On the minimum number of eigenvalues of trees of diameter seven

TL;DR

This work addresses the problem of determining $q(T)$, the minimum number of distinct eigenvalues for real symmetric matrices whose underlying graph is a tree with diameter seven. It uses seed-based decompositions and the eigenvalue-location algorithm of Diagonalize to classify which seeds are defective; in particular, it shows that the diameter-seven seeds $S_7^{(7)}$, $S_7^{(8)}$, and $S_7^{(9)}$ are defective, correcting earlier claims. The authors then provide explicit eight-eigenvalue realizations for all diameter-seven unfoldings of these seeds, proving that $q(T)=8$ for those trees and ruling out $7$ as possible. Collectively, these results refine the landscape of diminimal trees at diameter seven and establish a robust framework for locating eigenvalues in tree realizations via seed unfoldings and a structured block construction.

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.

Figures (13)

• Figure 1: The tree in (B) is an unfolding of the tree $T$ in (A).
• Figure 2: The twelve seeds of diameter $7$, denoted $S_7^{(1)},\ldots,S_7^{(12)}$.
• Figure 3: Algorithm Diagonalize
• Figure 4: The seed $S_7^{(8)}$ and a sequence of diameter-preserving unfoldings that produce a tree $T$ that is not diminimal.
• Figure 5: Notation for $T_1$ and $T_2$ (on the left) and for $T_3$ and $T_4$ (on the right).

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof : Proof of Lemma \ref{['no_zero']}
• Theorem 3.4