Counting vertices and cubes in median graphs of circular split systems

Median graphs are a natural generalisation of trees and hypercubes that are closely related to distributive lattices and graph retracts. In the past decade, they have become of increasing interest to the biological community, where, amongst other things, they are applied to the study of evolutionary relationships within populations.Two simple measures of complexity for a median graph are the number of vertices and the number of maximal induced subcubes. These numbers can be useful in biological applications, and they are also of purely mathematical interest. However, they can be hard to compute in general. Here we present some special families of median graphs where it is possible to find formulae and recursions for these numbers.