> 1.   I am interested in generating the Prufer code for the following simple
> graph which I generate as follows:

> Needs["Combinatorica`"]
> Needs["GraphUtilities`"]

> In[137]= gr1 = MakeSimple[
>   ToCombinatoricaGraph[{2 -> 4, 4 -> 3, 3 -> 1},
>    Method -> "CircularEmbedding"]];

> 2.  I can visualize the graph as follows, and all is well (it is as I
> expect, with no edge between vertices 1 and 2):

> In[140]= ShowGraph[gr1, EdgeColor -> Orange, VertexColor -> LightYellow,
>  VertexLabelPosition -> Center, VertexStyle -> Disk[Large]]

> 3.  However, if I now generate the Prufer code for gr1,  I get a strange
> result.

> In[141]= LabeledTreeToCode@gr1

> Out[141]= {2, 3}

> 4. Firstly, the above Prufer code is clearly wrong:  it surely should be
> {3,4}?

> Secondly, if I use EdgeList, I also get an apparently wrong answer:

> In[143]= EdgeList[gr1]

> {{1, 2}, {2, 1}, {2, 3}, {3, 2}, {3, 4}, {4, 3}}

> Why the edge between vertices 1 and 2?

> 5.  If I 'work backwards' I do not get back the original graph:

> In[144]= pcgr1 = CodeToLabeledTree[{2, 3}];

> ShowGraph[pcgr1, VertexLabel -> True, EdgeColor -> Orange,
>  VertexColor -> LightYellow, VertexLabelPosition -> Center,
>  VertexStyle -> Disk[Large]]

> In[145]= EdgeList[pcgr1]

> Out[145]= {{1, 2}, {2, 1}, {2, 3}, {3, 2}, {3, 4}, {4, 3}}

> The  EdgeList is correct for pcgr1, but pcgr1 is not identical with gr1!

> 6.  Finally, if I work with the expected Prufer code, I get a graph and
> EdgeList consistent
> with what I expected for gr1 but did not get:

> In[148]= gr3 = CodeToLabeledTree[{3, 4}];

> In[149]= ShowGraph[gr3, EdgeColor -> Orange, VertexLabel -> True,
>  VertexColor -> LightYellow, VertexLabelPosition -> Center,
>  VertexStyle -> Disk[Large]]

> In[150]= EdgeList[gr3]

> Out[150]=  {{1, 3}, {2, 4}, {3, 1}, {3, 4}, {4, 2}, {4, 3}}

> 7.  I am working with Mathematica 7.0.0 on a Mac Powerbook G4 (non-Intel)
> with OS X  v 10.5.8.

> Have I missed something or is this a bug?

> I am also wondering if others can reproduce
> this behaviour on other systems, in particular with Mathematica 7.0.1.

> Any help / workaround will be greatly appreciated.

> Thanks,

> Tom Dowling