EdgeLabelPosition is an option that allows the user to place an edge label in a certain position relative to the midpoint of the edge. LowerLeft is the default value of this ...

EdgeWeight is an option that allows the user to associate weights with edges. 1 is the default weight. EdgeWeight can be set as part of the graph data structure.

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.

FranklinGraph returns a 12-vertex graph that represents a 6-chromatic map on the Klein bottle. It is the sole counterexample to Heawood's map-coloring conjecture.

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.