Regular graphs with minimum spectral gap

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is (1+o(1))3n22π2. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected k-regular graph on n vertices is at least (1+o(1))2kπ23n2, and the bound is attained for at least one value of k. Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.