Optimal exponent in the eigenvalue bound for minor-free graphs

Determine whether the exponent c in the Spielman–Teng eigenvalue bound for n-vertex K_h-minor-free graphs of maximum degree Δ can be improved to the best-possible value 2; equivalently, establish whether every such graph satisfies λ₂(L_G) ≲ Δh²/n.

Background

Spielman and Teng conjectured that the second-smallest Laplacian eigenvalue of every n-vertex K_h-minor-free graph with maximum degree Δ is bounded by Δh^c/n for some absolute constant c. Earlier work established this bound with larger powers of h, and the paper improves the dependence to approximately Δh²(log h)²/n.

The paper therefore makes substantial progress toward the optimal exponent 2 but retains logarithmic factors in h. Consequently, removing those logarithmic factors, or proving the exact h² dependence requested by the open question, remains unresolved.