An Explicit Formula for the Eigenvectors of Acyclic Matrices and   Weighted Trees
  Let $A$ be an acyclic symmetric matrix of order $n$. There is a weighted forest $F$ whose adjacency matrix is $A$. In this paper, using some results on matching polynomials, we provide an explicit formula for eigenvectors of $A$. 
Asghar Bahmani and Dariush Kiani
math.CO