An interlacing theorem for matrices whose graph is a given tree

Let $A$ and $B$ be $(n\times n)$-matrices. For an index set $S\subset\{1,\ldots,n\}$, 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 $\eta(A,B)=\sum\limits_S\det A(S)\det B(S')$, where the summation is over all subsets of $\{1,\ldots,n\}$ and, by convention, $\det A(\varnothing)=\det B(\varnothing)=1$. C. R. Johnson conjectured that if $A$ and $B$ are Hermitian and $A$ is positive semidefinite, then the polynomial $\eta(\lambda A,-B)$ has only real roots. G. Rublein and R. B. Bapat proved that this is true for $n\leq3$. 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.