GraphIntersection[g_1, g_2, ...] constructs the graph defined by the edges that are in all the graphs g_1, g_2, ....
GraphProduct[g_1, g_2, ...] constructs the product of graphs g_1, g_2, and so forth.
GraphSum[g_1, g_2, ...] constructs the graph resulting from joining the edge lists of graphs g_1, g_2, and so forth.
GrayCodeKSubsets[l, k] generates k-subsets of l in Gray code order.
GrayGraph returns a 3-regular, 54-vertex graph that is edge-transitive but not vertex-transitive\[LongDash]the smallest known such example.
GroetzschGraph returns the smallest triangle-free graph with chromatic number 4. This is identical to MycielskiGraph[4].
HamiltonianPath[g] finds a Hamiltonian path in graph g if one exists. HamiltonianPath[g, All] gives all Hamiltonian paths of graph g.
HasseDiagram[g] constructs a Hasse diagram of the relation defined by directed acyclic graph g.
HeawoodGraph returns a smallest (6, 3)-cage, a 3-regular graph with girth 6.
HighlightedEdgeColors is an option to Highlight that determines which colors are used for the highlighted edges.