Nordhaus–Gaddum products of Laplacian eigenvalues: a counterexample and the sharp limiting profile

Unrefereed research candidate. Every graph has a complement, and Laplacian eigenvalues measure how well a graph is connected at different scales. Chen, Guo, Li and Wang (2025, Conjecture 20) conjectured that the product of the k-th largest Laplacian eigenvalue of a graph on n vertices and that of its complement is at most n(n−k) for k ≤ 3n/4. This paper shows the inequality is false and finds the true largest product to leading order at almost every rank. Counterexample. A 14-vertex graph built from the Fano plane (a 7-clique, 7 independent vertices, each joined to a line of three clique vertices) has μ6(G)μ6(complement) = 63 + 7√57 ≈ 115.85 > 112. Symmetric designs give an exact product formula and infinite families, and a design-free construction gives failures whenever 2(k+1) ≤ n < ⌊(k+1)2/4⌋. Limiting profile. Writing M(n,k) for the largest product and θ = k/n, M(n,k)/n2 tends to (1−θ/2)2 for θ < 1/2 and to 1/4 for θ > 1/2, jumping from 9/16 to 1/4 at the centre. Along the odd central sequence n = 2k+1 every limit point lies in [1/2, 9/16], and the limit 1/2 is conjectured. Necessary structure of asymptotic extremisers is also given. Credit: Chen, Guo, Li and Wang proved k = 1 and n/2 ≤ k ≤ 3n/4; Gibson (arXiv:2609.16049) refuted the equality clause with K2m,2m and left k < n/2 open; this paper addresses the inequality itself. A floating-point enumeration of all 12,293,433 graphs with n ≤ 10 found no violation above 10−9; this is not rigorous, and only the upper bound 14 on the smallest counterexample order is proved. The archive contains the paper (PDF and Markdown source), the exact certificate of the 14-vertex graph, exact and numerical verification scripts with recorded outputs, the enumeration results, a prior-art note and the raw Crossref and arXiv records from which every reference was copied. Assurance: written proofs, not machine-verified; finite-n bounds are not claimed sharp; the proved failures do not cover every pair with k < n/2 (Section 10); internal AI-assisted review only, not peer review, independent reproduction or formal verification. Creator: Anonymous. Text CC0-1.0; code MIT.