Prove the third-eigenvalue proportion conjecture

Prove that the supremum c₃ of λ₃(G)/|V(G)| over all finite simple graphs G equals 1/3.

Background

The paper proves λ₃(G) ≤ |V(G)|/3 for several substantial graph classes, including strongly regular graphs, regular line graphs, Cayley graphs on abelian groups, semi-Cayley graphs on abelian groups, and vertex-transitive graphs with abelian automorphism group. It also constructs families whose normalized third eigenvalue approaches 1/3.

Despite this evidence, the universal inequality remains unresolved in the paper. The conjecture is the broader form of the motivating problem attributed to Nikiforov and is not settled by the partial class-specific results.