Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o\left(1\right))\frac{3n^2}{2{\pi }^2}$. 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 $(1+o\left(1\right))\frac{2k{\pi }^2}{3n^2}$, 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.

Aldous和Fill推测，在具有正则关系的正则图上，随机游走的最大弛豫时间为 $ñ$ 顶点是 $（\mathrm{1个}+Ø（\mathrm{1个}））\frac{3{ñ}^{\mathrm{2个}}}{\mathrm{2个}{\pi }^{\mathrm{2个}}}$。这个猜想可以用如下的谱隙来表述：所连接的谱隙（代数连通性）$ķ$-正则图 $ñ$ 顶点至少是 $（\mathrm{1个}+Ø（\mathrm{1个}））\frac{\mathrm{2个}ķ{\pi }^{\mathrm{2个}}}{3{ñ}^{\mathrm{2个}}}$，并且边界至少达到的一个值 $ķ$。根据Brand，Guiduli和Imrich的先前工作，我们证明了三次图的这种猜想。我们还研究了在所有连接的四次图之间具有最小光谱间隙的四次（即4规则）图的结构。我们表明，它们必须具有从特定块构建的类似路径的结构。