I've been exploring a possibly novel graph/hyper-graph like structure where edges can connect other edges together and nodes are not needed. For the purposes of this question I'll refer to this sort of graph as "edge-connecting".
For example the following describes a labelled, "vertexless", "edge-connecting" graph.
{1, 2}
{1, 3}
{2, 3}
Each row of this table is a set and represents an edge. The first row states that "edge 1" connects edges 1 and 2. In the 2nd row "edge 2" connects edges 1 and 3, and so on.
I found a way to visually draw this graph (embedding in R^2), draw a curved or straight line segment for each edge and label it. Each segment terminates at the two edges it connects. I found it visually clear to have segments end against others at 90 degrees. As with non "edge-connecting" graphs some redrawing may be required for a satisfactory embedding.
To get all 6 such graphs with 3 edges, permute the rows in the table above. There are fewer than 6 if unlabelled rather than labelled. Being that the graph is not a multi-graph the following situations cannot arise:
1 1
This edges "ends" are the same,
connecting "edge 1" to itself.
.
2 2
1 2
"Edge 1" has a similar problem.
.
1 2
1 2
The edges are identical.
The set {1, 2} = {1, 2}.
Multiple separate empty edges seem to behave like nodes:
{}
{}
{1 2}
{2 4}
Four edges, two of them node like and connected by "edge 3".
Allowing edges that individually connect more than 2 edges makes the construction more like a hyper-graph. In this case a "simple vertexless edge-connecting hyper-graph".
1, 2, 3
1, 3, 4, 5
1, 2
2, 3
3, 4
Properties
It's simpler to just have edges and not nodes.
At least when labelled, there are enormously more of these constructions than ordinary graphs.
cite: Wolfram Alpha
Adding edges expands and changes the structure of the graph more than in a conventional graph.
Instead of separate Vertex and Edge transitivity, such graphs may simply have "transitivity".
There are no vertexless edge-connecting graphs with 1 or 2 edges.
Even more so there are no vertexless edge-connecting hypergraphs with 1 or 2 edges, as long as each edge connects the same number of edges.
They should work with existing concepts such as "simple", "multi", "fully connected", labelled/unlabelled, finite/infinite, addition, types of multiplication, transitivity, and graph colouring.
Question
Are these already named? Are there contradictions or other problems with the construction that I've missed?
Combinatorics
Counting labelled graphs:
2 ^ ((N * (N - 1)) / 2)
.
Counting labelled edge-connecting graphs with V vertex like edges + N larger edges.
((Triangle(N - 1) + V) choose N) * (N!)
hyper:
((Subsets(N) + V - 1) choose N) * (N!)
May I note that relations between edges is not a new concept. This is called a line graph in the literature. you may look at the definition at https://en.wikipedia.org/wiki/Line_graph
In addition, I noticed you posted this also in community.wolfram.com Mathematica has a function LineGraph that generates the corresponding line graph from a given graph
Saying "Edges only" is ignoring the fact that in the first place, an edge is define by its end points (nodes), and two edges are adjacent if they have an end point (node) in common. So, you cannot run away from that...
There are graph that have nodes and no edges. The opposite is not possible...