I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2004, VOLUME 10, NUMBER 3, PAGES 245-254
An interlacing theorem for matrices whose graph is a given tree
View as HTML
View as gif image
Let and be -matrices.
For an index set ,
denote by the principal
submatrix that lies in the rows and columns indexed
the complement of and define , where the summation
is over all subsets of and, by
C. R. Johnson conjectured that if and are Hermitian and
positive semidefinite, then the polynomial has only real roots.
G. Rublein and R. B. Bapat proved that this is true for
Bapat also proved this result for any with the condition that
both and are tridiagonal.
In this paper, we generalize some little-known results concerning the
characteristic polynomials and adjacency matrices of trees to matrices
whose graph is a given tree and prove the conjecture for
under the additional assumption that both and are matrices whose graph
is a tree.
Last modified: February 28, 2005