Last update: 1/10/06
Abstract
A metrized graph is essentially a finite weighted graph whose edges are thought of as line segments. Metrized graphs have appeared in various guises throughout the mathematical literature, and have a number of applications. In this expository paper, we study the Laplacian operator on a metrized graph and some important functions related to it. These functions include the "j-function" jz(x,y) and the "effective resistance function" r(x,y). We discuss the relationship between metrized graphs and electrical networks, which provides some physical intuition for the concepts bring dealt with. We also discuss the relation between the Laplacian on a metrized graph and the combinatorial Laplacian matrix. Finally, we introduce the "canonical measure" on a metrized graph, which arises naturally when considering the Laplacian of r(x,y), and use it to obtain a new proof of Foster's network theorem.