  The graph of a Hecke operator encodes all information about the action of
this operator on automorphic forms over a global function field. These graphs
were introduced by Lorscheid in his PhD thesis for $\text{PGL}_{2}$ and we
generalized to $\text{GL}_{n}$ in the paper "On graphs of Hecke operators".
After reviewing some general properties, we explain the connection to the Hall
algebra of the function field. In the case of an elliptic function field, we
can use structure results of Burban-Schiffmann and Fratila to develop an
algorithm which explicitly calculate these graphs. We apply this algorithm to
determine some structure constants and provide explicitly the rank two case in
the last section.
