The following won't work for multigraphs, but it is faster for the other graphs:
elist[g_Graph] := With[
{sa = UpperTriangularize @ WeightedAdjacencyMatrix[g, EdgeWeight->Range@EdgeCount@g]},
sa["NonzeroPositions"][[InversePermutation @ sa["NonzeroValues"]]]
]
Check:
Grid @ Table[
With[
{
r1=AbsoluteTiming[Flatten[elist[Symbol@g]]-1],
r2=AbsoluteTiming[igEdgeList[Symbol@g]]
},
{g,r1[[1]], r2[[1]], r1[[2]]===r2[[2]]}
],
{g, {"g1","g2","g3","g4","wg1","wg2","h1","h2","h3","h4","wh1","wh2"}}
]
g1 0.052843 0.276878 True
g2 0.051412 0.283879 True
g3 0.048165 0.222559 True
g4 0.050206 0.239958 True
wg1 0.049296 0.228189 True
wg2 0.048674 0.263723 True
h1 0.081907 0.11832 True
h2 0.084709 0.241035 False
h3 0.077764 0.124032 True
h4 0.082046 0.21298 True
wh1 0.080524 0.118146 True
wh2 0.077027 0.232665 False
The two discrepancies occur because some of the edges are reversed. For example:
Sort/@elist[h2] - 1 === Sort /@ Partition[igEdgeList[h2],2]
Sort/@elist[wh2] - 1 === Sort /@ Partition[igEdgeList[wh2],2]
(* True *)
(* True *)
