A Spectral Turán Problem for a Fixed Tree
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arXiv
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| Format: | Preprint |
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2025
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| author | Desai, Dheer Noal Kaul, Hemanshu Kudarzi, Bahareh |
| author_facet | Desai, Dheer Noal Kaul, Hemanshu Kudarzi, Bahareh |
| contents | We study the spectral Turán problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say $T \in \mathcal{T}_{m,l+1}^δ$ if $T$ is a tree of order $m$, where the order of the smaller partite set $A$ of $T$ is $l+1$, and $δ$ is the minimum degree of the vertices in $A$. The motivation for this parametrization comes from the recent proof of the spectral Erdős-Sós conjecture. For a given fixed tree $T$, we describe $\mathrm{SPEX}(n,T)$ and consequently, bound $\mathrm{spex}(n,T)$ in terms of $m,l,δ$ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree $T \in \mathcal{T}_{m,l+1}^δ$ into graphs of the form $\overline{K}_l \vee m S_δ$. We give bounds on $\mathrm{spex}(n,T)$ within an error of $Θ(n^{-1/2})$ and $Θ(n^{-1})$ that are based on our embedding results for the given $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14908 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Spectral Turán Problem for a Fixed Tree Desai, Dheer Noal Kaul, Hemanshu Kudarzi, Bahareh Combinatorics 05C50, 05C35, 05C05 We study the spectral Turán problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say $T \in \mathcal{T}_{m,l+1}^δ$ if $T$ is a tree of order $m$, where the order of the smaller partite set $A$ of $T$ is $l+1$, and $δ$ is the minimum degree of the vertices in $A$. The motivation for this parametrization comes from the recent proof of the spectral Erdős-Sós conjecture. For a given fixed tree $T$, we describe $\mathrm{SPEX}(n,T)$ and consequently, bound $\mathrm{spex}(n,T)$ in terms of $m,l,δ$ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree $T \in \mathcal{T}_{m,l+1}^δ$ into graphs of the form $\overline{K}_l \vee m S_δ$. We give bounds on $\mathrm{spex}(n,T)$ within an error of $Θ(n^{-1/2})$ and $Θ(n^{-1})$ that are based on our embedding results for the given $T$. |
| title | A Spectral Turán Problem for a Fixed Tree |
| topic | Combinatorics 05C50, 05C35, 05C05 |
| url | https://arxiv.org/abs/2505.14908 |