Distinguishing finite metric spaces via similarity spectra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916069317279744 |
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| author | O'Hara, Jun |
| author_facet | O'Hara, Jun |
| contents | We study spectra and characteristic polynomials of similarity matrices associated with finite metric spaces, where the similarity matrix of a finite metric space $X=\{x_1,\dots,x_n\}$ is given by $\displaystyle Z(q)=(q^{d(x_i,x_j)})_{i,j},$ where $d(x_i,x_j)$ denotes the distance between $x_i$ and $x_j$. %
We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the normalized $q$-spectrum, defined respectively from $Z(q)$ and its normalized transition matrix. In the case of graphs, these invariants recover the adjacency spectrum and the Laplacian spectrum in the limit $q\to0$.
Our main result shows that the $q$-spectrum determines a large class of finite metric spaces under a natural nondegeneracy condition. We also prove that all four-point metric spaces are determined by their $q$-spectra. %
The key observation is that the coefficients of the characteristic polynomial of $Z(q)$ encode cycle structures of the underlying metric space. We further investigate the normalized $q$-spectrum and present computational examples comparing these invariants with classical graph spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinguishing finite metric spaces via similarity spectra O'Hara, Jun Metric Geometry 05C50, 05C60, 51F99 We study spectra and characteristic polynomials of similarity matrices associated with finite metric spaces, where the similarity matrix of a finite metric space $X=\{x_1,\dots,x_n\}$ is given by $\displaystyle Z(q)=(q^{d(x_i,x_j)})_{i,j},$ where $d(x_i,x_j)$ denotes the distance between $x_i$ and $x_j$. % We introduce two spectral invariants of finite metric spaces, the $q$-spectrum and the normalized $q$-spectrum, defined respectively from $Z(q)$ and its normalized transition matrix. In the case of graphs, these invariants recover the adjacency spectrum and the Laplacian spectrum in the limit $q\to0$. Our main result shows that the $q$-spectrum determines a large class of finite metric spaces under a natural nondegeneracy condition. We also prove that all four-point metric spaces are determined by their $q$-spectra. % The key observation is that the coefficients of the characteristic polynomial of $Z(q)$ encode cycle structures of the underlying metric space. We further investigate the normalized $q$-spectrum and present computational examples comparing these invariants with classical graph spectra. |
| title | Distinguishing finite metric spaces via similarity spectra |
| topic | Metric Geometry 05C50, 05C60, 51F99 |
| url | https://arxiv.org/abs/2502.08980 |