$k$-NIM trees: Characterization and Enumeration

TL;DR

This paper characterizes and counts $k$-NIM trees, a class of trees with specific eigenvalue multiplicity properties, revealing that such trees only exist for $k=1,2,3$, with stars being the only 3-NIM trees.

Contribution

It provides a graph-theoretic characterization and enumeration of $k$-NIM trees for all $k\, extgreater=1$, extending known results for NIM trees.

Findings

$k$-NIM trees exist only for $k=1,2,3$

Characterization of $k$-NIM trees for each $k$

Only stars are 3-NIM trees

Abstract

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.