  For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of
$K_{n,n}$, we construct an infinite sequence of graphs based on the pair
$(G,J)$. The First graph in the sequence is $G$, then at each stage replacing
every vertex of the previous graph by a copy of $G$ and every edge of the
previous graph by a copy of $J$ the new graph is constructed. We call these
graphs {\em self-similar} graphs. We are interested in delineating those pairs
$(G,J)$ for which the chromatic numbers of the graphs in the sequence are
bounded. Here we have some partial results. When $G$ is a complete graph and
$J$ is a special matching we show that every graph in the resulting sequence is
an {\em expander} graph.
