Integral trees with given nullity
  A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique. 
E. Ghorbani, A. Mohammadian, B. Tayfeh-Rezaie
math.CO