  We consider the range of random walks up to time n, R_n, on graphs satisfying
a uniform condition. This condition is characterized by potential theory. Not
only all vertex transitive graphs but also many non-regular graphs satisfy the
condition. We show certain weak laws of R_n from above and below. We also show
that there is a graph such that it satisfies the condition and a sequence of
the mean of R_n/n fluctuates. By noting the construction of the graph, we see
that under the condition, the weak laws are best in a sense.
