Eigenvalues of the Hodge Laplacian on digraphs

TL;DR

This paper introduces a novel method for computing eigenvalues of the Hodge Laplacian on directed graphs, utilizing normalized operators, product rules, and spectral relations to efficiently determine spectra on complex structures.

Contribution

It develops a new inductive approach for calculating Hodge Laplacian spectra on Cartesian powers of graphs using normalized operators and spectral relations.

Findings

Spectra of Hodge Laplacians on n-cubes and n-tori are computed.

The product rule applies to normalized Hodge operators, enabling inductive spectral calculations.

Spectral relations facilitate computation of Laplacian spectra from normalized operator spectra.

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 $\Delta _{p}^{(a)},$ 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.