EdgeQ (Built-in Mathematica Symbol)

EdgeQ[g, e] yields True if e is an edge in the graph g and False otherwise.

LineGraph (Built-in Mathematica Symbol)

LineGraph[g] gives the line graph of the graph g.

SimpleGraph (Built-in Mathematica Symbol)

SimpleGraph[g] gives the underlying simple graph from the graph g.

DepthFirstScan (Built-in Mathematica Symbol)

DepthFirstScan[g, s, {"event_1" -> f_1, "event_2" -> f_2, ...}] performs a depth-first scan of the graph g starting at the vertex s and evaluates f_i whenever "event_i" ...

VertexCount (Built-in Mathematica Symbol)

VertexCount[g] gives a count of the number of vertices in the graph g.VertexCount[g, patt] gives a count of the number of vertices that match the pattern patt.

New in 8.0: Alphabetical Listing (Mathematica Guide)

EdgeCount (Built-in Mathematica Symbol)

EdgeCount[g] gives a count of the number of edges in the graph g.EdgeCount[g, patt] gives a count of the number of edges that match the pattern patt.

Summary of New Features in Mathematica 8 (Mathematica Guide)

FindShortestPath (Built-in Mathematica Symbol)

FindShortestPath[g, s, t] finds the shortest path from source vertex s to target vertex t in the graph g.FindShortestPath[g, s, All] generates a ShortestPathFunction[...] ...

GraphDistance (Built-in Mathematica Symbol)

GraphDistance[g, s, t] gives the distance from source vertex s to target vertex t in the graph g.GraphDistance[g, s] gives the distance from s to all vertices of the graph g.