The number and structure of connected graphs with a fixed degree sequence

TL;DR

This paper analyzes the enumeration and structure of connected graphs with a fixed degree sequence in the sparse regime, linking them to the configuration model and giant component properties.

Contribution

It provides an asymptotic count of such graphs, relates them to the configuration model, and studies their local structure and rare event probabilities.

Findings

Number of connected graphs with fixed degree sequence identified up to exponential order

Connected graphs viewed as giant components in a larger configuration model

Results on local structure and rare event probabilities for these graphs

Abstract

We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs up to the exponential order. We do this by viewing a connected graph with a given degree distribution as the realization of the giant component in a larger configuration model, and carefully choosing the degree distribution of the larger graph so that it is likely that its giant component has the required degree distribution. To ensure that the connected graph has exactly the correct degrees, we use a switching argument. Additionally, we obtain results on rare event probabilities and describe the local structure of a uniform connected graph with a fixed degree sequence.