Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs

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.

[1]  W. Parry.  Growth Series of Coxeter Groups and Salem Numbers , 1993 .

[2]  Michael Doob,et al.  On graphs whose spectral radius does not exceed $(2+\sqrt5)^1/2$ , 1982 .

[3]  A. Neumaier.  The second largest eigenvalue of a tree , 1982 .

[4]  Chris Smyth.  Salem numbers of negative trace , 2000, Math. Comput..

[5]  P. Rowlinson.  ALGEBRAIC GRAPH THEORY (Graduate Texts in Mathematics 207) By CHRIS GODSIL and GORDON ROYLE: 439 pp., £30.50, ISBN 0-387-95220-9 (Springer, New York, 2001). , 2002 .

[6]  James W. Cannon,et al.  Growth functions of surface groups , 1992 .

[7]  James McKee,et al.  Salem Numbers of Trace -2 and Traces of Totally Positive Algebraic Integers , 2004, ANTS.

[8]  D. Cvetkovic,et al.  The largest eigenvalue of a graph: A survey , 1990 .

[9]  There are Salem Numbers of Every Trace , 2003, math/0308213.

[10]  R. Salem,et al.  A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan , 1944 .

[11]  The Lehmer Polynomial,et al.  The Lehmer Polynomial and Pretzel Links , 2022 .

[12]  Kenneth J. Giuliani,et al.  Small Salem Numbers , 1997 .

[13]  On the spectral radius of Coxeter transformations of trees , 2005 .

[14]  A. Hoffman.  On limit points of spectral radii of non-negative symmetric integral matrices , 1972 .

[15]  A. Neumaier,et al.  The graphs with spectral radius between 2 and 2+5 , 1989 .

[16]  Nick Lord,et al.  Pisot and Salem Numbers , 1991 .

[17]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[18]  Chistopher J. Smyth,et al.  Totally positive algebraic integers of small trace , 1984 .

[19]  Changan.  SOME PROPERTIES OF THE SPECTRUM OF GRAPHS , 1999 .

[20]  William J. Floyd,et al.  Symmetries of planar growth functions , 1988 .

[21]  Growth of planar Coxeter groups, P.V. numbers, and Salem numbers , 1992 .