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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2507.22200 |
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| _version_ | 1866915417152290816 |
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| author | Berkolaiko, Gregory Bronski, Jared C. Goresky, Mark |
| author_facet | Berkolaiko, Gregory Bronski, Jared C. Goresky, Mark |
| contents | An oscillation formula is established for the $k$-th eigenvector (assumed to be simple and with non-zero entries) of a weighted graph operator. The formula directly attributes the number of sign changes exceeding $k-1$ to the cycles in the graph, by identifying it as the Morse index of a weighted cycle intersection form introduced in the text.
Two proofs are provided for the main result. Additionally, it is related to the nodal--magnetic theorem of Berkolaiko and Colin de Verdière and to a similar identity of Bronski, DeVille and Ferguson obtained for the linearization of coupled oscillator network equations around a known solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oscillation of graph eigenfunctions Berkolaiko, Gregory Bronski, Jared C. Goresky, Mark Spectral Theory An oscillation formula is established for the $k$-th eigenvector (assumed to be simple and with non-zero entries) of a weighted graph operator. The formula directly attributes the number of sign changes exceeding $k-1$ to the cycles in the graph, by identifying it as the Morse index of a weighted cycle intersection form introduced in the text. Two proofs are provided for the main result. Additionally, it is related to the nodal--magnetic theorem of Berkolaiko and Colin de Verdière and to a similar identity of Bronski, DeVille and Ferguson obtained for the linearization of coupled oscillator network equations around a known solution. |
| title | Oscillation of graph eigenfunctions |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2507.22200 |