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