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arXiv · 2208.05450

$k$-NIM trees: Characterization and Enumeration

Charles R. Johnson·George Tsoukalas·Greyson C. Wesley·Zachary Zhao
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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.

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Charles R. Johnson, George Tsoukalas, Greyson C. Wesley, Zachary Zhao. 2022-08-10. $k$-NIM trees: Characterization and Enumeration. https://arxiv.org/abs/2208.05450

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