  We investigate existence of global in time solutions and blow-up of solutions
to the semilinear heat equation posed on infinite graphs. The source term is a
general function $f(u)$. We always assume that the infimum of the spectrum of
the Laplace operator $\lambda_1(G)$ on the graph is positive. According to an
interaction between the behavior of $f$ close to $0$ and the value
$\lambda_1(G)$, we get the existence of a global in time solution or blow-up of
any nonnegative solution, provided that the initial datum is nontrivial.
