”It seems there must be a geometric component in there”
No, there isn’t.
”if they were defined just in terms of abstract sets of vertices and edges between them, it would be rather trivial to show if two graphs are the same”
No, it isn’t, unless you know a novel argument that hasn’t been thought of yet by mathematicians.
It is fairly simple for graphs with very few vertices or very few edges, but in general, it isn’t easy at all.
No, there isn’t.
”if they were defined just in terms of abstract sets of vertices and edges between them, it would be rather trivial to show if two graphs are the same”
No, it isn’t, unless you know a novel argument that hasn’t been thought of yet by mathematicians.
It is fairly simple for graphs with very few vertices or very few edges, but in general, it isn’t easy at all.