  A graph theoretic perspective is taken for a range of phenomena in continuum
physics in order to develop representations for analysis of large scale,
high-fidelity solutions to these problems. Of interest are phenomena described
by partial differential equations, with solutions being obtained by
computation. The motivation is to gain insight that may otherwise be difficult
to attain because of the high dimensionality of computed solutions. We consider
graph theoretic representations that are made possible by low-dimensional
states defined on the systems. These states are typically functionals of the
high-dimensional solutions, and therefore retain important aspects of the
high-fidelity information present in the original, computed solutions. Our
approach is rooted in regarding each state as a vertex on a graph and
identifying edges via processes that are induced either by numerical solution
strategies, or by the physics. Correspondences are drawn between the sampling
of stationary states, or the time evolution of dynamic phenomena, and the
analytic machinery of graph theory. A collection of computations is examined in
this framework and new insights to them are presented through analysis of the
corresponding graphs.
