# $k$-NIM Trees: Characterization and Enumeration

Published in *TBD, Submitted*, 2022

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 ≥ 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.