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I stumbled (here, in the expression for the block diagram) upon a version of Graph that I don't see in the documentation over here. That documentation admits three overloads (the following are in my own words):

  1. Graph[{edges}]
  2. Graph[{vertices}, {edges}]
  3. Graph[{decoratedVertices}, {decoratedEdges}]

For instance, the following example invokes overload 2:

Graph[{1,2,3}, {{1, 2}, {1, 3}, {2, 3}}]

enter image description here

If I add a Null in an undocumented place, I get arrows:

Graph[{1,2,3}, { {{1, 2}, {1, 3}, {2, 3}}, Null } ]

Notice there is an extra level of List in the list of vertices. It's as though the list of vertices contains a nested list of vertices and a Null vertex.

This seems to produce similar results to

Graph[{1, 2, 3}, {1 \[DirectedEdge] 2, 1 \[DirectedEdge] 3, 2 \[DirectedEdge] 3}]

Anyone happen to know what's going on? I'm ok with blindly using the cool feature, but I wonder what the other options for this API might be.

enter image description here

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This seems to give you the format:

Graph[ {vertices}, { directededges, undirectededges} ]

For example, try changing up what you have to include non-null entries:

Graph[{1, 2, 3}, { {{1, 2},{2, 3},{1, 3}}, {{3, 2}} }]

Graph[{1, 2, 3}, { {{1, 2},{2, 3}}, {{3, 2},{1, 3}} }]

etc...

It still falls under category "2" from your list, more or less. I don't see this in the documentation, but that may be because it's just a bit of an odd feature.

So, as further example, you could take:

Graph[{1, 2, 3}, {{{2, 3}}, {{1, 2}, {3, 1}}}]

which is the same as:

Graph[{1 <-> 2, 2 -> 3, 3 <-> 1}]

However, one important difference I noticed is that it's actually hard to use the syntax you've discovered because it doesn't play nicely with other options:

Graph[{1 <-> 2, 2 -> 3, 3 <-> 1},, EdgeStyle -> {3 <-> 1 -> Red}]

will return a nice graph with one red edge and one directed edge. However:

Graph[{1, 2, 3}, {{{2, 3}}, {{1, 2}, {3, 1}}},EdgeStyle -> {{3,1} -> Red}]

will not do this. (I haven't found a variation on EdgeStyle type options that allow you to specify styles for this double-list of directed & undirected edges.) I suspect that this double-list format to specify directed vs. undirected edges may be a sort of legacy format that was built into the Graph command in previous versions, although I don't find documentation offhand showing this syntax in the old Combiantorica package (although there's a huge volume of documentation for Combinatorica and the transition to the newer built-in functions, not to mention Combinatorica may have included undocumented properties).

Here's another example to play with:

m = 5;
de = Table[{RandomInteger[{1, m}], RandomInteger[{1, m}]}, {i, 1, m^2/4}];
ue = Table[{RandomInteger[{1, m}], RandomInteger[{1, m}]}, {i, 1, m^2/4}];
Graph[Range[m], {de, ue}]

This just makes random lists of edges. You can vary m to make the graph bigger or smaller.

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