  A gain graph is a graph whose edges are orientably labelled from a group. A
weighted gain graph is a gain graph with vertex weights from an abelian
semigroup, where the gain group is lattice ordered and acts on the weight
semigroup. For weighted gain graphs we establish basic properties and we
present general dichromatic and forest-expansion polynomials that are Tutte
invariants (they satisfy Tutte's deletion-contraction and multiplicative
identities). Our dichromatic polynomial includes the classical graph one by
Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with
positive integer weights, and that of rooted integral gain graphs by Forge and
Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that
remains to be found.
  An evaluation of one example of our polynomial counts proper list colorations
of the gain graph from a color set with a gain-group action. When the gain
group is Z^d, the lists are order ideals in the integer lattice Z^d, and there
are specified upper bounds on the colors, then there is a formula for the
number of bounded proper colorations that is a piecewise polynomial function of
the upper bounds, of degree nd where n is the order of the graph.
  This example leads to graph-theoretical formulas for the number of integer
lattice points in an orthotope but outside a finite number of affinographic
hyperplanes, and for the number of n x d integral matrices that lie between two
specified matrices but not in any of certain subspaces defined by simple row
equations.
