Positivity 8: 209–213, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Resolution of the Symmetric Nonnegative Inverse
Eigenvalue Problem for Matrices Subordinate to a
and CHARLES R. JOHNSON
Departmento de Matemática, Universidade de Coimbra, Apartado 3008, 3000 Coimbra, Portugal
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
(Received 2 March 1999; accepted 27 November 2000)
Abstract. There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G
on n vertices, with eigenvalues
if and only if
0, in which m is the matching number of G.
Other observations are also made about the symmetric nonnegative inverse eigenvalue problem with
respect to a graph.
1 Nonnegative Inverse Eigenvalue Problems
The nonnegative inverse eigenvalue problem (NIEP) asks which collections of n
complex numbers (repeats allowed) occur as the eigenvalues of an n-by-n, entry-
wise nonnegative matrix. This problem has attracted considerable attention over
50+ years  and, despite many exciting partial results, remains quite unresolved.
The companion symmetric nonnegative inverse eigenvalue problem (SNIEP) in
which the realizing nonnegative matrix is required to be symmetric and the eigen-
values are (of course) real is also open and has also been the subject of attention e.g.
[2, 6], etc. The intermediate real nonnegative inverse eigenvalue problem (RNIEP)
asks which collections of n real numbers occur as the eigenvalues of an n-by-n
nonnegative matrix and is now known to be a properly different problem from the
2 Graph Theoretic Versions
Mathematically, it is natural to consider graph theoretic versions (of the nonneg-
ative inverse eigenvalue problems), in which a non-edge requires a 0 entry in the
realizing matrix. Speciﬁcally, given a directed (undirected) graph G on n vertices,
which may be taken to be 1n, we say that an n-by-n matrix A = a
subordinate to G if a
=0, i = j, implies that ij (resp. ij is an edge of
This research was supported in part by Project 574/94 of Fundação Luso Americana para
o Desenvolvimento, by CMUC/FCT, by Projecto PRAXIS 2/2.1/MAT/458/94 and by PRAXIS