CycleGraph[n] gives the cycle graph with n vertices C_n.
ReverseGraph[g] gives the reverse graph of the directed g.
BipartiteGraphQ[g] yields True if the graph g is a bipartite graph and False otherwise.
GraphML
(Mathematica Import/Export Format) GraphML graph data format. Standard exchange for graphs. GraphML is an acronym derived from Graph Markup Language. XML-based format. Represents typed, attributed, directed, ...
GraphUnion[g_1, g_2] gives the graph union of the graphs g_1 and g_2.GraphUnion[g_1, g_2, ...] gives the graph union of g_1, g_2, ....
VertexList[g] gives the list of vertices for the graph g.VertexList[g, patt] gives a list of vertices that match the pattern patt.
EdgeList[g] gives the list of edges for the graph g. EdgeList[g, patt] gives a list of edges that match the pattern patt.
IsomorphicGraphQ[g_1, g_2] yields True if the graphs g_1 and g_2 are isomorphic, and False otherwise.
CayleyGraph[group] returns a Cayley graph representation of group.
UndirectedGraph[g] gives an undirected graph from the directed graph g.