EmptyGraph[n] generates an empty graph on n vertices. An option Type that can take on values Directed or Undirected is provided. The default setting is Type -> Undirected.

EquivalenceRelationQ[r] yields True if the matrix r defines an equivalence relation. EquivalenceRelationQ[g] tests whether the adjacency matrix of graph g defines an ...

EulerianQ[g] yields True if graph g is Eulerian, meaning there exists a tour that includes each edge exactly once.

FindCycle[g] finds a list of vertices that define a cycle in graph g.

FromCycles[{c_1, c_2, ...}] gives the permutation that has the given cycle structure.

GetEdgeLabels[g] returns the list of labels of the edges of g. GetEdgeLabels[g, e] returns the list of labels in graph g of the edges in e.

GetEdgeWeights[g] returns the list of weights of the edges of g. GetEdgeWeights[g, e] returns the list of weights in graph g of the edges in e.

GetVertexLabels[g] returns the list of labels of vertices of g. GetVertexLabels[g, v] returns the list of labels in graph g of the vertices specified in list v.

GetVertexWeights[g] returns the list of weights of vertices of g. GetVertexWeights[g, v] returns the list of weights in graph g of the vertices in v.

Girth[g] gives the length of a shortest cycle in a simple graph g.