Graph center

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.

A graph with central points colored red. These are the three vertices A such that d(A, B) ≤ 3 for all vertices B. Each black vertex is a distance of at least 4 from some other vertex.

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]