In all the Markov diagrams I see, the transitions from state A to B always total to one.
Just one of many examples, this image from this website.
I've also seen examples that don't have values, but have variables. For example this diagram:
What I'm wondering is: if $P_{AB}$ is the only path from $A$ to $B$, does $P_{AB}$ have to be $1$?
The transition probabilities from one node to the other nodes have to sum up to $1$. This is true even if there is only one path out of a certain state.
In what follows let $A^n$ the $n^{th}$ state and let $A_{i}^{n+1}$ denote the $(n+1)^{st}$ state of the Markov chain. Also, suppose that $P(A^n)>0$. $(i=1,2, ... m)$ Then $$\sum_{1} ^{m}P(A_{i}^{n+1}|A^n)=\sum_{1} ^{m}\frac{P(A_{i}^{n+1}\cap A^n)}{P(A^n)}=\frac{1}{P(A^n)}\sum_{1} ^{m}P(A_{i}^{n+1}\cap A^n)=\frac{P(A^n)}{P(A^n)}=1.$$