Experimental Mathematics

Salem numbers, Pisot numbers, Mahler measure, and graphs

James McKee and Chris Smyth

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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 {\small$n$}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 {\small $\frac12(1+\sqrt{5})$}. Finally, we list all small Salem numbers known to be definable using a graph.

Article information

Source
Experiment. Math., Volume 14, Issue 2 (2005), 211-229.

Dates
First available in Project Euclid: 30 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.em/1128100133

Mathematical Reviews number (MathSciNet)
MR2169524

Zentralblatt MATH identifier
1082.11066

Subjects
Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Keywords
Pisot numbers Salem numbers Mahler measure graph spectra

Citation

McKee, James; Smyth, Chris. Salem numbers, Pisot numbers, Mahler measure, and graphs. Experiment. Math. 14 (2005), no. 2, 211--229. https://projecteuclid.org/euclid.em/1128100133


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