Brandt’s least signless-Laplacian eigenvalue conjecture

Brandt posed the conjecture in 1997 for regular triangle-free graphs: $\lambda_1(G)+\lambda_n(G)\le \frac{4n}{25}$.

Known results

• A preprint claims that every triangle-free graph satisfies $q_n(G)\le \frac{15n}{94}<0.1596n$, which would prove Brandt’s conjecture for regular graphs.
• The same work reports computer-assisted bounds $q_n(G)<0.15467n$ and $\lambda_1(G)+\lambda_n(G)<0.15442n$ in the regular case.
• The sharp bound remains open; the Higman–Sims graph gives $q_n(G)=0.14n$.

October 2026 narrower advance

Qi Zhou’s preprint proves a nonregular bound for $\{C_3,C_5\}$-free graphs using flag inequalities and local Rayleigh constraints. It strengthens results in that narrower class but does not settle Brandt’s original conjecture.

Current status (as of October 2026): Brandt’s conjecture is claimed solved by a preprint but remains unverified; the sharp bound and broader triangle-free problem remain open.