@J.M. - this is a great application of FindIndependentEdgeSet.
It's interesting to note that you can convert a BuckyballGraph to a molecule using the XYZ importer,
In[109]:= bucky = BuckyballGraph[];
xyzString = StringJoin[IntegerString[VertexCount[bucky]], "\n\n",
ExportString[Prepend[#, "C"] & /@ GraphEmbedding[bucky], "Table"]
];
ImportString[xyzString, "XYZ"] // OutputForm
Out[111]//OutputForm= Molecule[<C60>]
In[112]:= Counts[%["BondType"]]
Out[112]= <|"Aromatic" -> 90|>
