Aldous–Fill spectral-gap conjecture for regular graphs

Let $G$ be a connected $k$-regular graph on $n$ vertices. Its spectral gap, or algebraic connectivity, is the second smallest eigenvalue of its Laplacian matrix. Equivalently, for regular graphs it is $k$ times the second smallest eigenvalue of the normalized Laplacian.

Aldous–Fill spectral-gap conjecture. The spectral gap of a connected $k$-regular graph on $n$ vertices is at least

$$(1+o(1))\frac{2k\pi^2}{3n^2},$$

and the bound is attained for at least one value of $k$.

This is an equivalent spectral-gap formulation of the conjecture on maximum relaxation time. The paper proves it for cubic graphs, while the general assertion remains open in the supplied text.