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))2Ο23n2β. 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))3n22kΟ2β,
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
graph on n vertices is (1+o(1))n24Ο2β. From this result, the
Aldous--Fill conjecture follows for k=4.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1907.0373