Chip-Firing on Infinite $k$-ary Trees

Dillan Agrawal; Selena Ge; Jate Greene; Tanya Khovanova; Dohun Kim; Rajarshi Mandal; Tanish Parida; Anirudh Pulugurtha; Gordon Redwine; Soham Samanta; Albert Xu

Chip-Firing on Infinite $k$-ary Trees

Combinatorics 2025-01-14 v1

Authors: Dillan Agrawal , Selena Ge , Jate Greene , Tanya Khovanova , Dohun Kim , Rajarshi Mandal , Tanish Parida , Anirudh Pulugurtha , Gordon Redwine , Soham Samanta , Albert Xu

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Abstract

We use an infinite $k$ -ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with $N$ chips at the root. We describe the stable configurations. We calculate the number of fires for each vertex and the total number of fires. We study a sequence of the number of root fires for a given $k$ as a function of $N$ and study its properties. We do the same for the total number of fires.

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@article{arxiv.2501.06675,
  title  = {Chip-Firing on Infinite $k$-ary Trees},
  author = {Dillan Agrawal and Selena Ge and Jate Greene and Tanya Khovanova and Dohun Kim and Rajarshi Mandal and Tanish Parida and Anirudh Pulugurtha and Gordon Redwine and Soham Samanta and Albert Xu},
  journal= {arXiv preprint arXiv:2501.06675},
  year   = {2025}
}

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22 pages, 1 figure

Chip-Firing on Infinite $k$-ary Trees

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