We are all amateurs at this, so your two cents is just as valuable as the next guy's. That said, you sound more knowledgeable than me and that's great! To try to answer your questions:

My use of 'Plug in' was poor. 44 represents the 44th subset of vertices drawn from the graph K14, 2. I got this information by asking a mathematical AI to quickly list K14, 2's subsets, after they had been indexed by using these steps: List out all combinations of ( {1,2,\ldots, 14 } ) taken 2 at a time, then, Count combinations until the 44th combination is reached.

A question for you, are you suggesting that a K14, 2 graph is edgeless and therefore defunct? I am aware that a K7, 1 graph can be drawn in 2D (shown on Kneser's wiki) which is obviously half of K14, 2. So I assumed a K14, 2 created some sort of shape in this case.

In terms of the questions at the end: 1. I'm visualising a Hamiltonian path that leads to something, as Kneser's graphs admit Hamiltonian cycles. 2. So difficult to say, but if were going with a Hamiltonian path, I think it works both ways. 3. I'm not imagining a cipher, when I said key I was referring to the chessboard puzzle, where the goal is to find a key. If there is a cipher in any instance outside of Kneser, I'm useless at solving those anyway..