# The connectivity of graphs of graphs with self-loops and a given degree sequence

## TL;DR

This paper characterizes when double edge swaps can connect all loopy graphs with a given degree sequence and introduces an MCMC sampler for uniform sampling of such graphs.

## Contribution

It provides the first characterization of degree sequences for loopy graphs where double edge swaps are insufficient and develops an efficient algorithm to identify these sequences.

## Key findings

- Double edge swaps do not always connect all loopy graphs.
- A classification scheme for degree sequences is introduced.
- An MCMC sampler for loopy graphs is developed.

## Abstract

`Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swaps can transform, for any fixed degree sequence, any two graphs inside the classes of simple graphs, multigraphs, and pseudographs, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. The same classification scheme to characterize degree sequences can be used to prove that, for all degree sequences, loopy graphs are connected by a combination of double and triple edge swaps. Thus, we contribute the first MCMC sampler that uniformly samples loopy graphs with any given sequence.