I am reading a book and I have a problem understanding why a relation holds. Assume that we have a time-homogeneous random walk on a connected graph $G=(V,E)$.
For $o\in V$, the roundtrip from $o$ is the random walk from o that stops upon the first return to $o$. Let $j_o(e)$ be the expected number of times that $e\in E$ is traversed in a roundtrip from o. Let also $\pi(x)$ be the expected number of visits to $x\in V$ in a roundtrip from o. Then,
$$j_o(x\to y)=\pi(x) p(x\to y).$$
Why does this hold? Maybe I have to use the law of total expectation in a way that I don't see.