Graph center
The center (or Jordan center) of a graph is the set of all vertices of minimum eccentricity, 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. 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. Another algorithm has been proposed based on matrix calculus. 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).
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
[edit]- ^ 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
- ^ McHugh, James A., Algorithmic Graph Theory Archived 2010-08-01 at the Wayback Machine
- ^ Weisstein, Eric W. "Graph center". MathWorld.
- ^ 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.