Report
Eigenvalues of the Hodge Laplacian on digraphs
| العنوان: | Eigenvalues of the Hodge Laplacian on digraphs |
|---|---|
| المؤلفون: | Grigor'yan, Alexander, Lin, Yong, Yau, S. -T., Zhang, Haohang |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Algebraic Topology |
| الوصف: | 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. |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2406.09814 |
| رقم الانضمام: | edsarx.2406.09814 |
| قاعدة البيانات: | arXiv |
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