  Let $\frak{p}$ be a prime ideal of $\mathbb{F}_q[T]$. Let $J_0(\frak{p})$ be
the Jacobian variety of the Drinfeld modular curve $X_0(\frak{p})$. Let $\Phi$
be the component group of $J_0(\frak{p})$ at the place $1/T$. We use graph
Laplacians to estimate the order of $\Phi$ as $\mathrm{deg}(\frak{p})$ goes to
infinity. This estimate implies that $\Phi$ is not annihilated by the
Eisenstein ideal of the Hecke algebra $\mathbb{T}(\frak{p})$ acting on
$J_0(\frak{p})$ once the degree of $\frak{p}$ is large enough. We also obtain
an asymptotic formula for the size of the discriminant of
$\mathbb{T}(\frak{p})$ by relating this discriminant to the order of $\Phi$; in
this problem the order of $\Phi$ plays a role similar to the Faltings height of
classical modular Jacobians. Finally, we bound the spectrum of the adjacency
operator of a finite subgraph of an infinite diagram in terms of the spectrum
of the adjacency operator of the diagram itself; this result has applications
to the gonality of Drinfeld modular curves.
