Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs

Abstract

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 nth 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 ½ (1+√5). Finally, we list all small Salem numbers known to be definable using a graph.

Keywords:

• Primary 11R06, 05C50
• Pisot numbers
• Salem numbers
• Mahler measure
• graph spectra