  For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$
acting on the space of $n$-differentials on a confinite Riemann surface $X$
which has ramification points. The trace formula for the resolvent kernel is
developed along the line \`a la Selberg. Using the trace formula, we compute
the regularized determinant of $\Delta_n+s(s+2n-1)$, from which we deduce the
regularized determinant of $\Delta_n$, denoted by $\det\!'\Delta_n$. Taking
into account the contribution from the absolutely continuous spectrum,
$\det\!'\Delta_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when
$n\geq 2$. Here $Z(s)$ is the Selberg zeta function of $X$. When $n=0$ or
$n=1$, $Z(n)$ is replaced by the leading coefficient of the Taylor expansion of
$Z(s)$ around $s=0$ and $s=1$ respectively. The constants $\mathcal{C}_n$ are
calculated explicitly. They depend on the genus, the number of cusps, as well
as the ramification indices, but is independent of the moduli parameters.
