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 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]
^ abWasserman, 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.
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...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect...
mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful...
graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject...
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes...
In graph theory, a factor of a graph G is a spanning subgraph, i.e., a subgraph that has the same vertex set as G. A k-factor of a graph is a spanning...
the middle of an object Center (algebra), used in various contexts Center (group theory) Center (ring theory) Graphcenter, the set of all vertices of...
discipline of graph theory, a wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle. A wheel graph with n vertices...
of a single bullet graph: Below is the same example, this time with labels to identify each part of the bullet graph. The dark center line represents the...
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional...
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each...
held on May 24, 2007, at the San Francisco Design Center in San Francisco. The notion of the social graph was introduced. The 2008 F8 event was held on July...
pentagrammic prisms. In the mathematical field of graph theory, a rhombicosidodecahedral graph is the graph of vertices and edges of the rhombicosidodecahedron...
subfield of graph theory, a centered tree is a tree with only one center, and a bicentered tree is a tree with two centers. Given a graph, the eccentricity...
Grapher is a computer program bundled with macOS since version 10.4 that is able to create 2D and 3D graphs from simple and complex equations. It includes...
A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It...
In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if...
A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that...
graphs are a way of modeling a system of tasks to be scheduled and timing constraints that must be respected by the schedule. They are mixed graphs,...
In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and...
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the...
In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally...
In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently...