Salem numbers, Pisot numbers, graphs, and Mahler measure
by
James McKee
Royal Holloway, University of London
Coauthors: Chris Smyth

We use graphs to define sets of Salem and Pisot numbers, and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers n the smallest known element of the n-th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph, and find all graphs of Mahler measure less than (1+sqrt5)/2. We start the task of extending this work from graph adjacency matrices to all integer symmetric matrices by classifying all such matrices having all eigenvalues in the interval [-2, 2].

Paper reference: arXiv:math.NT/0503480

Date received: November 30, 2005