Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 4 / b / i / Solution

Both chains are reversible. Since $G$ is $d$-regular, simple random walk $X$ is reversible with uniform invariant distribution. In the weighted graph every vertex has total incident weight $d+\theta$, so $Y$ is also reversible with the same uniform invariant distribution.