Algebraic Potential Theory on Graphs

This paper encompasses a motley collection of ideas from severalareas of mathematics, including, in no particular order, randomwalks, the Picard group, exchange rate networks, chip-firinggames, cohomology, and the conductance of an electrical network.The linking threads are the discrete Laplacian on a graph andthe solution of the associated Dirichlet problem. Thirty yearsago, this subject was dismissed by many as a trivial specialisationof cohomology theory, but it has now been shown to have hiddendepths. Plumbing these depths leads to new theoretical advances,many of which throw light on the diverse applications of thetheory. 1991 Mathematics Subject Classification 05C50.