Diminimal families of arbitrary diameter

TL;DR

This paper constructs families of diminimal trees with any fixed diameter, providing explicit matrices with minimal eigenvalue counts and spectra, advancing understanding of eigenvalue properties related to tree structures.

Contribution

It introduces constructive methods to generate diminimal trees of any diameter and computes explicit matrices with minimal eigenvalues for these trees.

Findings

Families of diminimal trees of any fixed diameter are constructed.

Explicit symmetric matrices with exactly d+1 distinct eigenvalues are provided.

The approach enables spectrum computation for these trees.

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.