Chip-Firing and Rotor-Routing on Directed Graphs

Abstract

Abstract. We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

“…(15) is an integer, and as a consequence of Corollary 5.5, in all relevant cases the exponent of the highest power of p ∈ P dividing this integer is at most t, so that an integer µ p,t for which (15) holds indeed can be found. Next, Theorem 2.5 states thatẽ p t Mp − λ p,t e Mp is in S(m, d) if and only if p t M p − λ p,t M p ≡ 0 mod m, or, equivalently, if p t ≡ λ p,t mod m, which holds since it is just the second requirement in (15). By the same theorem, obviouslyẽ Mp = π p (m)e Mp is also in S(m, d), and the same holds forẽ M 2 andẽ m/2 as defined in (16 (15).…”

“…In that case, S(G, v) does not depend on the vertex v and is equal to the critical group S(G) of G, essentially because in that case, not only the columns, but also the rows of the Laplacian ∆ of G sum to zero; for a detailed proof, see [15] (note that the proof as given there does not use the assumption that the directed graph is connected). For more details on sandpile groups and the critical group of a directed graph, we refer for example to [15] or [33].…”

“…For more details on sandpile groups and the critical group of a directed graph, we refer for example to [15] or [33].…”

“…The sandpile group S(G, v) of a directed graph or digraph G at a vertex v is an abelian group obtained from the reduced Laplacian ∆ v of G; by the Matrix Tree Theorem [32], its order is equal to the number of directed trees rooted at v, see for example [19]. If the graph G is Eulerian, then the sandpile group does not depend on the vertex v and is equal to the critical group S(G) of G [15]. Most of the literature on sandpile groups is concerned with undirected graphs, which can be considered as a special case, namely for directed graphs that are obtained by replacing each undirected edge in a graph by a pair of directed edges oriented in opposite directions.…”

“…For more details and background, see, for example, [21,29,1] for the undirected case and [15,33] for the directed case.…”