ON TWO CONJECTURES REGARDING AN INVERSE
EIGENVALUE PROBLEM FOR ACYCLIC SYMMETRIC MATRICES
FRANCESCO BARIOLI AND SHAUN M. FALLAT
Abstract. 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.
Key words. Symmetric matrices, Acyclic matrices, Eigenvalues, Graphs, Binary trees.
AMS subject classifications. 15A18, 15A48, 05C50.
1. Introduction. Spectral Graph Theory is the study of the eigenvalues of var-
ious structured matrices associated with graphs. This subject lies at the crossroads
of Linear Algebra and Graph Theory, and has become a prominent area of study
for both disciplines. Of particular interest here is the so-called "inverse eigenvalue
Essentially our goal is to construct a certain type of matrix from some specified
spectral information. In our case part of this spectral information will be contained
in an underlying graph. If A is any n × n symmetric matrix, then the graph of A,