Thank you, Eric. Your post contained commands (TakeList, Rule, BlockMap) that were new for me. I used your suggestion to construct a SparseArray whose AdjacencyMatrix is the requested graph.

With[{d = 4}, 
 AdjacencyGraph[
  SparseArray[(Flatten[
         BlockMap[Inner[List, #[[1]], Partition[#[[2]], 2, 1], List] &,
           TakeList[Range[d (d + 1)/2], Range[d]], 2, 1], 2][[#]] -> 
       1) & /@ Range[d (d - 1)], d (d + 1)/2], 
  VertexLabels -> Automatic]]

One way:

Block[{n = 6}, (* n >= 1 *)
  Flatten@Table[
    {Thread[DirectedEdge[
       k (k - 1)/2 // Range[1 + #, # + k] &,
       k (k + 1)/2 // Range[1 + #, # + k] &]
      ],
     Thread[DirectedEdge[
       k (k - 1)/2 // Range[1 + #, # + k] &,
       k (k + 1)/2 // Range[2 + #, # + k + 1] &]
      ]}, {k, n - 1}]
  ] // Graph[#, VertexLabels -> Automatic] &