The edges of a graph are pairs (or ordered pairs if we are dealing with digraphs) of vertices that are connected in a graph. Any edge must be a member of the collection of subsets of the vertex set of a graph.
i1 : D = digraph({{1,2},{2,1},{3,1}},EntryMode=>"edges");
i2 : edges D
o2 = {{1, 2}, {2, 1}, {3, 1}}
o2 : List
i3 : G = cycleGraph 4;
i4 : edges G
o4 = {set {0, 1}, set {0, 3}, set {1, 2}, set {2, 3}}
o4 : List
See also
vertexSet -- Returns the vertices of a graph or digraph