Connectivity for random graphs from a weighted bridge-addable class

TL;DR

This paper studies the connectivity properties of weighted random graphs within a broad class of bridge-addable graphs, providing improved bounds and generalizations on their structure and component behavior.

Contribution

It extends existing results to weighted distributions and offers new bounds on the expected size of remaining components after removing the largest one.

Findings

Expected number of vertices outside the largest component is less than 2.

Provides bounds on the probability of connectivity in weighted bridge-addable classes.

Generalizes previous results to weighted graph distributions.

Abstract

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general 'bridge-addable' class of graphs - if a graph is in the class and u and v are vertices in different components then the graph obtained by adding an edge (bridge) between u and v must also be in the class. Various bounds are known concerning the probability of a random graph from such a class being connected or having many components, sometimes under the additional assumption that bridges can be deleted as well as added. Here we improve or amplify or generalise these bounds. For example, we see that the expected number of vertices left when we remove a largest component is less than 2. The generalisation is to consider 'weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.