Spectral analysis of the wreath product of a complete graph with a cocktail party graph

Graph products and the corresponding spectra are often studied in the literature. A special attention has been given to the wreath product of two graphs, which is derived from the homonymous product of groups. Despite a general formula for the spectrum is also known, such a formula is far from giving an explicit spectrum of the compound graph. Here, we consider the latter product of a complete graph with a cocktail party graph, and by making use of the theory of circulant matrices we give a direct way to compute the (adjacency) eigenvalues

Harmonic analysis of finite lamplighter random walks

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.Comment: 29 page

Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs

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