Every totally real algebraic integer is a tree eigenvalue

Justin Salez (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Every graph eigenvalue is in particular a totally real algebraic integer, i.e. a zero of some real-rooted monic polynomial with integer coefficients. Conversely, the fact that every such number occurs as an eigenvalue of some finite graph is a remarkable result, conjectured forty years ago by Hoffman, and proved twenty years later by Bass, Estes and Guralnick. This note provides an independent, elementary proof of a stronger statement, namely that the graph may always be chosen to be a tree.

Mots clés: tree; adjacency matrix; eigenvalue; totally real algebraic integer

Domaines: Combinatoire [math.CO]

Justin Salez. Every totally real algebraic integer is a tree eigenvalue. 2013. ⟨hal-00789806v1⟩