Classify graphs attaining the known third-eigenvalue construction

Classify all graphs G satisfying λ₃(G) = |V(G)|/3 − 1 by proving that every such graph is isomorphic to H_{a,b} for some non-negative integers a and b, where H_{a,b} is the closed vertex multiplication of C₆ by [a,b,a,b,a,b].

Background

The paper introduces the family H_{a,b}, obtained by replacing the six vertices of C₆ with cliques of alternating sizes a and b. This family contains the previously known constructions 3K_n and the closed blow-up of C₆, and the authors prove that its third eigenvalue is exactly |V(H_{a,b})|/3 − 1.

The proposed classification would show that these constructions exhaust all graphs attaining the paper’s principal lower-bound example with the exact additive term −1. It is stronger than the broader conjecture that λ₃(G) ≤ |V(G)|/3, because it concerns the structure of equality cases for the sharper expression |V(G)|/3 − 1.

References

First, we demonstrate a family of graphs, denoted $H_{a,b}$, satisfying $\lambda_3(G) = \frac{|V(G)|}{3} - 1$, which generalises the only previously-known constructions of $3K_n$ and the closed blow-up of $C_6$. In particular, these include the first non-regular examples of such graphs and we propose the following conjecture. If $\lambda_3(G) = \frac{|V(G)|}{3} - 1$, then $G$ is isomorphic to $H_{a,b}$ for some non-negative integers $a,b$.

— On graphs with large third eigenvalue  (2501.02563 - Leonida et al., 5 Jan 2025) in Section 2, immediately following Definition 2.1