Metric properties of graphs

Given a connected graph, we can form a metric space from the graph in which the distance between any two points is determined by the shortest path between them. The shortest path between two points need not be unique, unless the graph is a geodetic graph. This metric space is always going to be a discrete space, and a very particular type of discrete space which is generated by the unit distances between its points. The most interesting metric property of any vertex in a graph is its eccentricity which is the greatest distance between the vertex and any other point. The radius and the diameter of a graph are distance related properties determined by the minimum and the maximum eccentricity respectively.