Salem numbers, Pisot numbers, Mahler measure, and graphs. (English) Zbl 1082.11066

McKee, James; Smyth, Chris

Exp. Math. 14, No. 2, 211-229 (2005).

As usual, let \(S\) and \(T\) denote the sets of Pisot and Salem numbers. This interesting paper is an attempt to shed light on the conjecture that the set \(S \cup T\) is a closed set whose set of limit points is the set \(S\).
The authors introduce two subsets \(S_{\text{graph}} \subset S\) and \(T_{\text{graph}} \subset T\) defined in terms of the eigenvalues of certain graphs. They deduce many interesting properties of these numbers and in particular that \(S_{\text{graph}} \cup T_{\text{graph}}\) is a closed set whose set of limit points is the set \(S_{\text{graph}}\). However, the inclusions \(S_{\text{graph}} \subset S\) and \(T_{\text{graph}} \subset T\) are shown to be proper and hence the corresponding result for \(S \cup T\) remains open. They do not completely characterize the graphs that produce Salem numbers but they characterize all the trees with this property.
The authors introduce a natural notion of the Mahler measure of a graph and show that this is either equal to \(1\) or else bounded below by Lehmer’s number \(\tau_1\), the positive zero of Lehmer’s polynomial \(z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1\). They give a version of a theorem of M.-J. Bertin [Bull. Sci. Math., II. Ser. 104, 3–17 (1980; Zbl 0458.12003)] about the higher derived sets of \(S\), by showing that \((k + \sqrt{k^2+4})/2\) is in the \((2k-1)\)st derived set of \(S_{\text{graph}}\) and that \(k+1\) is in the \((2k)\)th derived set of \(S_{\text{graph}}\). It would be of interest to determine the order type of \(S_{\text{graph}}\). Is it the same as the order type of \(S\)? [see D. W. Boyd and R. D. Mauldin, Topology Appl. 69, No. 2, 115–120 (1996; Zbl 0857.11050)].
The paper should be of interest to those interested in eigenvalues of graphs as well as those whose interest is more number theoretical.

Reviewer: David W. Boyd (Vancouver)

MSC:

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

Keywords:

Pisot number; Salem number; Mahler measure; graph eigenvalue

Citations:

Zbl 0458.12003; Zbl 0857.11050