The center (or Jordan center [1]) of a graph is the set of all vertices of minimum eccentricity, [2] that is, the set of all vertices u where the greatest distance d(u,v) to other vertices v is minimal. Equivalently, it is the set of vertices with eccentricity equal to the graph's radius. [3] Thus vertices in the center (central points) minimize the maximal distance from other points in the graph.
This is also known as the vertex 1-center problem and can be extended to the vertex k-center problem.
Finding the center of a graph is useful in facility location problems where the goal is to minimize the worst-case distance to the facility. For example, placing a hospital at a central point reduces the longest distance the ambulance has to travel.
The center can be found using the Floyd–Warshall algorithm. [4] [5] Another algorithm has been proposed based on matrix calculus. [6]
The concept of the center of a graph is related to the closeness centrality measure in social network analysis, which is the reciprocal of the mean of the distances d(A,B). [1]