可平面图的直径与谱隙
Diameter and spectral gap for planar graphs
摘要：
本文证明了有限可平面图$X$的谱间隙是由$λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$限定的，其中$C$只依赖于$X$的度.然后，我们给出一系列这样的图，表明上述估计不能得到改善。这就否定了Benjamini和Curien关于可平面图上简单随机游动的混合时间的一个问题。
英文摘要：
We prove that the spectral gap of a finite planar graph $X$ is bounded by $λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Curien on the mixing times of the simple random walk on planar graphs.
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