Line graph: one vertex per edge with adjacency for shared endpoints
LineGraph[g]
LineGraph[{v -> w, …}]
{g = CycleGraph[6], LineGraph[g]}
→ {Graph[<6>, <6>], Graph[<6>, <6>]}{g = CompleteGraph[4], LineGraph[g]}
→ {Graph[<4>, <6>], Graph[<6>, <12>]}{g = GridGraph[{3, 4}], LineGraph[g]}
→ {Graph[<12>, <17>], Graph[<17>, <34>]}