ELA, Volume 11, pp. 41-50, February 2004, abstract.

On Two Conjectures Regarding an Inverse Eigenvalue 
Problem for Acyclic Symmetric Matrices

Francesco Barioli and Shaun M. Fallat

For a given acyclic graph G, an important problem is to 
characterize all of the eigenvalues over all symmetric 
matrices with graph G. Of particular interest is the 
connection between this standard inverse eigenvalue 
problem and describing all the possible associated 
ordered multiplicity lists, along with determining the 
minimum number of distinct eigenvalues for a symmetric 
matrix with graph G. In this note two important open 
questions along these lines are resolved, both in the 
negative.