Professor X’s Mutations of Quivers

Patterns arise naturally throughout mathematics, one of which is known as Graph Theory. A graph is a collection of vertices, called nodes, and line segments connecting them, called edges. Graphs are often used in the form of networks to analyze things such as flow of information in social media, biological systems, and chip design in computers.

If we start with a graph called a quiver, we can follow certain rules to mutate that graph, and at each step arrive at a different quiver. For example, we can start with 4 nodes and directed edges between them, such as

A mutation resulting in a tree

In this project, we will analyze graphs and quivers with the help of Sage and a Java applet (which generates pictures similar to those above, and the sequences on the right). We will see how mutations of quivers relate to the Fibonacci numbers and other sequences of numbers. We hope to then find sequences and a quiver together with a set mutations that produces that sequence.

An infinite mutation