  We investigate stochastic reaction-diffusion equations on finite metric
graphs. On each edge in the graph a multiplicative cylindrical Gaussian noise
driven reaction-diffusion equation is given. The vertex conditions are the
standard continuity and generalized, non-local Neumann-Kirchhoff-type law in
each vertex. The reaction term on each edge is assumed to be an odd degree
polynomial, not necessarily of the same degree on each edge, with possibly
stochastic coefficients and negative leading term. The model is a
generalization of the problem in [14] where polynomials with much more
restrictive assumptions are considered and no first order differential operator
is involved. We utilize the semigroup approach from [15] to obtain existence
and uniqueness of solutions with sample paths in the space of continuous
functions on the graph.
