Solving Dirichlet and Poisson problems on graphs by means of equilibrium measures

Authors:
Enrique Bendito;Ángeles Carmona;Andrés M. Encinas
Affiliations:
Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Spain;Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Spain;Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Spain
Venue:
European Journal of Combinatorics
Year:
2003

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Abstract

We aim here at obtaining an explicit expression of the solution of the Dirichlet and Poisson problems on graphs. To this end, we consider the Laplacian of a graph as a kernel on the vertex set, V , in the framework of Potential Theory. Then, the properties of such a kernel allow us to obtain for each proper vertex subset the equilibrium measure that solves the so-called equilibrium problem . As a consequence, the Green function of the Dirichlet problems, the generalized Green function of the Poisson problems and the solution of the condenser principle are obtained solely in terms of equilibrium measures for suitable subsets. In particular, we get a formula for the effective resistance between any pair of vertices of a graph. Specifically, r xy = 1/n (v x (y) + v y (x)) , where v z denotes the equilibrium measure for the set V - {z} . In any case, the equilibrium measure for a proper subset is accomplished by solving a Linear Programming Problem.