# Regular Graphs with Minimum Spectral Gap

## TL;DR

This paper proves Aldous and Fill's conjecture on the spectral gap for cubic graphs and characterizes the structure of quartic graphs with minimal spectral gap, advancing understanding of spectral properties in regular graphs.

## Contribution

It confirms the spectral gap conjecture for cubic graphs and describes the structure of quartic graphs with minimal spectral gap.

## Key findings

- Spectral gap bound is proven for cubic graphs.
- Quartic graphs with minimal spectral gap have a path-like structure.
- The conjecture relates relaxation time to spectral gap in regular graphs.

## 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 $k$. Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.