小于1的树的拉普拉斯特征值个数
The number of Laplacian eigenvalues of trees less than one
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中文总结 AI 辅助
本文研究树的拉普拉斯特征值中小于1的个数与支配数的关系，刻画了二者相等时的所有树，并证明几乎所有树满足更强的下界。
中文摘要 AI 辅助
设$m_T[0,1)$表示树$T$中严格小于$1$的拉普拉斯特征值的个数。Guo、Xue和Liu\cite{GXL}证明了直径为$d$的每棵树$T$满足$m_T[0,1)\ge\lceil(d+1)/3\rceil$，并且当$d\equiv2\pmod3$时该界是紧的。经典结论是，直径为$d$的每个图的支配数至少为$\lceil(d+1)/3\rceil$。本文中，对于任意直径$d$，$m_T[0,1)=\lceil(d+1)/3\rceil$当且仅当$\gamma(T)=\lceil(d+1)/3\rceil$；我们刻画了满足此充要条件的所有树，并证明了几乎所有树都满足$m_T[0,1)\ge\lceil(d+1)/3\rceil+1$。
英文摘要
Let $m_T[0,1)$ denote the number of Laplacian eigenvalues of a tree $T$ that are strictly less than $1$. Guo, Xue and Liu \cite{GXL} proved that every tree $T$ of diameter $d$ satisfies $m_T[0,1)\ge \lceil(d+1)/3\rceil$, and that this bound is sharp when $d\equiv2\pmod3$. It is classical that every graph of diameter $d$ has domination number at least $\lceil(d+1)/3\rceil$. In this paper, for any diameter $d$, $m_T[0,1)=\lceil(d+1)/3\rceil$ if and only if $γ(T)=\lceil(d+1)/3\rceil$; we characterize all trees satisfying this necessary and sufficient condition, and also prove that almost all trees satisfy $m_T[0,1)\ge\lceil(d+1)/3\rceil+1$.
发表机构
- Information Engineering University（信息工程大学）
- McCombs School of Business, The University of Texas at Austin（德克萨斯大学奥斯汀分校麦库姆斯商学院）
- School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
