GraphicQ[s] yields True if the list of integers s is a graphic sequence, and thus represents a degree sequence of some graph.

GraphOptions[g] returns the display options associated with g. GraphOptions[g, v] returns the display options associated with vertex v in g. GraphOptions[g, {u, v}] returns ...

GraphUnion[g_1, g_2, ...] constructs the union of graphs g_1, g_2, and so forth. GraphUnion[n, g] constructs n copies of graph g, for any nonnegative integer n.

GreedyVertexCover[g] returns a vertex cover of graph g constructed using the greedy algorithm. This is a natural heuristic for constructing a vertex cover, but it can produce ...

HamiltonianCycle[g] finds a Hamiltonian cycle in graph g if one exists. HamiltonianCycle[g, All] gives all Hamiltonian cycles of graph g.

HamiltonianQ[g] yields True if there exists a Hamiltonian cycle in graph g, or in other words, if there exists a cycle that visits each vertex exactly once.

Harary[k, n] constructs the minimal k-connected graph on n vertices.

HideCycles[c] canonically encodes the cycle structure c into a unique permutation.

IdenticalQ[g, h] yields True if graphs g and h have identical edge lists, even though the associated graphics information need not be the same.

InDegree[g, n] returns the in-degree of vertex n in directed graph g. InDegree[g] returns the sequence of in-degrees of the vertices in directed graph g.