On the density of triangles and squares in regular finite and unimodular random graphs

TL;DR

This paper characterizes the possible densities of triangles and squares in r-regular finite and unimodular random graphs, providing explicit descriptions and approximation results that connect finite and infinite graph structures.

Contribution

It offers an explicit characterization of triangle and square densities in r-regular graphs and shows how unimodular random graphs can be approximated by finite graphs with respect to these densities.

Findings

Explicit description of triangle and square densities in finite r-regular graphs

Approximation of unimodular random graphs by finite graphs based on densities

Characterization of moments of spectral measures for r-regular graphs

Abstract

We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs.