Repelling curvature via $ε-$repelling Laplacian on positive connected signed graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866914464328056832 |
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| author | Lin, Yong Wan, Shi |
| author_facet | Lin, Yong Wan, Shi |
| contents | The paper defines a positive semidefinite operator called $ε-$repelling Laplacian on a positive connected signed graph where $ε$ is an arbitrary positive number less than a constant $ε_0$ related to the graph's consensus problem. Then we investigate the upper bound of the second smallest eigenvalue of $ε-$repelling Laplacian. Besides, we use the pseudoinverse of $ε-$repelling Laplacian to construct a simplex as well as $ε-$repelling cost whose square root turns out to be a distance among the vertices of the simplex. We also extend the node and edge resistance curvature proposed by K.Devriendt et al. to node and edge $ε-$repelling curvature and derive the corresponding Lichnerowicz inequalities on any positive connected signed graph. Moreover, it turns out that edge $ε-$repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is $ε-$repelling cost rather than the length of the shortest path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Repelling curvature via $ε-$repelling Laplacian on positive connected signed graphs Lin, Yong Wan, Shi Spectral Theory The paper defines a positive semidefinite operator called $ε-$repelling Laplacian on a positive connected signed graph where $ε$ is an arbitrary positive number less than a constant $ε_0$ related to the graph's consensus problem. Then we investigate the upper bound of the second smallest eigenvalue of $ε-$repelling Laplacian. Besides, we use the pseudoinverse of $ε-$repelling Laplacian to construct a simplex as well as $ε-$repelling cost whose square root turns out to be a distance among the vertices of the simplex. We also extend the node and edge resistance curvature proposed by K.Devriendt et al. to node and edge $ε-$repelling curvature and derive the corresponding Lichnerowicz inequalities on any positive connected signed graph. Moreover, it turns out that edge $ε-$repelling curvature is no more than the Lin-Lu-Yau curvature of the underlying graph whose transport cost is $ε-$repelling cost rather than the length of the shortest path. |
| title | Repelling curvature via $ε-$repelling Laplacian on positive connected signed graphs |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2506.10492 |