The Avalanche Polynomial of a Graph

TL;DR

This paper introduces a multivariate avalanche polynomial for graphs, generalizing the univariate version, and characterizes it for various graph classes, revealing its ability to uniquely identify certain graphs.

Contribution

It defines and characterizes the multivariate avalanche polynomial, extending previous work and providing new insights into graph identification and avalanche distributions.

Findings

Multivariate avalanche polynomial generalizes the univariate version.

The polynomial uniquely identifies trees among graphs.

Characterizations are provided for trees, cycles, wheels, and complete graphs.

Abstract

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.