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.
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]
References
- ↑ 1.0 1.1 Wasserman, Stanley, and Faust, Katherine (1994), Social Network Analysis: Methods and Applications, page 185. Cambridge: Cambridge University Press. ISBN:0-521-38269-6
- ↑ McHugh, James A., Algorithmic Graph Theory
- ↑ Weisstein, Eric W.. "Graph center". http://mathworld.wolfram.com/GraphCenter.html.
- ↑ Floyd, Robert W. (June 1962). "Algorithm 97: Shortest Path". Communications of the ACM. 5 (6): 345 https://doi.org/10.1145/367766.368168
- ↑ Warshall, Stephen (January 1962). "A theorem on Boolean matrices". Journal of the ACM. 9 (1): 11–12 https://doi.org/10.1145/321105.321107
- ↑ "A new algorithm for graph center computation and graph partitioning according to the distance to the center". October 2019. https://hal.archives-ouvertes.fr/hal-02304090.
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