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Mesoscopic physics on graphsLaboratoire de Physique des Solides, associe au CNRS Universite Paris-Sud, Bâtiment 510, Orsay, 91405, France , Orsay, Франция This report is a summary of recent work on the properties of phase coherent diffusive conductors, especially in the geometry of networks — also called graphs — made of quasi-$\mathrm{1D}$ diffusive wires. These properties are written as a function of the spectral determinant of the diffusion equation (the product of its eigenvalues). For a network with $N$ nodes, this spectral determinant is related to the determinant of an $N\times N$ matrix which describes the connectivity of the network. I also consider the transmission through networks made of $\mathrm{1D}$ ballistic wires and show how the transmission coefficient can be written in terms of an $N\times N$ matrix very similar to the above one. Finally I present a few considerations on the relation between the magnetism of noninteracting systems and the magnetism of interacting diffusive systems.
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