Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph

The concepts of the "center" and the "median vertex" of a graph are generalized to the "absolute center" and the "absolute median" of a weighted graph a graph with weights attached to its vertices as well as to its branches. These results are used to find the optimum location of a "switching center" in a communication network and to locate the best place to build a "police station" in a highway system. It is shown that the optimum location of a switching center is always at a vertex of the communication network while the best location for the police station is not necessarily at an intersection. Procedures for finding these locations are given.