论文标题
$ K $ -NIM树：表征和枚举
$k$-NIM trees: Characterization and Enumeration
论文作者
论文摘要
在这些真实的对称矩阵中，其图是给定的树$ t $的，众所周知，特征值可以实现的最大多重性$ m（t）$是$ t $的路径覆盖号。我们说，如果每当特征值达到$ k-1 $的$ k-1 $，则一棵树为$ k $ -nim，小于最大多重性的$ k-1 $，所有其他多重性都是$ 1 $。 $ 1 $ - nim树被称为nim树，并且已经知道了nim树的特征。在这里，我们为每$ k \ geq 1 $的$ k $ nim树提供图形理论表征，并计算它们。从以下特征角度来看，$ k $ nim树仅在$ n $顶点上存在于$ k = 1,2,3 $时。如果$ k = 3 $，则仅$ 3 $ nim树是简单的星星。
Among those real symmetric matrices whose graph is a given tree $T$, the maximum multiplicity $M(T)$ that can be attained by an eigenvalue is known to be the path cover number of $T$. We say that a tree is $k$-NIM if, whenever an eigenvalue attains a multiplicity of $k-1$ less than the maximum multiplicity, all other multiplicities are $1$. $1$-NIM trees are known as NIM trees, and a characterization for NIM trees is already known. Here we provide a graph-theoretic characterization for $k$-NIM trees for each $k\geq 1$, as well as count them. It follows from the characterization that $k$-NIM trees exist on $n$ vertices only when $k=1,2,3$. In case $k=3$, the only $3$-NIM trees are simple stars.
