Eigenvalues of the Hodge Laplacian on digraphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916923921399808 |
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| author | Grigor'yan, Alexander Lin, Yong Yau, S. -T. Zhang, Haohang |
| author_facet | Grigor'yan, Alexander Lin, Yong Yau, S. -T. Zhang, Haohang |
| contents | 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 $\mathcal{L}_{p}=\partial ^{\ast }\partial$ also acting on $Ω_{p}$. Knowing the spectra of $Δ_{p}^{(a)}$ for all values of $p$, we compute the spectra of $\mathcal{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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of the Hodge Laplacian on digraphs Grigor'yan, Alexander Lin, Yong Yau, S. -T. Zhang, Haohang Combinatorics Algebraic Topology 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 $\mathcal{L}_{p}=\partial ^{\ast }\partial$ also acting on $Ω_{p}$. Knowing the spectra of $Δ_{p}^{(a)}$ for all values of $p$, we compute the spectra of $\mathcal{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. |
| title | Eigenvalues of the Hodge Laplacian on digraphs |
| topic | Combinatorics Algebraic Topology |
| url | https://arxiv.org/abs/2406.09814 |