Quartic Graphs with Minimum Spectral Gap

Abstract

Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1))\frac{3n^2}{2{\pi }^2}$. 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 $(1+o(1))\frac{2k{\pi }^2}{3n^2}$, 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 graphs on $n$ vertices is $(1+o(1))\frac{4{\pi }^2}{n^2}$. From this result, the Aldous--Fill conjecture follows for $k=4$.

arXiv comment31 pages, final version, to appear in Journal of Graph Theory