Bound the penultimate eigenvalue of vertex multiplications of T

Establish that every vertex multiplication \tilde{T} of the specified 11-vertex graph T satisfies λ_{n−1}(\tilde{T})/|V(\tilde{T})| ≥ −1/3.

Background

Under the assumption that a minimizer of λ_{n−1} has a simple final eigenvalue, the paper reduces the analysis to vertex multiplications of an explicit 11-vertex graph T. The resulting reduction shows that a suitable vertex multiplication of T approximates the global weighted minimum up to an additive O(√n) error.

The conjectured bound would control the normalized penultimate eigenvalue for every vertex multiplication of T. Together with the paper’s blow-up lemma, such a bound would support the global weighted lower bound of −1/3 and consequently the corresponding third-eigenvalue conjecture.