Quartic Graphs with Minimum Spectral Gap

TL;DR

This paper determines the structure of connected quartic graphs with the smallest spectral gap, confirming the Aldous-Fill conjecture for 4-regular graphs by establishing the minimum spectral gap as approximately 4π²/n².

Contribution

It characterizes the structure of quartic graphs with minimal spectral gap and proves the Aldous-Fill conjecture for 4-regular graphs.

Findings

Minimum spectral gap for connected quartic graphs is approximately 4π²/n².

Structural characterization of quartic graphs with minimal spectral gap.

Confirmation of the Aldous-Fill conjecture for k=4.

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 $k=4$.