Hitting time of the bridge vertex between two cliques ($h_1=n^2-n+1,h_o=n^2$)

For simple random walk on two disjoint complete graphs of size $n\ge 2$ joined through one new vertex $w$, let $h_1$ be the expected hitting time of $w$ from either vertex attached to $w$, and $h_o$ the expected hitting time from any other clique vertex. First-step analysis gives $h_1=1+(n-1)h_o/n$ and $h_o=h_1+n-1$, so $h_1=n^2-n+1$ and $h_o=n^2$. The expected hitting time from $w$ is zero. In discrete time the random walk is an aperiodic Markov chain only for $n\ge 3$.