Quadratic inequalities between the largest eigenvalues of a graph
Abstract
We prove a sharp quadratic inequality between the largest two eigenvalues λ1 ≥ λ2 of a graph with n vertices. We also prove a quadratic inequality between the second and third largest eigenvalues λ2 ≥ λ3. These results in particular imply the bounds λ1 + λ2 ≤ 8/7 n − 2, λ3 ≤ n/3 − 1 and λ2 + λ3 ≤ 2/3 n − 2. In fact we determine the closure of the set of possible ((λ1+1)/n, (λ2+1)/n) ∈ ℝ2 and ((λ2+1)/n, (λ3+1)/n) ∈ ℝ2. More generally, we prove quadratic bounds in the case of symmetric matrices in [0,1]n × n, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in [−1,1]n × n. These bounds are proved by transforming the problem into extremal geometric questions in ℝ3 and ℝ2. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.
Related
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- Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
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- Generalized Nordhaus--Gaddum Inequalities for Eigenvalues
BibTeX
@misc{paulin2026quadratic,
title = {{Quadratic inequalities between the largest eigenvalues of a graph}},
author = {Roland Paulin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37980},
eprint = {2609.37980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37980},
}
RIS
TY - UNPB
TI - Quadratic inequalities between the largest eigenvalues of a graph
AU - Roland Paulin
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37980
UR - https://arxiv.org/abs/2609.37980
AB - We prove a sharp quadratic inequality between the largest two eigenvalues λ_1 ≥ λ_2 of a graph with n vertices. We also prove a quadratic inequality between the second and third largest eigenvalues λ_2 ≥ λ_3. These results in particular imply the bounds λ_1 + λ_2 ≤ 8/7 n − 2, λ_3 ≤ n/3 − 1 and λ_2 + λ_3 ≤ 2/3 n − 2. In fact we determine the closure of the set of possible ((λ_1+1)/n, (λ_2+1)/n) ∈ ℝ^2 and ((λ_2+1)/n, (λ_3+1)/n) ∈ ℝ^2. More generally, we prove quadratic bounds in the case of symmetric matrices in [0,1]^n × n, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in [−1,1]^n × n. These bounds are proved by transforming the problem into extremal geometric questions in ℝ^3 and ℝ^2. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.
ER -