Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866913655029760000 |
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| author | Grudsky, S. M. Maximenko, E. A. Soto-González, A. |
| author_facet | Grudsky, S. M. Maximenko, E. A. Soto-González, A. |
| contents | We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $α\in\mathbb{C}$, with $\operatorname{Re}(α)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(α) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(α)<0$ and $n>\operatorname{Re}(α-1)/\operatorname{Re}(α)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(α)^2/(2\operatorname{Re}(α)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07514 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge Grudsky, S. M. Maximenko, E. A. Soto-González, A. Spectral Theory Rings and Algebras 05C50, 47B36, 15A18, 15B05, 41A60, 65F15 We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order $n$, where one edge has weight $α\in\mathbb{C}$, with $\operatorname{Re}(α)<0$, and all the others have weights $1$. This paper is a sequel of a previous one where we considered $\operatorname{Re}(α) \in[0,1]$ (Eigenvalues of laplacian matrices of the cycles with one weighted edge, Linear Algebra Appl. 653, 2022, 86--115). We prove that for $\operatorname{Re}(α)<0$ and $n>\operatorname{Re}(α-1)/\operatorname{Re}(α)$, one eigenvalue is negative while the others belong to $[0,4]$ and are distributed as the function $x\mapsto 4\sin^2(x/2)$. Additionally, we prove that as $n$ tends to $\infty$, the outlier eigenvalue converges exponentially to $4\operatorname{Re}(α)^2/(2\operatorname{Re}(α)-1)$. We give exact formulas for the half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as $n$ tends to $\infty$, both for the eigenvalues belonging to $[0,4]$ and the outlier. We also compute the eigenvectors and their norms. |
| title | Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge |
| topic | Spectral Theory Rings and Algebras 05C50, 47B36, 15A18, 15B05, 41A60, 65F15 |
| url | https://arxiv.org/abs/2308.07514 |