 Solution

A chain is reversible with respect to $\pi$ when ${\pi }_i{P}_{ij}={\pi }_j{P}_{ji}$ for all states. For random walk on a finite connected undirected graph, ${\pi }_i=\mathrm{deg}(i)/(2|E|)$ satisfies this because both sides equal $1/(2|E|)$ on an edge and zero otherwise.