2024/09/26 by Ryuji Inagaki, Tanya Khovanova, Inagaki, Ryota +2 · 1 citation
Computer Science · Biochemistry, Genetics and Molecular Biology · #Interconnection Networks and Systems #Cellular Automata and Applications #DNA and Biological Computing
paper · pdf · doi:10.48550/arxiv.2410.00039
Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can fire if the number of chips placed on it is at least its degree. In our case, a vertex can fire if it has at least 3 chips, and it fires by dispersing 1 chip to each neighbor. Motivated by a 2023 paper by Musiker and Nguyen on this setting of chip-firing, we give an upper bound for the number of stable configurations when we place 2^ℓ - 1 labeled chips at the root. When starting with N chips at the root where N is a positive integer, we determine the number of times each vertex fires when N is not necessarily of the form 2^ℓ - 1. We also calculate the total number of fires in this case.