Stationary Distribution and Return Times on a Small Graph

Consider the graph $G$ on four vertices $\{A, B, C, D\}$ with edges $\{A{-}B,\, A{-}C,\, A{-}D,\, B{-}C\}$, so the degree sequence is $d(A)=3$, $d(B)=2$, $d(C)=2$, $d(D)=1$. A simple random walk moves at each step to a uniformly random neighbor.

(a) Find the stationary distribution $\pi$.

(b) Compute the expected return time $E_v[T_v^+]$ for each vertex $v$.