Paul, I forgot to mention, I've already done the hard work on this notebook, just run this one, the long version, it only takes a few minutes at most: https://www.notebookarchive.org/ppt-graph-nb--2021-06-6y23baq/

JP

James, Thank you for the link, however I can't seem to get it to run, there seems to be some information missing so I can't see the end result. I did notice that you were using Reduce to get the triples which in my view is very slow, so I have taken the liberty of amending my method of producing PPTs to produce what I think was the desired output of your notebook.

pyths[x_Integer, max_Integer] := (t = Select[Divisors[x^2], # <= x &];
   m = Sort[
    Select[(x^2 - t^2)/(2 t), 
     IntegerQ[#] && # > x && Sqrt[x^2 + #^2] <= max && 
       GCD[x, #] == 1 &]]; 
  Table[{x, m[[i]], Sqrt[x^2 + m[[i]]^2]}, {i, 1, Length[m]}])

t1 = Flatten[Table[pyths[i, 1720], {i, 3, 1720}], 1];

t1 // Map[Most] // 
 ListPlot[{#, Map[Reverse, #]}, AspectRatio -> 1, ImageSize -> 600, 
   PlotStyle -> {Red, Black}] &

On my pc if they were all computed together runs in 270ms. Maybe this helps you.

You may also find my method of generating PPTs useful. It generates all ppt's {a, b, c}, with the short leg as the input, where a<b<c. It is easily modified if you want all ppts with any leg equal to the input value by changing this part # > x to # > 0.

pyths[x_Integer] := (t = Select[Divisors[x^2], # <= x &]; 
  m = Sort[Select[(x^2 - t^2)/(2 t), IntegerQ[#] && # > x &]]; 
  Table[{x, m[[i]], Sqrt[x^2 + m[[i]]^2]}, {i, 1, Length[m]}])

or primitive ppt's

primpyths[x_Integer] := (t = Select[Divisors[x^2], # <= x &]; 
  m = Sort[Select[(x^2 - t^2)/(2 t), 
     IntegerQ[#] && # > x && GCD[x, #] == 1 &]]; 
  Table[{x, m[[i]], Sqrt[x^2 + m[[i]]^2]}, {i, 1, Length[m]}])

pyths[1000]

{{1000, 1050, 1450}, {1000, 1875, 2125}, {1000, 2400, 2600}, {1000, 
  3045, 3205}, {1000, 4950, 5050}, {1000, 6210, 6290}, {1000, 9975, 
  10025}, {1000, 12480, 12520}, {1000, 15609, 15641}, {1000, 24990, 
  25010}, {1000, 31242, 31258}, {1000, 49995, 50005}, {1000, 62496, 
  62504}, {1000, 124998, 125002}, {1000, 249999, 250001}}

primpyths[1000]

{{1000, 15609, 15641}, {1000, 249999, 250001}}

Thanks, I'm not familiar with genPTunder. I'll look into it. Your graph looks like you have all Pythagorean triples, not just the primitive ones.