  This paper aims to compute and estimate the eigenvalues of the Hodge
Laplacians on directed graphs. We have devised a new method for computing Hodge
spectra with the following two ingredients.
  (I) We have observed that the product rule does work for the so-called
normalized Hodge operator, denoted by $\Delta _{p}^{(a)},$ where $a$ refers to
the weight that is used to redefine the inner product in the spaces $\Omega
_{p}$. This together with the K\"{u}nneth formula for product allows us to
compute inductively the spectra of all normalized Hodge operators $\Delta
_{p}^{(a)}$ on Cartesian powers including $n$-cubes and $n$-tori.
  (II) We relate in a certain way the spectra of $\Delta _{p}$ and
$\Delta_{p}^{(a)}$ to those of operators $\mathcal{L}_{p}=\partial ^{\ast
}\partial$ also acting on $\Omega _{p}$. Knowing the spectra of
$\Delta_{p}^{(a)}$ for all values of $p$, we compute the spectra of
$\mathcal{L}_{p} $ and then the spectra of $\Delta _{p}.$ This program yields
the spectra of all operators $\Delta _{p}$ on all $n$-cubes and $n$-tori.
