10
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I suspect this is just a bug in Mathematica. Here is the same graph embedded with different options for GraphLayout:

edges =
{
UndirectedEdge[1, 2],
UndirectedEdge[2, 3],
UndirectedEdge[3, 5],
UndirectedEdge[4, 7],
UndirectedEdge[4, 8],
UndirectedEdge[5, 6],
UndirectedEdge[5, 7]
};
{Graph[edges, GraphLayout -> "GridEmbedding", ImageSize -> Medium], 
 Graph[edges, GraphLayout -> "LayeredEmbedding", ImageSize -> Medium]}

The output is shown below.

Output

The two graphs appear to have different connectivity. What is up with this?

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  • $\begingroup$ add the options EdgeShapeFunction -> "CurvedArc" and VertexLabels -> "Name" to both Graphs to avoid overlapping edges. $\endgroup$ – kglr Jan 17 at 1:03
  • 1
    $\begingroup$ Ahh, the EdgeShapeFunction is what I need I suppose. In my actual case rather than MWE, I did have labelled vertices, but it still didn't occur to me that Mathematica was drawing the edges to exactly overlap. Thank you! $\endgroup$ – Andrew Patrick Turner Jan 17 at 1:10
  • 3
    $\begingroup$ Don't use GridEmbedding for anything else but a GridGraph. It's a dumb method that puts vertices on a grid in the order they appear in the vertex list. It will almost always result in overlapping edges unless your graph has a very specific structure and vertex ordering (i.e. it's what GridGraph generates, possibly with some missing edges) $\endgroup$ – Szabolcs Jan 17 at 8:05
  • $\begingroup$ The "CurvedArc" option is pretty neat; however, where is it documented? I went through the v12 documentation and it doesn't appear at all as near as I can tell. $\endgroup$ – Mark Kotanchek Jan 30 at 17:38
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Edges 4 <-> 7 and 4 <-> 8 overlap in the first graph. You can make the edges curved using the option EdgeShapeFunction -> "CurvedArc" to avoid overlapping edges: 

Row[{Graph[edges, 
    GraphLayout -> "GridEmbedding", ImageSize -> Medium, VertexLabels -> "Name"], 
  Graph[edges, 
    GraphLayout -> "GridEmbedding", ImageSize -> Medium, VertexLabels -> "Name", 
    EdgeShapeFunction -> "CurvedArc"], 
  Graph[edges, 
    GraphLayout -> "LayeredEmbedding", ImageSize -> Medium, VertexLabels -> "Name"], 
  Graph[edges, 
    GraphLayout -> "LayeredEmbedding", ImageSize -> Medium, VertexLabels -> "Name", 
    EdgeShapeFunction -> "CurvedArc"]}, Spacer[10]]

enter image description here

Alternatively, used "CurvedArc" only for the edges 4 <-> 7 and 4 <-> 8:

Graph[edges,
 GraphLayout -> "GridEmbedding", ImageSize -> Medium, VertexLabels -> "Name", 
 EdgeShapeFunction -> {UndirectedEdge[4, _] -> "CurvedArc"}]

enter image description here

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