Let $X$ be a curve over $\F_q$ with function field $F$. In this paper, we
define a graph for each Hecke operator with fixed ramification. A priori, these
graphs can be seen as a convenient language to organize formulas for the action
of Hecke operators on automorphic forms. However, they will prove to be a
powerful tool for explicit calculations and proofs of finite dimensionality
results.

We develop a structure theory for certain graphs $G_x$ of unramified Hecke
operators, which is of a similar vein to Serre's theory of quotients of Bruhat
Tits trees. To be precise, $G_x$ is locally a quotient of a Bruhat Tits tree
and has finitely many components. An interpretation of $G_x$ in terms of rank 2
bundles on $X$ and methods from reduction theory show that $G_x$ is the union
of finitely many cusps, which are infinite subgraphs of a simple nature, and a
nucleus, which is a finite subgraph that depends heavily on the arithmetics of
$F$.

We describe how one recovers unramified automorphic forms as functions on the
graphs $G_x$. In the exemplary cases of the cuspidal and the toroidal
condition, we show how a linear condition on functions on $G_x$ leads to a
finite dimensionality result. In particular, we re-obtain the
finite-dimensionality of the space of unramified cusp forms and the space of
unramified toroidal automorphic forms.

In an Appendix, we calculate a variety of examples of graphs over rational
function fields.