具有最小谱隙的四次图
Quartic Graphs with Minimum Spectral Gap
摘要：
Aldous和Fill证明了具有$n$顶点的连通正则图上随机游动的最大松弛时间为$(1+o(1)) \frac{3n^2}{2π^2}$ .这个猜想可以用谱间隙的概念来表示：$k$ -正则图在$n$顶点上的谱间隙（代数连通度）至少为$(1+o(1))\frac{2kπ^2}{3n^2}$，并且至少对$k$的一个值给出了上界.我们确定了$n$顶点上具有最小谱间隙的连通四次图的结构，从而证明了$n$顶点上连通四次图的最小谱间隙是$(1+o(1))\frac{4π^2}{n^2}$ .根据这个结果，对$k=4$给出了Aldous-Fill猜想。
英文摘要：
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2π^2}$. 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 $(1+o(1))\frac{2kπ^2}{3n^2}$, and the bound is attained for at least one value of $k$. We determine the structure of connected quartic graphs on $n$ vertices with minimum spectral gap which enable us to show that the minimum spectral gap of connected quartic graphs on $n$ vertices is $(1+o(1))\frac{4π^2}{n^2}$. From this result, the Aldous--Fill conjecture follows for $k=4$.
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