Chip-firing and Riemann-Roch theory for directed graphs.

*(English)*Zbl 1274.05189
Nešetřil, Jarik (ed.) et al., Extended abstracts of the sixth European conference on combinatorics, graph theory and applications, EuroComb 2011, Budapest, Hungary, August 29 – September 2, 2011. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 38, 63-68 (2011).

Summary: Baker and Norine developed a graph-theoretic analogue of the classical Riemann-Roch theorem. Amini and Manjunath extended their criteria to all full-dimensional lattices orthogonal to the all ones vector. We show that Amini and Manjunath’s criteria hold for all full-dimensional lattices orthogonal to some positive vector and study some combinatorial examples of such lattices. Two distinct generalizations of the chip-firing game of Baker and Norine to directed graphs are provided. We describe how the “row” chip-firing game is related to the sandpile model and the “column” chip-firing game is related to directed \(G\)-parking functions. We finish with a discussion of arithmetical graphs, introduced by Lorenzini, viewing them as a class of vertex weighted graphs whose Laplacian is orthogonal to a positive vector and describe how they may be viewed as a special class of unweighted strongly connected directed graphs.

For the entire collection see [Zbl 1242.05003].

For the entire collection see [Zbl 1242.05003].

##### MSC:

05C20 | Directed graphs (digraphs), tournaments |

05C10 | Planar graphs; geometric and topological aspects of graph theory |

05C57 | Games on graphs (graph-theoretic aspects) |

##### References:

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