Generalized crested products of Markov chains

We define a finite Markov chain, called generalized crested product, which naturally appears as a generalization of the first crested product of Markov chains. A complete spectral analysis is developed and the $k$-step transition probability is given. It is important to remark that this Markov chain describes a more general version of the classical Ehrenfest diffusion model. As a particular case, one gets a generalization of the classical Insect Markov chain defined on the ultrametric space. Finally, an interpretation in terms of representation group theory is given, by showing the correspondence between the spectral decomposition of the generalized crested product and the Gelfand pairs associated with the generalized wreath product of permutation groups.Comment: 18 page

Weighted spanning trees on some self-similar graphs

We compute the complexity of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.Comment: 21 page

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

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