  Recently, several papers have been devoted to the analysis of lamplighter
random walks, in particular when the underlying graph is the infinite path
$\mathbb{Z}$. In the present paper, we develop a spectral analysis for
lamplighter random walks on finite graphs. In the general case, we use the
$C_2$-symmetry to reduce the spectral computations to a series of eigenvalue
problems on the underlying graph. In the case the graph has a transitive
isometry group $G$, we also describe the spectral analysis in terms of the
representation theory of the wreath product $C_2\wr G$. We apply our theory to
the lamplighter random walks on the complete graph and on the discrete circle.
These examples were already studied by Haggstrom and Jonasson by probabilistic
methods.
