Discrete Laplacian on Graphs, Dirichlet Problem, and Minimization of Energy

Abstract

This report is primarily interested in discrete Laplacian on graphs and Dirichlet prob- lem. We investigate two cases of graphs, their Dirichlet problems, and the method of finding solutions to such problems by minimization of their corresponding energy functionals. In the case of an infinite, locally-finite, undirected, simple graph, we find that the minimiza- tion of its energy functional yields a solution to its Dirichlet or boundary-value problem. In the case of finite, directed, locally-finite decision tree, we propose a type of such tree specified by a chosen vertex with no predecessors and by a chosen boundary set of vertices that have no successors. We then further consider a special case of such construction where all the paths of the tree terminate at the same level. We find that the minimization of such special case’s energy functional yields a solution to its Dirichlet problem only when some additional conditions on and close to the boundary (level of termination) are met.