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.