Eigenvalue bounds of the Kirchhoff Laplacian
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910455485693952 |
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| author | Knill, Oliver |
| author_facet | Knill, Oliver |
| contents | We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10968 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Eigenvalue bounds of the Kirchhoff Laplacian Knill, Oliver Combinatorics Discrete Mathematics 05Cxx, 05Exx, 68Rxx We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver. |
| title | Eigenvalue bounds of the Kirchhoff Laplacian |
| topic | Combinatorics Discrete Mathematics 05Cxx, 05Exx, 68Rxx |
| url | https://arxiv.org/abs/2205.10968 |