An interlacing theorem for matrices whose graph is a given tree

Abstract

Let A and B be (n×n)-matrices. For an index set S ⊂ {1, …, n}, denote by A(S) the principal submatrix that lies in the rows and columns indexed by S. Denote by S′ the complement of S and define η(A, B) = \(\mathop \sum \limits_S \) det A(S) det B(S′), where the summation is over all subsets of {1, …, n} and, by convention, det A(∅) = det B(∅) = 1. C. R. Johnson conjectured that if A and B are Hermitian and A is positive semidefinite, then the polynomial η(λA,-B) has only real roots. G. Rublein and R. B. Bapat proved that this is true for n ⩽ 3. Bapat also proved this result for any n with the condition that both A and B 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 any n under the additional assumption that both A and B are matrices whose graph is a tree.