Adam Clearwaters's 2016 summer research project in mathematics, on "Abelian graphs" is available here.

### Outline

For my Summer Research Scholarship I have worked on understanding and extending the results of Abelian Spiders by Calegari-Guo [CG15].
Let $\Gamma$ be a connected finite graph, let $n \in \mathbb{ N}$ be fixed, and let $v_1,\ldots,v_n$ be vertices
of $\Gamma$. For any $n$-tuple of non-negative integers $a = (a_1, \ldots , a_n)$, the spider graph $\Gamma_a$ is
defined to be the graph obtained from $\Gamma$ by adjoining a 2-valent tree of length $a_i$ to vertex $v_i$. Most of my
own work this summer has been on $n$-spokes, which are spider graphs obtained from the graph consisting of a single
vertex and no edges, along with the $n$-tuple $a = (a_1,\ldots,a_n)$ where we suppose that $1 \leq a_1 \leq \cdots
\leq a_n$.

Of primary interest is the question of if a given graph is abelian. A graph $\Gamma$ is abelian if $\mathbb{Q}
(\lambda^2)$ is an abelian extension of $\mathbb{Q}$, where $\lambda$ is the Perron-Frobenius eigenvalue of $\Gamma$, the
largest eigenvalue of the adjacency matrix of $\Gamma$. Equivalently, $\Gamma$ is abelian if $\lambda$ is a cyclotomic
integer, which is an algebraic integer that is an element of $\mathbb{Q}(\zeta_N )$ where $\zeta_N = \exp(2\pi i/N)$ for
some $N \in \mathbb{ N}$.

This project consisted of three sections. During an initial reading of [CG15] I undertook a period of background reading on basic graph theory, algebraic number theory and some field theory, in order to understand the definitions and claims made in the papers [ST02, GR01]. My attention then turned to $n$-spokes, where I reconstructed much of the section of [CG15] on 3-spokes. In doing so, I gained some experience in computing with Mathematica. Finally, a number of lemmas and calculations for 3-spokes were replicated for the case of 4-spokes, making progress towards the goal of finding all abelian 4-spokes.