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