On the minimum number of eigenvalues of trees of diameter seven

TL;DR

This paper investigates the minimum number of distinct eigenvalues for symmetric matrices associated with trees of diameter seven, correcting previous results and building on seed-based construction methods.

Contribution

It corrects a previous result and advances understanding of eigenvalue realizations for diameter seven trees using seed-based approaches.

Findings

Corrected a previous result on eigenvalues of diameter seven trees.

Established new bounds or characterizations for q(T) in these trees.

Extended seed-based construction methods to diameter seven trees.

Abstract

The underlying graph $G$ of a symmetric matrix $M=(m_{ij})\in \mathbb{R}^{n\times n}$ is the graph with vertex set $\{v_1,\ldots,v_n\}$ such that a pair $\{v_i,v_j\}$ with $i\neq j$ is an edge if and only if $m_{ij}\neq 0$. Given a graph $G$, let $q(G)$ be the minimum number of distinct eigenvalues in a symmetric matrix whose underlying graph is $G$. A symmetric matrix $M$ is said to be a realization of $q(G)$ if it has underlying graph $G$ and $q(G)$ distinct eigenvalues. In the case of trees, a paper by Johnson and Saiago [Johnson, C.R, and Saiago, C.M, Diameter Minimal Trees, Linear and Multilinear Algebra 64(3) (2015), 557--571.] proposed an approach by which realizations of large trees are constructed from realizations of smaller trees with the same diameter, known as seeds, which has proved to be very successful. In this paper, we discuss realizations of $q(T)$ for trees of diameter seven based on the seed that defines it, correcting a result in the aforementioned paper.