A new characterization of the set of Laplacian spectral radii of trees
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913022360944640 |
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| author | Dong, Fengming Zhang, Ruixue |
| author_facet | Dong, Fengming Zhang, Ruixue |
| contents | For any positive integer $r$ and real number $α>1$, let ${\mathscr L}_r(α)$ denote the set of positive real numbers defined recursively: $α-1\in {\mathscr L}_r(α)$, and for any multi-subset $\{q_1,q_2,\dots,q_s\}$ of ${\mathscr L}_r(α)$, where $0<s<r$, $β:=α-1-s-\sum_{i=1}^sq_i^{-1}$ belongs to ${\mathscr L}_r(α)$ as long as $β>0$. We show that $(α-1)^{-1}\in {\mathscr L}_r(α)$ if and only if there exists a tree $T$ with its maximum degree $Δ(T)\le r$ and Laplacian spectral radius $μ(T)=α>1$. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers $α\ge 2$ such that $(α-1)^{-1}\in {\mathscr L}_r(α)$ for $r=\lfloorα\rfloor-1$. Applying this conclusion, we then show that for any integer $n$, there exists a tree $T$ with $Δ(T)<n$ and $μ(T)=n+1$ if and only if $n\ge 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09972 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new characterization of the set of Laplacian spectral radii of trees Dong, Fengming Zhang, Ruixue Combinatorics 05C05, 05C20, 05C50 For any positive integer $r$ and real number $α>1$, let ${\mathscr L}_r(α)$ denote the set of positive real numbers defined recursively: $α-1\in {\mathscr L}_r(α)$, and for any multi-subset $\{q_1,q_2,\dots,q_s\}$ of ${\mathscr L}_r(α)$, where $0<s<r$, $β:=α-1-s-\sum_{i=1}^sq_i^{-1}$ belongs to ${\mathscr L}_r(α)$ as long as $β>0$. We show that $(α-1)^{-1}\in {\mathscr L}_r(α)$ if and only if there exists a tree $T$ with its maximum degree $Δ(T)\le r$ and Laplacian spectral radius $μ(T)=α>1$. It follows that the set of Laplacian spectral radii of non-trivial trees is exactly the set of real numbers $α\ge 2$ such that $(α-1)^{-1}\in {\mathscr L}_r(α)$ for $r=\lfloorα\rfloor-1$. Applying this conclusion, we then show that for any integer $n$, there exists a tree $T$ with $Δ(T)<n$ and $μ(T)=n+1$ if and only if $n\ge 4$. |
| title | A new characterization of the set of Laplacian spectral radii of trees |
| topic | Combinatorics 05C05, 05C20, 05C50 |
| url | https://arxiv.org/abs/2604.09972 |