Abstract. The theory of resistive electrical networks is invoked to develop a novel view: if fixed resistors are assigned to each edge of a connected graph, then the effective resistance between pairs of vertices is a graphical distance. Such an approach has been developed in Texas A&M University in Galveston by Douglas Klein in the late 1990s. Some basic definitions and results are presented, including several theorems. A number of sum rules for this metric are established. "Global" and "local" types of sum rules are identified.