This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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Graph Measures & Metrics
Mathematica supports a broad range of measures that characterize graphs, from simple measures, such as the number of vertices and edges that tell the size and sparsity of a graph, to vertex degrees, which tell how locally well-connected each vertex is. Other measures include the geodesic distances in a graph or centrality measures that give a measure of how central in the overall graph each vertex is; for example, PageRank and HITS are measures used to order web page importance as returned from a search engine.
Basic Measures
VertexCount the number of vertices
EdgeCount the number of edges
Degree Measures
VertexDegree the number of edges for each vertex
VertexInDegree the number of in-edges for each vertex
VertexOutDegree the number of out-edges for each vertex
Distance Measures
GraphDistance the length of the shortest path between two vertices
Centrality Measures
ClosenessCentrality closeness centrality for each vertex
BetweennessCentrality betweenness centrality for each vertex