PreprintarXiv licencearXiv
Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach
Abstract
For any graph G, we associate a family of real symmetric matrices, S(G), where for any A ∈ S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) represent the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we provide an alternative technique to establish that q(G) ≤ 4 for any threshold graph G as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further
Related
- Graphs with three distinct distance eigenvalues, one of which is zero
- Absence of non-threshold embedded eigenvalues for nonlinear Schrödinger equations
- On Hamming Eigenvalues and Hamming Energy of Graphs
- On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs
- Convergence of cyclic steepest descent for quadratics with two distinct eigenvalues
BibTeX
@misc{breen2026threshold,
title = {{Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach}},
author = {Jane Breen and Shaun Fallat and Johnna Parenteau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10623},
eprint = {2609.10623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10623},
}
RIS
TY - UNPB
TI - Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach
AU - Jane Breen
AU - Shaun Fallat
AU - Johnna Parenteau
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.10623
UR - https://arxiv.org/abs/2609.10623
AB - For any graph G, we associate a family of real symmetric matrices, S(G), where for any A ∈ S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) represent the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we provide an alternative technique to establish that q(G) ≤ 4 for any threshold graph G as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further
ER -