Laplacian Spectrum and Domination in Trees
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918191567994880 |
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| author | Rajendraprasad, Deepak Sankaranarayanan, Durga R. |
| author_facet | Rajendraprasad, Deepak Sankaranarayanan, Durga R. |
| contents | For a finite simple undirected graph $G$, let $γ(G)$ denote the size of a smallest dominating set of $G$ and $μ(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities. Hedetniemi, Jacobs and Trevisan [Eur. J. Comb. 2016] showed that for any graph $G$, $μ(G) \leqslant γ(G)$. Cardoso, Jacobs and Trevisan [Graphs Combin. 2017] asks whether the ratio $γ(T)/μ(T)$ is bounded by a constant for all trees $T$. We answer this question by showing that this ratio is less than $4/3$ for every tree. We establish the optimality of this bound by constructing an infinite family of trees where this ratio approaches $4/3$. We also improve this upper bound for trees in which all the vertices other than leaves and their parents have degree at least $k$, for every $k \geqslant 3$. We show that, for such trees $T$, $γ(T)/μ(T) < 1 + 1/((k-2)(k+1))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20318 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Laplacian Spectrum and Domination in Trees Rajendraprasad, Deepak Sankaranarayanan, Durga R. Spectral Theory Combinatorics 05C50, 05C69, 15A18, 15A42 For a finite simple undirected graph $G$, let $γ(G)$ denote the size of a smallest dominating set of $G$ and $μ(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities. Hedetniemi, Jacobs and Trevisan [Eur. J. Comb. 2016] showed that for any graph $G$, $μ(G) \leqslant γ(G)$. Cardoso, Jacobs and Trevisan [Graphs Combin. 2017] asks whether the ratio $γ(T)/μ(T)$ is bounded by a constant for all trees $T$. We answer this question by showing that this ratio is less than $4/3$ for every tree. We establish the optimality of this bound by constructing an infinite family of trees where this ratio approaches $4/3$. We also improve this upper bound for trees in which all the vertices other than leaves and their parents have degree at least $k$, for every $k \geqslant 3$. We show that, for such trees $T$, $γ(T)/μ(T) < 1 + 1/((k-2)(k+1))$. |
| title | Laplacian Spectrum and Domination in Trees |
| topic | Spectral Theory Combinatorics 05C50, 05C69, 15A18, 15A42 |
| url | https://arxiv.org/abs/2510.20318 |