For a tree T, $U\left(T\right)$ denotes the minimum number of eigenvalues of multiplicity 1 among all real symmetric matrices, whose graph is T. It is known that $U\left(T\right)\ge 2$. A tree is linear if all its vertices of degree at least 3 lie on a single induced path, and k-linear if there are k of these high degree vertices. $U\left(T\right)$ is known for generalized stars (1-linear trees), and $U\left(T\right)$ is determined here for 2-linear trees. Three (and higher) linear trees offer considerable additional complications.

中文翻译：

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图为二元线性树的实对称矩阵中重数为1的特征值的最小个数

对于树T，$ü\left(吨\right)$表示所有实对称矩阵中重数为 1 的特征值的最小个数，其图形为T。众所周知$ü\left(吨\right)\ge \mathrm{2个}$. 如果一棵树的所有度数至少为 3 的顶点都位于一条诱导路径上，则该树是线性的，如果存在k 个这些高度数的顶点，则为k线性树。$ü\left(吨\right)$以广义恒星（1-线性树）而闻名，并且$ü\left(吨\right)$此处为 2 线性树确定。三棵（或更多）线性树提供了相当多的额外复杂性。