realtion between trace of product of two matrices and their eigenvalues

$\displaystyle A$ is a symmetric matrix containing only 0s and 1s; in particular all the diagonal entries are 0. $\displaystyle B$ is a symmetric positive definite matrix. I am interested in $\displaystyle T = \mathrm{trace}(AB)$. Is there a simple relation between $\displaystyle T$ and the eigenvalues of $\displaystyle A$ and $\displaystyle B$?

Re: realtion between trace of product of two matrices and their eigenvalues

Nope.

Counter example:

$\displaystyle A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$

$\displaystyle B=\begin{pmatrix}1&0\\0&2\end{pmatrix}$ respectively $\displaystyle B=\begin{pmatrix}-1/2&3/2\\3/2&-1/2\end{pmatrix}$

Both versions of B have the same positive eigenvalues, but the trace of AB is different.