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 ...

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.

LexicographicSubsets[l] gives all subsets of set l in lexicographic order. LexicographicSubsets[n] returns all subsets of {1, 2, ..., n} in lexicographic order.

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

Partitions[n] constructs all partitions of integer n in reverse lexicographic order. Partitions[n, k] constructs all partitions of the integer n with maximum part at most k, ...

UnrankPermutation[r, l] gives the r\[Null]^th permutation in the lexicographic list of permutations of list l. UnrankPermutation[r, n] gives the r\[Null]^th permutation in ...

VertexLabelPosition is an option that allows the user to place a vertex label in a certain position relative to the vertex.

VertexNumberColor is an option that can be used in ShowGraph to associate different colors to vertex numbers.

VertexNumber is an option that can take on values True or False. This can be used in ShowGraph to display or suppress vertex numbers.

All the functionality in Algebra`Quaternions` is available in the newly created Quaternions Package.