  The graph of a Hecke operator encodes all information about the action of
this operator on automorphic forms. Let $X$ be a curve over $\mathbb{F}_q$, $F$
its function field and $\mathbb{A}$ the adele ring of $F$. In this paper we
will exhibit the first properties for the graph of Hecke operators for
$\mathrm{GL}_n(\mathbb{A}),$ for every $n \geq 1.$ This includes a description
of the graph in terms of coherent sheaves on $X.$ We provide a numerical
condition for two vertices to be connected by an edge. Moreover, we describe
how to calculate these graphs in the case of the projective line $X =
\mathbb{P}^1(\mathbb{F}_q).$
