  A large class of Thermodynamic Bethe Ansatz equations governing the
Renormalization Group evolution of the Casimir energy of the vacuum on the
cylinder for an integrable two-dimensional field theory, can often be encoded
on a tensor product of two graphs. We demonstrate here that in this case the
two graphs can only be of $ADE$ type. We also give strong numerical evidence
for a new large set of Dilogarithm sum Rules connected to $ADE\times ADE$ and a
simple formula for the ultraviolet perturbing operator conformal dimensions
only in terms of rank and Coxeter numbers of $ADE\times ADE$. We conclude with
some remarks on the curious case $ADE\times D$. [Talk given by F.R. at the
Cargese Workshop "New Developments in String Theory, Conformal Models and
Topological Field Theory" (May 1993)]
