Graph center

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.

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 A where the greatest distance d(A,B) to other vertices B 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. 1 2 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. ↑ Weisstein, Eric W., "Graph center", MathWorld.
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