  A common generalization for the chromatic polynomial and the flow polynomial
of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning
for the coefficients of $T$ was discovered by Tutte at the beginning of its
definition. However, for a long time the combinatorial meaning for the values
of $T$ is missing, except for a few values such as $T(G;i,j)$, where $1\leq
i,j\leq 2$, until recently for $T(G;1,0)$ and $T(G;0,1)$. In this third one of
a series of papers, we introduce product valuations, cartesian product
arrangements, and multivariable characteristic polynomials, and apply the
theory of product arrangement to the tension-flow group associated with graphs.
Three types of tension-flows are studied in details: elliptic, parabolic, and
hyperbolic; each type produces a two-variable polynomial for graphs. Weighted
polynomials are introduced and their reciprocity laws are obtained. The dual
versions for the parabolic case turns out to include Whitney's rank generating
polynomial and the Tutte polynomial as special cases. The product arrangement
part is of interest for its own right. The application part to graphs can be
modified to matroids.
