A random r-regular graph is a graph, which denotes the probability space of all r-regular graphs is even. It is therefore a particular kind of random graph, but the regularity restriction significantly alters the properties that will hold, since most graphs are not regular.

Properties of random regular graphs

As with more general random graphsregular graphs hold asymptotically almost surely, a random r-regular graph of large size is asymptotically almost surely r-connected. increases.

is the least integer satisfying

then, asymptotically almost surely, a random r-regular graph has diameter at most d. There is also a (more complex) lower bound on the diameter of r-regular graphs, so that almost all r-regular graphs (of the same size) have almost the same diameter.

are asymptotically independent Poisson random variables with means

Algorithms for random regular graphs

It is non-trivial to implement the random selection of r-regular graphs efficiently and in an unbiased way, since most graphs are not regular. The pairing model (also configuration model) is a method which takes nr points, and partitions them into n buckets with r points in each of them. Taking a random matching of the nr points, and then contracting the r points in each bucket into a single vertex, yields an r-regular graph or multigraph. If this object has no multiple edges or loops (i.e. it is a graph), then it is the required result. If not, a restart is required.

A refinement of this method was developed by Brendan McKay and Nicholas Wormald.

References