I will use the notation from Mac Lane's Categories for the Working Mathematician. To avoid confusion and possible mistakes, I will add screenshots of the material in question, rather than trying to phrase preliminaries in my own words.
I don't understand why every edge of $G_{n,b}$ contains only one instance of $\alpha$ or $\alpha^{-1}$. I reckon it should be deduced from associator conditions of the monoidal category $B$ stating that, for any $a,b,c,d \in B$, we have $$(1_a\otimes\alpha_{b,c,d})\circ\alpha_{a,b\otimes c, d}\circ (\alpha_{a,b,c}\otimes 1_d) = \alpha_{a,b,c\otimes d}\circ \alpha_{a\otimes b, c,d},$$ but I can't quite phrase a possible proof of this fact.
Note that $G_{n,b}$ is a graph, not a category. Its edges are defined to be the basic arrows, which are defined to admit exactly one occurrence of $\alpha$ or $\alpha^{-1}$. So there is nothing to prove. You seem to be jumping ahead to proving that $G_{n,b}$ indexes a commutative diagram; keep reading.