Diminimal families of arbitrary diameter

2023/02/02 by Luiz Emílio Allem, Rodrigo O. Braga, Allem, Luiz Emilio +9 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2302.00835

openalex publication_date 2023/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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)≥ 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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