Characterize graphs attaining equality in the λ4(G) bound

Characterize all graphs G on n vertices for which the fourth largest adjacency eigenvalue satisfies λ4(G) = ((1 + √5)/12) n − 1.

Background

Using the identification with projection constants and the known value λ(3) = (1 + √5)/2, the paper proves λ4(G) ≤ ((1 + √5)/12) n − 1 for all graphs, and cites Linz’s construction (closed blowups of the icosahedral graph) that achieves equality.

The authors explicitly leave the complete characterization of all equality cases open for future work.

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