I've been playing around writing some software to generate Feynman diagrams for QED, respecting the vertex "rules" described here, and avoiding creating isomorphic duplicates.
So from a starter fermion arc
the only possibility - at least the way I create these things, by adding pairs of vertices by splitting fermion arcs and linking the fresh vertices with boson arcs - is:
(boson arcs are in red) and from that there are 3 possibilities
and then 15
and then 105, and then 945 (for 10 interior vertices) and so on (I just killed it after a few hours runtime and it was working it's way through the many thousands of 12 interior vertex possibilities).
My question is: given all these diagrams, is there some simple procedure I can apply and have some fundamental dimensionless constant of the universe (or at least of QED) pop out the other end ?
Background: my interest in this stuff got started with the iconography of these diagrams and their visual aspects (e.g how to layout such things nicely, and how to plot them with some wiggly lines as per the usual convention for the photons will get some attention next), and the original driving idea was to do some nice big poster which gives some visual impression of how the number of possibilities explodes. But it'd be the icing on the cake for the poster to also be able to show that some physical property of the universe follows as a direct consequence of some statistics or whatever of the diagrams (capped with the "Nobody knows how it can be like that" Feynman quote no doubt).
Disclaimer: I know mathematics and computing but most of my knowledge of this area of physics is from reading "pop science" books and I'm well aware they grossly simplify things, so maybe I'm hoping for too much from the diagrams alone.
Update: following helpful input from Vibert below, I'm now generating the below for 2 and 4 vertex diagrams (a grey background indicates "one particle reducible" diagrams); these now include the expected virtual particle loops (missed in my first attempt above). Diagrams which are isomorphic by simple reversal of all the arrows are omitted in the interests of saving space.
Outcome: Well after digging into this a bit more it seems there's no getting away from Doing The Math, integrals and all. I was originally hoping there might be something as simple and understandable as "well there are N diagrams in this category, M diagrams in that category and so the correction to the 1st order approximation is (N-M)/(N+M), which is more accurate". But material on Lamb Shift corrections gives an idea of what's actually involved (e.g the contribution from the simplest diagrams is ~1058 MHz, most of it from one case), while the next order's contributions sum to on the order of 100kHz.