VertexIndex[g, v] gives the integer index for the vertex v in the graph g.

AddEdges[g, edgeList] gives graph g with the new edges in edgeList added. edgeList can have the form {a, b} to add a single edge {a, b} or the form {{a, b}, {c, d}, ...}, to ...

GeneralizedPetersenGraph[n, k] returns the generalized Petersen graph, for integers n > 1 and k > 0, which is the graph with vertices {u_1, u_2, ..., u_n} and {v_1, v_2, ..., ...

GraphRadius[g] gives the minimum eccentricity of the vertices in the graph g.

VertexEccentricity[g, s] gives the length of the longest shortest path from the source s to every other vertex in the graph g.

VertexRenderingFunction is an option for GraphPlot and related functions that gives a function to generate the graphics primitives to use in rendering each vertex.

Contract[g, {x, y}] gives the graph resulting from contracting the pair of vertices {x, y} of graph g.

MakeSimple[g] gives the undirected graph, free of multiple edges and self-loops derived from graph g.

RegularGraph[k, n] constructs a semirandom k-regular graph on n vertices, if such a graph exists.

RobertsonGraph returns a 19-vertex graph that is the unique (4, 5)-cage graph.