Eigenvalues of the Hodge Laplacian on digraphs

2024/06/14 by Grigor'yan, Alexander, Lin, Yong, Yau, S. -T. +1
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.09814

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 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 L_p=∂ ^∗ ∂ also acting on Ω_p. Knowing the spectra of Δ_p^(a) for all values of p, we compute the spectra of 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.

Related

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