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 K\"{u}nneth 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 Δₚ⁽ᵃ⁾ to those of operators Lₚ=∂ ∂ also acting on Ω ₚ. Knowing the spectra of Δₚ⁽ᵃ⁾ for all values of p, we compute the spectra of Lₚ and then the spectra of Δ ₚ. This program yields the spectra of all operators Δ ₚ on all n-cubes and n-tori.