Distance, eccentricity, radius, diameter, center, and median: Basic properties of a graph

Basic Properties of Graphs

distance	Distance between u and v in the graph G is the length of the shortest u – v path in G if such a path exists. If no u – v path exist, define d(u, v) = ¥ .
eccentricity	For each vertex v in a graph G, the eccentricity, e(v), is the distance from v to the vertex furthest from v, that is, e(v) = max {d(v, u) \| u Î V(G)}.
radius	Radius rad G of a connected graph G is defined as rad G = min {e(v) \| v Î V(G)}.
diameter	Diameter diam G of a connected graph G is defined as diam G = max {e(v) \| v Î V(G)}.
center	Center C(G) is the subgraph induced by the vertices of G whose eccentricity equals the radius of G.
distance of a vertex	Distance d(v) of a vertex v is the sum of the distances from v to each vertex of G.
median	Median M(G) is the subgraph induced by the set of vertices having minimum distance. Center and median may be disjoint.

Two applications:

In order to determine the best location for a new hospital we have mapped out the distances between all major intersections in the city. The best location is one that minimizes the distance to any portion of the city from the hospital, i.e. the center of this graph. Your job is to determine the center of this graph.
The local newspaper is interested in establishing the best location for distribution of the morning paper. The lead paperboys deliver by bike one load of papers to each intersection so we must therefore pick a location minimizing the average of the travel distances needed to reach each intersection. Your job is to find the median of this graph.

"Weighted" adjacency matrix for G