  This article studies a discrete geometric structure on triangulated manifolds
and an associated curvature flow (combinatorial Yamabe flow). The associated
evolution of curvature appears to be like a heat equation on graphs, but it can
be shown to not satisfy the maximum principle. The notion of a parabolic-like
operator is introduced as an operator which satisfies the maximum principle,
but may not be parabolic in the usual sense of operators on graphs. A maximum
principle is derived for the curvature of combinatorial Yamabe flow under
certain assumptions on the triangulation, and hence the heat operator is shown
to be parabolic-like. The maximum principle then allows a characterization of
the curvature as well was a proof of long term existence of the flow.
