کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4656773 | 1632982 | 2015 | 8 صفحه PDF | دانلود رایگان |
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×nn×n symmetric matrices with independent Bernoulli (cn) entries are exactly the totally real algebraic integers. This settles an open problem raised by Ben Arous (2010).
Journal: Journal of Combinatorial Theory, Series B - Volume 111, March 2015, Pages 249–256