Uniqueness of equality cases for the λ3(G) bound

Prove that for graphs G on n vertices with n divisible by 3, the only graphs satisfying λ3(G) = n/3 − 1 are the Leonida–Li graphs H_{a,b}.

Background

The main theorem recovers Tang’s sharp bound λ3(G) ≤ n/3 − 1. The authors assert that, by tracing their inequalities, the only extremal graphs achieving equality are the Leonida–Li family H_{a,b}.

However, they do not provide the proof and explicitly defer it to future work, making the uniqueness of equality cases an open problem.

References

We can trace through the chain of inequalities in our master theorem to show that the Leonida-Li family of graphs $H_{a,b}$ are the only graphs attaining equality in $$\lambda_3(G) = \frac{n}{3}-1,\quad 3|n.$$ We leave the full proof of this, and the complete characterization of graphs attaining equality in $$\lambda_4(G) = \frac{1+\sqrt5}{12}n-1,$$ for future work.

— Graph Eigenvalues and Projection Constants  (2603.29280 - Wakhare, 31 Mar 2026) in Introduction