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?)