On the evolution in the configuration model

TL;DR

This paper provides detailed estimates for active and inactive half-edges in the configuration model for random regular graphs, using an analysis of a generalized urn model with negative eigenvalues.

Contribution

It introduces a novel analysis of a generalized urn model with negative eigenvalues to accurately estimate half-edge counts in the configuration model.

Findings

Precise estimates for active/inactive half-edges in the configuration model.

Analysis of a generalized urn model with negative eigenvalues.

Enhanced understanding of the structure of random regular graphs.

Abstract

We give precise estimates on the number of active/inactive half-edges in the configuration model used to generate random regular graphs. This is obtained by analyzing a more general urn model with negative eigenvalues.