A certain family of graphs crossed my way while performing some quantum mechanics calculations, and I am very curious whether they have been studied in mathematics before in a different context.

Also I would be interested in properties of these graphs, other that they are planar multi-graphs.

Here are the examples. Let's start with the simplest example. It is a complete $K_4$ graph with one additional edge, making one edge a double-edge (let's call it *Wing$_1$ graph*, for reasons that become clear soon):

Now here the second member of the family. These are two $K_4$ graphs which are connected at two vertices (let's call it *Wing$_2$ graph*):

And now we add more and more $K_4$ graphs to the graph, but without increasing the number of edges between the central two vertices. Here is an example of the *Wing$_5$* graph (from three different perspectives).

I am very curious whether this is family has been studied before, whether it belongs to a larger class of graphs with certain properties etc. (Also as a sort of meta-question: Is there a way how to find names of graphs in a simple way? Such as a OEIS for graphs?)

notisomorphic to), and for each and every result known about 'book graphs' investigate if and how the result changes on account of the one and onlymultiedgeat the 'spine' of the book; incidentally, such a multiedge in graph theory is called a mathworld.wolfram.com/DipoleGraph.html, while in theoretical physics I have repeatedly read it referred to as 'banana graph'. $\endgroup$ – Peter Heinig Mar 1 '18 at 18:50