Eigenvalues of the Hodge Laplacian on digraphs

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 ${\mathrm{\Delta }}_p^{(a)},$ where $a$ refers to the weight that is used to redefine the inner product in the spaces ${\mathrm{\Omega }}_p$. This together with the Künneth formula for product allows us to compute inductively the spectra of all normalized Hodge operators ${\mathrm{\Delta }}_p^{(a)}$ on Cartesian powers including $n$-cubes and $n$-tori. (II) We relate in a certain way the spectra of ${\mathrm{\Delta }}_p$ and ${\mathrm{\Delta }}_p^{(a)}$ to those of operators ${\mathcal{L}}_p={\mathrm{\partial }}^{\ast }\mathrm{\partial }$ also acting on ${\mathrm{\Omega }}_p$. Knowing the spectra of ${\mathrm{\Delta }}_p^{(a)}$ for all values of $p$, we compute the spectra of ${\mathcal{L}}_p$ and then the spectra of ${\mathrm{\Delta }}_p\mathrm{.}$ This program yields the spectra of all operators ${\mathrm{\Delta }}_p$ on all $n$-cubes and $n$-tori.