  We describe the infinite family of spider-web graphs $S_{k,M,N }$, $k \geq
2$, $M \geq 1$ and $N \geq 0$, studied in physical literature as tensor
products of well-known de Brujin graphs $B_{k,N}$ and cyclic graphs $C_M$ and
show that these graphs are Schreier graphs of the lamplighter groups $L_k =
Z/kZ \wr Z$. This allows us to compute their spectra and to identify the
infinite limit of $S_{k,M,N}$, as $N, M \to\infty$, with the Cayley graph of
the lamplighter group $L_k$.
  This is the final version of the article, taking in account comments from the
referees and with an extended introduction.
