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We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on
phenomena related to unboundedness of the Laplacians. This includes (failure of) essential
selfadjointness, absence of essential spectrum and stochastic incompleteness.
Sur un ensemble fini, on s'intéresse aux minorations linéaires
du trou spectral d'un noyau markovien réversible, en terme de
la constante isopérimétrique associée. On montre que la constante
optimale est l'inverse du cardinal de l'ensemble moins un, mais
on verra aussi comment il est possible de l'améliorer dans
certaines situations particulières (arbres pointés radiaux
à nombre fini de générations). Une application des inégalités
précédentes est de retrouver immédiatement le comportement...
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