Given an initial graph $G$, one may apply a rule $\R$ to $G$ which
replaces certain vertices of $G$ with other graphs called replacement
graphs to obtain a new graph $\R(G)$. By iterating this procedure on
each resulting graph, a sequence of graphs $\{\R^n(G)\}$ is obtained.
When the graphs in this sequence are normalized to have diameter one,
questions of convergence can be investigated. Sufficient conditions
for convergence
in
the Gromov-Hausdorff metric were given by J. Previte, M. Previte, and
M. Vanderschoot for such normalized
sequences of graphs when the replacement
rule $\R$ has more than one replacement graph.
M. Previte and H.S. Yang showed that under these conditions,
the limits of such sequences have topological dimension one.
In this paper, we compute the box
and Hausdorff dimensions of limit spaces of normalized sequences of
iterated vertex replacements when there is
more than one replacement graph. Since the limit spaces have
topological dimension one and typically have Hausdorff (and box)
dimension greater than one, they are fractals.
Finally, we
give examples of vertex replacement rules that yield fractals.