Diameter and spectral gap for planar graphs [article]

Larsen Louder, Juan Souto
2012 arXiv   pre-print
We prove that the spectral gap of a finite planar graph X is bounded by λ_1(X)< C(( X)/ 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.
arXiv:1204.4435v2 fatcat:ipickifdkbbldbgecczzfm5xii