# Graph center

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

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.

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 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]

## References

1. ^ a b Wasserman, Stanley, and Faust, Katherine (1994), Social Network Analysis: Methods and Applications, page 185. Cambridge: Cambridge University Press. ISBN 0-521-38269-6
2. ^ McHugh, James A., Algorithmic Graph Theory
3. ^