Graphcentered

Graphcentered refers to approaches, theories, or software that organize and analyze graphs through their central elements. In graph theory, the center of a connected graph is the set of vertices with minimum eccentricity, and the graph's radius is that minimum eccentricity, while the diameter is the maximum distance between any two vertices. A graphcentered perspective emphasizes identifying centers, measuring centrality, and designing operations around these core nodes.

Centrality measures such as degree, closeness, betweenness, and eigenvector centrality are commonly used to assess how prominent a node is within a graph. Center-based methods include center finding, center-based clustering, and center-driven network design. The k-center problem, where k centers must cover all vertices with minimal radius, is a classic computational task associated with graphcentered reasoning and is typically NP-hard for general graphs.

In practice, graphcentered approaches are used in facility location, network routing, social network analysis, and information diffusion studies. They also inform resilience and vulnerability assessments by focusing on central nodes whose failure would most disrupt connectivity.

Examples: in a tree, the center is the middle vertex or the two middle vertices on a longest path; in a cycle, the center consists of two opposite vertices. In complex networks, central nodes often coincide with high-betweenness or high-eigenvector centrality.

Variations include graphcentered software libraries or research projects that implement centrality calculations, center-based clustering algorithms, or visualization tools that highlight graph centers. See also: center of a graph, graph radius, graph diameter, centrality, k-center problem, network analysis.