On the class of graphs with strong mixing properties

TL;DR

This paper investigates the equivalence of key mixing properties in graphs, provides probabilistic estimates for these properties in random graphs, and derives asymptotic formulas for Eulerian orientations and circuits.

Contribution

It establishes the equivalence of algebraic connectivity, Cheeger constant, and spectral gap properties, and offers new probabilistic and asymptotic results related to these graph characteristics.

Findings

Proves the equivalence of three mixing properties in graphs.

Provides probability estimates for random graphs satisfying these properties.

Derives asymptotic formulas for Eulerian orientations and circuits.

Abstract

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph.