Minimum Vertex Potential Principle

Abstract

We consider the problem of minimizing a sum of vertex potentials of a finite directed graph for which 1) there is a directed path from a base vertex to every other vertex, 2) each arc is assigned a minimum rise potential constraint, and 3) Kirchoff's voltage law is satisfied. It is pointed out that a solution exists if and only if the sum of the minimum rise potential constraints about every cycle is nonpositive; and if a solution exists, an optimum is obtained by setting the potential at every vertex equal to the maximum sum of minimum rises taken over all paths to the vertex. Thus, the solution is easily obtained by finding a maximum distance tree. The intuitive simplicity of the result together with several ready physical applications make it seem worth mentioning.