  We study diffusions, variational principles and associated boundary value
problems on directed graphs with natural weightings. Using random walks and
exit times, we associate to certain subgraphs (domains) a pair of sequences,
each of which is invariant under the action of the automorphism group of the
underlying graph. We prove that these invariants differ by an explicit
combinatorial factor given by Stirling numbers of the first and second kind. We
prove that for any domain with a natural weighting, these invariants determine
the eigenvalues of the Laplace operator corresponding to eigenvectors with
nonzero mean. As a specific example, we investigate the relationship between
our invariants and heat content asymptotics, expressing both as special values
of a Dirichlet series.
