Chip-Firing and Riemann-Roch Theory for Directed Graphs

Abstract

We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a v0-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the "column" chip-firing game is related to directed G-parking functions and the "row" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, which after a simple transformation may be viewed as a special class of directed graphs which will always have the Riemann-Roch property for the column chip-firing game. Examples of arithmetical graphs are provided which demonstrate that either, both, or neither of the two Riemann-Roch conditions may be satisfied for the row chip-firing game.

Chip firing on Dynkin diagrams and McKay quivers

“…Now set u (0) := s α * (λ). Using induction on the cardinality of the set M (u (k) ) defined in Section 3.2, we can create a sequence u (0) , u (1) , . .…”

Arithmetical structures on bidents

“…We note that a sequence {r i } with 0 < r i+1 < r i and r i+1 ≡ −r i−1 (mod r i ) as in the proof of Proposition 2.4 is what is referred to as a Euclidean chain in [1].…”

“…This lemma is the same as [2, Lemma 13] except that it also allows for subdivision at vertex v +1 , and the proof follows directly from the definitions. As an example, observe that the arithmetical structure shown in Figure 5(d) can be obtained from the arithmetical structure shown in Figure 5(a) using any of b = (2, 2, 1), (2, 1, 3), or (1,3,3). Lemma 2.7 implies that the order of subdivision along the tail does not matter unless the subdivisions are adjacent to each other.…”

“…Lorenzini [9,Lemma 1.6] also shows that any finite, connected graph has a finite number of arithmetical structures; however, his proof does not give a bound on the number of such structures. Recent work in [1], [2], [4], and [5] involves studying arithmetical structures and their critical groups on various families of connected graphs. In [2], the authors show that the number of arithmetical structures on the path graph P n is given by the Catalan number C(n − 1) and that the number of arithmetical structures on the cycle graph C n is given by the binomial coefficient 2n−1 n−1 .…”

Chip-firing based methods in the Riemann–Roch theory of directed graphs

“…In 2007, Baker and Norine proved a graph-theoretic analogue of the classical Riemann-Roch theorem for algebraic curves [3]. This result inspired much research about Riemann-Roch theorems on tropical curves, lattices and directed graphs [1,2,10].…”