  We consider the range of the simple random walk on graphs with spectral
dimension two. We give a form of strong law of large numbers under a certain
uniform condition, which is satisfied by not only the square integer lattice
but also a class of fractal graphs. Our results imply the strong law of large
numbers on the square integer lattice established by Dvoretzky and Erd\"os
(1951). Our proof does not depend on spacial homogeneity of space and gives a
new proof of the strong law of large numbers on the lattice. We also show that
the behavior of appropriately scaled expectations of the range is stable with
respect to every "finite modification" of the two-dimensional integer lattice,
and furthermore we construct a recurrent graph such that the uniform condition
holds but the scaled expectations fluctuate. As an application, we establish a
form of law of the iterated logarithms for lamplighter random walks in the case
that the spectral dimension of the underlying graph is two.
