Trees with extremal Laplacian eigenvalue multiplicity
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| Format: | Preprint |
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2025
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| _version_ | 1866913950815223808 |
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| author | Gupta, Vinayak Lather, Gargi Balaji, R. |
| author_facet | Gupta, Vinayak Lather, Gargi Balaji, R. |
| contents | Let $T$ be a tree. Suppose $λ$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_{T}(λ)$. It is known that $m_{T}(λ) \leq p(T)-1$, where $p(T)$ is the number of pendant vertices of $T$. In this paper, we characterize all trees $T$ for which there exists an eigenvalue $λ$ such that $m_{T}(λ)=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $α$ and $β$ are two distinct pendant vertices, then the distance $d(α,β)$ satisfies $d(α, β) \equiv 2q ~{\rm{mod}}~(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(α,β) \equiv 2\,\mbox{mod}\, 3$ for all distinct pendant vertices $α$ and $β$ of $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15472 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trees with extremal Laplacian eigenvalue multiplicity Gupta, Vinayak Lather, Gargi Balaji, R. Combinatorics 05C05 Let $T$ be a tree. Suppose $λ$ is an eigenvalue of the Laplacian matrix of $T$ with multiplicity $m_{T}(λ)$. It is known that $m_{T}(λ) \leq p(T)-1$, where $p(T)$ is the number of pendant vertices of $T$. In this paper, we characterize all trees $T$ for which there exists an eigenvalue $λ$ such that $m_{T}(λ)=p(T)-1$. We show that such trees are precisely either paths, or there exists an integer $q$ such that if $α$ and $β$ are two distinct pendant vertices, then the distance $d(α,β)$ satisfies $d(α, β) \equiv 2q ~{\rm{mod}}~(2q+1)$. As a consequence, we show that $1$ is an eigenvalue of $L_T$ with multiplicity $p(T)-1$ if and only if $d(α,β) \equiv 2\,\mbox{mod}\, 3$ for all distinct pendant vertices $α$ and $β$ of $T$. |
| title | Trees with extremal Laplacian eigenvalue multiplicity |
| topic | Combinatorics 05C05 |
| url | https://arxiv.org/abs/2507.15472 |