1. Every totally real algebraic integer is a tree eigenvalue.
- Author
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Salez, Justin
- Subjects
- *
ALGEBRA , *INTEGERS , *TREE graphs , *EIGENVALUES , *GRAPH theory - Abstract
Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep and remarkable result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and elementary proof of a stronger statement, namely that the graph may actually be chosen to be a tree. As a by-product, our result implies that the atoms of the limiting spectrum of n × n symmetric matrices with independent Bernoulli ( c n ) entries are exactly the totally real algebraic integers. This settles an open problem raised by Ben Arous (2010). [ABSTRACT FROM AUTHOR]
- Published
- 2015
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