), no. 4, 317–326.  https://doi.org/10.22108/toc.2021.128705.1855
[15] C. Puente, J. Romeu, R. Pous, X. Garcia, and F. Benitez, Fractal multiband antenna based on the sierpinski triangle, Electron. Lett. 32 (1996), no. 1, 1–2.  https://doi.org/10.1049/el:19960033
[16] M.J. Punitha and A. Sagaya Suganya, 2-pebbling property of butterfly-derived graphs, Int. J. Dyn. Syst. Differ. Equ. 11 (2021), no. 5-6, 566–578.  https://doi.org/10.1504/IJDSDE.2021.120050
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 28 March 2025

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