AI 中文总结
研究简单连通图\(G\)，探讨其拉普拉斯特征值重数\(m_G(\lambda)\)与圈空间维数\(c(G)\)、拟悬挂顶点数\(q(G)\)的关系，给出\(m_G(\lambda)=2c(G)+q(G)-1\)成立时图\(G\)的完整刻画。
AI 中文摘要
本文中，\(G\)是简单连通图，\(m_G(\lambda)\)表示\(\lambda\)作为拉普拉斯矩阵\(L(G)\)特征值的重数。\(p(G)\)表示\(G\)的悬挂顶点数，\(q(G)\)表示\(G\)的拟悬挂顶点数，\(c(G)\)表示\(G\)的圈空间维数。Li等人证明了若\(G\)是\(G\not\cong K_{1,n - 1}\)的树且\(\lambda\neq1\)，则\(m_G(\lambda)\leq q(T) - 1\)。对于一般图\(G\)且\(\lambda\neq1\)，\(m_G(\lambda)\leq c(G)+q(G)\)，当且仅当\(G\cong K_{1,n - 1}\)或\(G\cong C_n\)且\(\lambda\notin\{0,4\}\)时取等号。由此推出若\(c(G)+q(G)\geq2\)，则\(m_G(\lambda)\leq2c(G)+q(G)-1\)。当\(c(G)=0\)时，此结果与Li等人关于树的结果一致。本文给出了使\(m_G(\lambda)=2c(G)+q(G)-1\)成立的图\(G\)的完整刻画。
英文摘要
In this paper, \( G \) is a simple connected graph, and \( m_G(λ) \) denotes the multiplicity of \( λ\) as an eigenvalue of the Laplacian matrix \( L(G) \). Let \( p(G) \) denote the number of pendant vertices of \( G \), \( q(G) \) the number of quasi-pendant vertices of \( G \), and \( c(G) \) the dimension of the cycle space of \( G \). Li et al. [Discrete Mathematics, 2026] proved that if \( G \) is a tree with \( G \not\cong K_{1,n-1} \) and \( λ\neq 1 \), then \[ m_G(λ) \le q(T) - 1. \] Moreover, for a general graph \( G \) with \( λ\neq 1 \), Li et al. also proved in the same paper that \[ m_G(λ) \le c(G) + q(G), \] with equality if and only if \( G \cong K_{1,n-1} \) or \( G \cong C_n \) with \( λ\notin \{0, 4\} \). A natural consequence is that if \( c(G) + q(G) \ge 2 \), then \[ m_G(λ) \le 2c(G) + q(G) - 1. \] When \( c(G) = 0 \), this reduces to the result of Li et al. for trees. In this paper, we give a complete characterization of graphs \( G \) attaining the equality \[ m_G(λ) = 2c(G) + q(G) - 1. \]
