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arXiv · 2302.00835

Diminimal families of arbitrary diameter

Luiz Emilio Allem·Rodrigo Orsini Braga·Carlos Hoppen·Elismar da Rosa Oliveira·Lucas Siviero Sibemberg·Vilmar Trevisan
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Abstract

Given a tree $T$, let $q(T)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $T$. It is well known that $q(T)\geq d(T)+1$, where $d(T)$ is the diameter of $T$, and a tree $T$ is said to be diminimal if $q(T)=d(T)+1$. In this paper, we present families of diminimal trees of any fixed diameter. Our proof is constructive, allowing us to compute, for any diminimal tree $T$ of diameter $d$ in these families, a symmetric matrix $M$ with underlying graph $T$ whose spectrum has exactly $d+1$ distinct eigenvalues.

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Luiz Emilio Allem, Rodrigo Orsini Braga, Carlos Hoppen, Elismar da Rosa Oliveira, Lucas Siviero Sibemberg, Vilmar Trevisan. 2023-02-02. Diminimal families of arbitrary diameter. https://arxiv.org/abs/2302.00835

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