The center (or

Jordan Jordan ( ar, الأردن; tr. ' ), officially the Hashemite Kingdom of Jordan,; tr. ' is a country in Western Asia. It is situated at the crossroads of Asia, Africa, and Europe, within the Levant region, on the East Bank of the Jordan Rive ...

center Wasserman, Stanley, and Faust, Katherine (1994), ''Social Network Analysis: Methods and Applications'', page 185. Cambridge: Cambridge University Press. ) of a

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

is the set of all vertices of minimum

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...

, 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 In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

. 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 The vertex ''k''-center problem is a classical NP-hardness, NP-hard problem in computer science. It has application in Facility location problem, facility location and Cluster analysis, clustering. Basically, the vertex ''k''-center problem model ...

. Finding the center of a graph is useful in

facility location problem The study of facility location problems (FLP), also known as location analysis, is a branch of operations research and computational geometry concerned with the optimal placement of facilities to minimize transportation costs while considering fact ...

s 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 In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with p ...

.Warshall, Stephen (January 1962). "A theorem on Boolean matrices". Journal of the ACM. 9 (1): 11–12 https://doi.org/10.1145/321105.321107 Another algorithm has been proposed based on matrix calculus.{{Cite web, url=https://hal.archives-ouvertes.fr/hal-02304090, title=A new algorithm for graph center computation and graph partitioning according to the distance to the center, date=October 2019 The concept of the center of a graph is related to the

closeness centrality In a connected graph, closeness centrality (or closeness) of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph. Thus, the m ...

measure in

social network analysis Social network analysis (SNA) is the process of investigating social structures through the use of networks and graph theory. It characterizes networked structures in terms of ''nodes'' (individual actors, people, or things within the network) ...

, which is the reciprocal of the mean of the distances ''d''(''A'',''B'').

References

Graph theory objects