Finding a Criterion for Allowed Reactions in ABC Toy Theory

[Someone1987]

Homework Statement

In an ABC toy theory, the only allowed vertex is one that couples A, B and C. Thus there are no AAA, ABB, ACC...ect vertices allowed. My question is suppose a diagram has n_a external A lines, n_b external B lines, and n_c external C lines. Develop a simple criterion for determining whether it is an allowed reaction.

Homework Equations

The Attempt at a Solution

I'm just a little unsure of what the question wants. Does it want a mathematical formula or just a statement in plain english. I don't see a way of developing a mathematical formula that can encompass all the possible diagrams (decays and scattering) one can draw. The best criterion I can think of is that each identical external line must be attached to separate vertices. Another way of saying this is that no vertex can have two A's, two B,'s or two C's attached to it. Although this is correct, it doesn't seem like its the best criterion one can develop. I'd appreciate any additional insights anyone can provide. Thanks.

 

[Avodyne]

Either a formula or a statement in plain English is probably acceptable.

Here's a way to think about it: let i_a, i_b, and i_c be the number of internal lines of each type. Can you find a relation between n_a, i_a, and the number of vertices v?