A characterization of graphs $G$ with $m_G(λ)= 2c(G) + q_s(G) - 1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912147631505408 |
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| author | Xu, Songnian Zhen, Wenhao Wong, Dein |
| author_facet | Xu, Songnian Zhen, Wenhao Wong, Dein |
| contents | Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and let $|Q_s(G)| = q_s(G)$. For $G \in \mathbb{G}_s$, Li et al. (2024) proved that when $λ$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(λ) \leq 2c(G) + q_s(G) - 1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(λ) = 2c(G) + q_s(G) - 1$ when $G \in \mathbb{G}_{s}$ and $λ\notin σ(P_s)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of graphs $G$ with $m_G(λ)= 2c(G) + q_s(G) - 1$ Xu, Songnian Zhen, Wenhao Wong, Dein Spectral Theory Combinatorics 05C50, 05C05, 05C69 Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and let $|Q_s(G)| = q_s(G)$. For $G \in \mathbb{G}_s$, Li et al. (2024) proved that when $λ$ is not an eigenvalue of $P_s$ and $G$ is neither a cycle nor a starlike tree $T_k$, it holds that $m_G(λ) \leq 2c(G) + q_s(G) - 1$ and characterized the extremal graphs when $G$ is a tree. In this article, we characterize the extremal graphs for which $m_G(λ) = 2c(G) + q_s(G) - 1$ when $G \in \mathbb{G}_{s}$ and $λ\notin σ(P_s)$. |
| title | A characterization of graphs $G$ with $m_G(λ)= 2c(G) + q_s(G) - 1$ |
| topic | Spectral Theory Combinatorics 05C50, 05C05, 05C69 |
| url | https://arxiv.org/abs/2411.04770 |