A characterization of graphs $G$ with nullity $n(G)-d(G)-1$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916449566588928 |
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| author | Xu, Songnian |
| author_facet | Xu, Songnian |
| contents | For a connected graph $G$ with order $n$, let $e(G)$ represent the number of its distinct eigenvalues, and let $d$ denote its diameter. We denote the eigenvalue multiplicity of $μ$ in $G$ by $m_G(μ)$. It is well established that the inequality $e(G) \geq d + 1$ implies that when $μ$ is an eigenvalue of $P_{d+1}$, it follows that $m_G(μ) \leq n - d$; otherwise, for any real number $μ$, we have $m_G(μ) \leq n - d - 1$. A graph is termed minimal if $e(G) = d + 1$. In 2013, Wong et al. characterized all minimal graphs for which $m_G(0) = n - d$. In this article, we provide a complete characterization of the graphs $G$ such that $m_G(0) = n - d - 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17837 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of graphs $G$ with nullity $n(G)-d(G)-1$ Xu, Songnian Spectral Theory 05C50 For a connected graph $G$ with order $n$, let $e(G)$ represent the number of its distinct eigenvalues, and let $d$ denote its diameter. We denote the eigenvalue multiplicity of $μ$ in $G$ by $m_G(μ)$. It is well established that the inequality $e(G) \geq d + 1$ implies that when $μ$ is an eigenvalue of $P_{d+1}$, it follows that $m_G(μ) \leq n - d$; otherwise, for any real number $μ$, we have $m_G(μ) \leq n - d - 1$. A graph is termed minimal if $e(G) = d + 1$. In 2013, Wong et al. characterized all minimal graphs for which $m_G(0) = n - d$. In this article, we provide a complete characterization of the graphs $G$ such that $m_G(0) = n - d - 1$. |
| title | A characterization of graphs $G$ with nullity $n(G)-d(G)-1$ |
| topic | Spectral Theory 05C50 |
| url | https://arxiv.org/abs/2410.17837 |