  We study the dynamics of the Fisher-KPP equation on the infinite homogeneous
tree and Erd\H{o}s-R\'eyni random graphs. We assume initial data that is zero
everywhere except at a single node. For the case of the homogeneous tree, the
solution will either form a traveling front or converge pointwise to zero. This
dichotomy is determined by the linear spreading speed and we compute critical
values of the diffusion parameter for which the spreading speed is zero and
maximal and prove that the system is linearly determined. We also study the
growth of the total population in the network and identify the exponential
growth rate as a function of the diffusion coefficient, $\alpha$. Finally, we
make predictions for the Fisher-KPP equation on Erd\H{o}s-R\'enyi random graphs
based upon the results on the homogeneous tree. When $\alpha$ is small we
observe via numerical simulations that mean arrival times are linearly related
to distance from the initial node and the speed of invasion is well
approximated by the linear spreading speed on the tree. Furthermore, we observe
that exponential growth rates of the total population on the random network can
be bounded by growth rates on the homogeneous tree and provide an explanation
for the sub-linear exponential growth rates that occur for small diffusion.
