2016/05/09 by Demara Austin, Austin, Demara, M. M. Chambers +7
Computer Science · Mathematics · Physics and Astronomy · #05A15 #05C30 #05C31 (Secondary) #05C99 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1605.02713
openalex publication_date 2016/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The (univariate) avalanche polynomial of a graph, introduced by Cori, Dartois and Rossin in 2006, captures the distribution of the length of (principal) avalanches in the abelian sandpile model. This polynomial has been used to show that the avalanche distribution in the sandpile model on a multiple wheel graph does not follow the expected power law function. In this article, we introduce the (multivariate) avalanche polynomial that enumerates the toppling sequences of all principal avalanches. This polynomial generalizes the univariate avalanche polynomial and encodes more information. In particular, the avalanche polynomial of a tree uniquely identifies the underlying tree. In this paper, the avalanche polynomial is characterized for trees, cycles, wheels, and complete graphs.