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.

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.

Index[p] gives the index of permutation p, the sum of all subscripts j such that p[[j]] is greater than p[[j + 1]].

KnightsTourGraph[m, n] returns a graph with mn vertices in which each vertex represents a square in an m*n chessboard and each edge corresponds to a legal move by a knight ...

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

LoopPosition is an option to ShowGraph whose values tell ShowGraph where to position a loop around a vertex. This option can take on values UpperLeft, UpperRight, LowerLeft, ...

MakeUndirected[g] gives the underlying undirected graph of the given directed graph g.

MaximumClique[g] finds a largest clique in graph g. MaximumClique[g, k] returns a k-clique, if such a thing exists in g; otherwise it returns {}.