Abelian spiders and real cyclotomic integers

Abstract:

If $\Gamma$ is a finite graph, then the largest eigenvalue $\lambda$ of the adjacency matrix of $\Gamma$ is a totally real algebraic integer ($\lambda$ is the Perron-Frobenius eigenvalue of $\Gamma$). We say that $\Gamma$ is abelian if the field generated by $\lambda ^2$ is abelian. Given a fixed graph $\Gamma$ and a fixed set of vertices of $\Gamma$, we define a spider graph to be a graph obtained by attaching to each of the chosen vertices of $\Gamma$ some $2$-valent trees of finite length. The main result is that only finitely many of the corresponding spider graphs are both abelian and not Dynkin diagrams, and that all such spiders can be effectively enumerated; this generalizes a previous result of Calegari, Morrison, and Snyder. The main theorem has applications to the classification of finite index subfactors. We also prove that the set of Salem numbers of “abelian type” is discrete.