Abstract

AbstractAldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with vertices is . This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected ‐regular graph on vertices is at least , and the bound is attained for at least one value of . We determine the structure of connected quartic graphs on vertices with a minimum spectral gap which enables us to show that the minimum spectral gap of connected quartic graphs on vertices is . From this result, the Aldous–Fill conjecture follows for .

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