  We investigate nonexistence of nontrivial nonnegative solutions to a class of
semilinear parabolic equations with a positive potential, posed on weighted
graphs. Assuming an upper bound on the Laplacian of the distance and a suitable
weighted space-time volume growth condition, we show that no global solutions
exists. We also discuss the optimality of the hypotheses, thus recovering a
critical exponent phenomenon of Fujita type.
