Diameter and spectral gap for planar graphs

Details Diameter and spectral gap for planar graphs Diameter and spectral gap for planar graphs

2012-04-19

arxiv.org/abs/1204.4435v1

Mathematics

Geometric Topology

16 pages, 1 figure

Scientific paper

We prove that the spectral gap of a finite planar graph $X$ is bounded by $\lambda_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ 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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