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Diameter and spectral gap for planar graphs

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arxiv 1204.4435 v2 pith:T7HKMPLL submitted 2012-04-19 math.GT math.PR

Larsen Louder , Juan Souto This is my paper · ORCID
classification math.GTmath.PR
keywords graphsplanardiamspectralaboveanswerbenjaminibounded
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abstract

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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Cited by 1 Pith paper

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    A learned triangle-selection module rewires graphs for GNNs, improving node classification over prior rewiring methods on 9 of 10 benchmarks.

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