In <a href="http://front.math.ucdavis.edu/math.CO/0411341">"Cluster algebras of finite type and positive symmetrizable matrices"</a> by Barot, Geiss and Zelevinsky we can see that positive definite _quasi_-Cartan matrices (defined the same as Cartan matrices but relaxing the condition of non-positivity of elements off the diagonal) actually come from positive definite Cartan matrices. That is, each positive definite quasi-Cartan matrix is equivalent to a positive definite Cartan matrix, where equivalence in the symmetric case is defined as $A\sim B$ if there is a matrix $E$ with determinant $\pm 1$ so that $A=E^TBE$. This is proved in proposition 2.9.

This means that the graphs you are looking for are classified up to the equivalence relation above. Since a such a matrix $E$ as above can be constructed easily for the case of trees for example this means that all trees that answer your question are also ADE.

A classification result that includes the trees example above is given in "Sincere weakly positive unit quadratic forms" and again, this classification contains graphs that are obtained from ADE by some modifications, and the resulting graphs are described with 2-colored edges.