On the multiplicity of 1 as a Laplacian eigenvalue of a graph
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2025
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| _version_ | 1866916832853622784 |
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| author | Tian, Fenglei Wong, Dein |
| author_facet | Tian, Fenglei Wong, Dein |
| contents | Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(λ)$ the multiplicity of $λ$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be obtained from $G$ by deleting some pendant vertices such that $p(\overline{G})=q(\overline{G})$. We first prove that $m_{L(G)}(1)=p(G)-q(G)+m_{L(\overline{G})}(1)$. Since deleting pendant path $P_3$ does not change the multiplicity of Laplacian eigenvalue 1 of a graph, we further focus on reduced graphs without pendant path $P_3$. Let $T$ be a reduced tree on $n(\geq 6)$ vertices without pendant path $P_3$, then it is proved that $$m_{L(T)}(1)\leq \frac{n-6}{4},$$ and all the trees attaining the upper bound are characterized completely. As an application, for a reduced unicyclic graph $G$ of order $n\geq 10$ without pendant path $P_3$, we get $$m_{L(G)}(1)\leq \frac{n}{4},$$ and all the unicyclic graphs attaining the upper bound are determined completely. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the multiplicity of 1 as a Laplacian eigenvalue of a graph Tian, Fenglei Wong, Dein Combinatorics 05C50 Let $G$ be a graph with $p(G)$ pendant vertices and $q(G)$ quasi-pendant vertices. Denote by $m_{L(G)}(λ)$ the multiplicity of $λ$ as a Laplacian eigenvalue of $G$. Let $\overline{G}$ be the reduced graph of $G$, which can be obtained from $G$ by deleting some pendant vertices such that $p(\overline{G})=q(\overline{G})$. We first prove that $m_{L(G)}(1)=p(G)-q(G)+m_{L(\overline{G})}(1)$. Since deleting pendant path $P_3$ does not change the multiplicity of Laplacian eigenvalue 1 of a graph, we further focus on reduced graphs without pendant path $P_3$. Let $T$ be a reduced tree on $n(\geq 6)$ vertices without pendant path $P_3$, then it is proved that $$m_{L(T)}(1)\leq \frac{n-6}{4},$$ and all the trees attaining the upper bound are characterized completely. As an application, for a reduced unicyclic graph $G$ of order $n\geq 10$ without pendant path $P_3$, we get $$m_{L(G)}(1)\leq \frac{n}{4},$$ and all the unicyclic graphs attaining the upper bound are determined completely. |
| title | On the multiplicity of 1 as a Laplacian eigenvalue of a graph |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2507.06184 |