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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2310.15449 |
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| _version_ | 1866913195638128640 |
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| author | Chen, Qian-Qian Guo, Ji-Ming |
| author_facet | Chen, Qian-Qian Guo, Ji-Ming |
| contents | Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $λ$ of $A(G)$ is denoted by $m_λ(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_λ(T)\leq β'(T)-1$ for any $λ\neq0$. Moreover, they characterized all trees with $m_λ(T)=β'(T)-1$, where $β'(G)$ is the induced matching number of $G$.
In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $λ$ of the connected graph $G$, $m_λ(G)\leq β'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $β'(G)\geq3$, we characterize all connected graphs with $m_λ(G)=β'(G)+c(G)-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15449 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A relation between multiplicity of nonzero eigenvalues and the matching number of graph Chen, Qian-Qian Guo, Ji-Ming Combinatorics Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $λ$ of $A(G)$ is denoted by $m_λ(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_λ(T)\leq β'(T)-1$ for any $λ\neq0$. Moreover, they characterized all trees with $m_λ(T)=β'(T)-1$, where $β'(G)$ is the induced matching number of $G$. In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $λ$ of the connected graph $G$, $m_λ(G)\leq β'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $β'(G)\geq3$, we characterize all connected graphs with $m_λ(G)=β'(G)+c(G)-1$. |
| title | A relation between multiplicity of nonzero eigenvalues and the matching number of graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.15449 |