 Killed-walk occupation voltage

For simple random walk on a finite connected loopless unweighted graph, start at $a\ne z$ and stop on first hitting $z$. Ifthen $v_{az}(z)=0$ and $L{v}_{az}={\delta }_a-{\delta }_z$ for the graph Laplacian $L$. The incoming-visit balance equation is $g_{az}(u)=1_{\{u=a\}}+{\sum }_{w\sim u}{g}_{az}(w)/\mathrm{deg}(w)$ for $u\ne z$; at $z$ the Laplacian value is $-1$ because all its coordinates sum to zero. Thus $v_{az}$ is the voltage for unit current from $a$ to $z$, and $v_{az}(a)=R_{\mathrm{eff}}(a,z)$. In particular, the expected number of directed transitions $u\to w$ before $T_z$ is $v_{az}(u)$.