Eigenvalues of the Hodge Laplacian on digraphs

Abstract

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