Let $A_G$ be the adjacency matrix of a graph $G=(V,E)$. The Laplacian of $G$ is defined to be the matrix $$ L_G=D-A_G $$ where $D$ is the diagonal matrix whose $(u,u)$ entry is equal to the degree of the vertex $u$ in $G$.
It is well known that the product of the non-zero eigenvalues of $L_G$ is equal to the number of spanning trees of $G$. Is there some similar results known about the product of non-zero eigenvalues of $A_G$?