Quartic graphs with minimum spectral gap

Aldous and Fill conjectured that the maximum relaxation time for random walk on a connected regular graph with n $n$ vertices is ( 1 + o ) 3 2 ? $(1+o(1))\frac{3{n}^{2}}{2{\pi }^{2}}$ . This conjecture can be rephrased in terms of spectral gap as follows: (algebraic connectivity) k $k$ -regular at least $(1+o(1))\frac{2k{\pi }^{2}}{3{n}^{2}}$ , bound attained one value We determine structure quartic graphs minimum which enables us to show 4 $(1+o(1))\frac{4{\pi }^{2}}{{n}^{2}}$ From this result, Aldous–Fill follows = $k=4$