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The paper by M. Baker and S. Norine in 2007 introduced a new parameter onconfigurations of graphs and gave a new result in the theory of graphs whichhas an algebraic geometry flavour. This result was called Riemann-Roch formulafor graphs since it defines a combinatorial version of divisors and their ranksin terms of configuration on graphs. The so called chip firing game on graphsand 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 inpurely combinatorial terms, which is more accessible and shorter than theoriginal one. An algorithm for the determination of the rank of configurationsis also given for the complete graph $K_n$. This algorithm has lineararithmetic complexity. The analysis of number of iterations in a less optimizedversion of this algorithm leads to an apparently new parameter which we callthe prerank. This parameter and the classical area parameter provide analternative description to some well known $q,t$-Catalan numbers. Restricted toa natural subset of configurations, the two natural statistics degree and rankin Riemann-Roch formula lead to a distribution which is described by agenerating function which, up to a change of variables, is a symmetric fractioninvolving two copies of Carlitz q-analogue of the Catalan numbers.

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