The Riemann-Roch theorem for graphs and the rank in complete graphs

TL;DR

This paper simplifies the Riemann-Roch theorem for graphs, provides an efficient algorithm for complete graphs, and explores new combinatorial parameters related to Catalan numbers and graph configurations.

Contribution

It offers a purely combinatorial presentation of Baker and Norine's Riemann-Roch theorem and introduces a linear-complexity algorithm for computing configuration ranks in complete graphs.

Findings

Algorithm for rank determination in complete graphs with linear complexity

Introduction of the prerank parameter related to configuration analysis

Connection of the statistics to q,t-Catalan numbers and symmetric generating functions

Abstract

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavour. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configuration on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we give a presentation of the theorem of Baker and Norine in purely combinatorial terms, which is more accessible and shorter than the original one. An algorithm for the determination of the rank of configurations is also given for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the classical area parameter provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank in Riemann-Roch formula lead to a distribution which is described by a generating function which, up to a change of variables, is a symmetric fraction involving two copies of Carlitz q-analogue of the Catalan numbers.