Prove the weighted penultimate-eigenvalue conjecture

Prove that for every n × n real symmetric matrix M with entries in [0,1], the penultimate eigenvalue satisfies λ_{n−1}(M) ≥ −n/3; equivalently, establish that inf_{M∈𝒮_n} λ_{n−1}(M) ≥ −n/3.

Background

The paper extends the graph eigenvalue problem from unweighted adjacency matrices to the compact set 𝒮n of symmetric matrices whose entries lie in [0,1], representing weighted undirected graphs, possibly with loops. The authors focus on minimizing λ{n−1}, which is related by Weyl’s inequality to maximizing the third eigenvalue.

The conjectured lower bound is the weighted analogue of the target constant 1/3 for the third eigenvalue. The paper proves several structural results about minimizers and establishes approximation and reduction statements, but does not prove the conjectured −n/3 bound in full generality.