Free Products of Graphs
A graph is a collection of vertices and edges with the edges being pairs of vertices. You can think of vertices as representing points in the plane and an edge paired with two vertices is represented by a line joined between these two points.
In certain areas of mathematics, we are often interested in the symmetries of a graph. We use a special type of mathematical structure called a group to study the symmetries of a graph. A group is a collection of objects that has some method of composing objects in the collection to form other objects in the collection. For instance, take the graph that looks like a square; four points in the plane and edges connecting the vertices in the natural way. If for instance, we apply two 90-degree rotations to this square graph consecutively, this would be the same as applying one 180-degree rotation. This is an example of how one can ‘compose’ symmetries of this square graph. We can compose reflections and other rotations of a square in a similar manner and the collection of all these rotations and reflections of the square graph form a particular group.
This is a basic example however, there are much more elaborate graphs that have very large and complex groups of symmetries associated with them. In my 2018/19 AMSI Vacation Research Scholarship project, we looked into a particular topic called Free Products of Graphs. A product of graphs is a method of taking two graphs, or some arbitrary number of graphs, and forming a new graph that has particular characteristics associated with it in terms of the graphs that we are taking the product of. A free product of graphs is a way of taking a collection of graphs each with a finite number of vertices (and edges) and forming a new graph with an infinite number of vertices (and edges) that has a lot of symmetries associated with it. The new free product graph has a particular characteristic in that it has a copy of each of the starting graphs attached at each of the vertices in the free product graph. It is also possible however, to take the free product of infinite graphs, and it is these free product graphs that we are most interested in.
The symmetry groups associated with these free product graphs are of particular interest to those who do research in the area of topological group theory and they form a special type of topological group called a totally disconnected, locally compact group. It is expected that these symmetry groups of free product graphs will form interesting examples of totally disconnected, locally compact groups.
Max Carter was a recipient of a 2018/19 AMSI Vacation Research Scholarship.