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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.04836 |
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| _version_ | 1866911302095470592 |
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| author | Oliveira, Elismar R. Szutkoski, Jonas Trevisan, VIlmar |
| author_facet | Oliveira, Elismar R. Szutkoski, Jonas Trevisan, VIlmar |
| contents | This paper investigates limit points of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a quadractic one-parameter family of matrices. First, we show that any value greater or equal to 1 is a deformed Laplacian limit point (for different values of the parameter $s$) using a simple family of trees. Second, we define $(T_k)_{k \in \mathbb{N}}$ the Shearer's sequence of caterpillars for $λ>1$ and we present a convergence criterion based on Shearer's approach. Our main result is that for any fixed value $λ_0>1$ there exists a unique value $0<s^* <\sqrt{λ_0} -1$ such that, and for any $s \in (0,s^*)$ the interval $[λ_0, \; +\infty)$ is entirely formed by $s$-deformed Laplacian limit points (for the same value of $s$). Finally, we provide some numerical data exploring the limit properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of deformed Laplacian limit points Oliveira, Elismar R. Szutkoski, Jonas Trevisan, VIlmar Combinatorics This paper investigates limit points of the deformed Laplacian matrix, which merges the Laplacian and signless Laplacian matrices of a graph through a quadractic one-parameter family of matrices. First, we show that any value greater or equal to 1 is a deformed Laplacian limit point (for different values of the parameter $s$) using a simple family of trees. Second, we define $(T_k)_{k \in \mathbb{N}}$ the Shearer's sequence of caterpillars for $λ>1$ and we present a convergence criterion based on Shearer's approach. Our main result is that for any fixed value $λ_0>1$ there exists a unique value $0<s^* <\sqrt{λ_0} -1$ such that, and for any $s \in (0,s^*)$ the interval $[λ_0, \; +\infty)$ is entirely formed by $s$-deformed Laplacian limit points (for the same value of $s$). Finally, we provide some numerical data exploring the limit properties. |
| title | Distribution of deformed Laplacian limit points |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.04836 |