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.
VertexCount — the number of vertices
EdgeCount — the number of edges
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
GraphDistance — the length of the shortest path between two vertices
ClosenessCentrality — closeness centrality for each vertex
BetweennessCentrality — betweenness centrality for each vertex
