Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

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.

Chip-Firing on Infinite $k$-ary Trees

Abstract

We use an infinite -ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with 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 as a function of and study its properties. We do the same for the total number of fires.
Paper Structure (18 sections, 20 theorems, 74 equations, 1 figure, 4 tables)

This paper contains 18 sections, 20 theorems, 74 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. History and Background
  3. Preliminaries
  4. The underlying graph
  5. Chip-firing process
  6. Base $k$
  7. Stable configurations
  8. The number of times each vertex fires
  9. The number of fires for a vertex on layer $i+1$
  10. The number of root fires
  11. Special case of $N = \frac{k^n-1}{k-1}$
  12. Differences
  13. The total number of fires
  14. The total number of fires
  15. Special case of $N = \frac{k^n-1}{k-1}$
  16. ...and 3 more sections

Key Result

Theorem 1

Let $N$ be the total number of chips. We have:

Figures (1)

  • Figure 1: An infinite undirected rooted 5-ary tree with a self-loop at the root

Theorems & Definitions (43)

  • Theorem 1: Theorem 2.3 in BLS91
  • Theorem 2: Theorem 2.2.2 in Kli19
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Example 1
  • Theorem 6
  • ...and 33 more