As a generalization of Example C.9, consider a random walk on an arbitrary undirected connected graph with a finite vertex set \(\mathscr{V}\). For any vertex \(v \in \mathscr{V}\), let \(d(v)\) be the number of neighbors of \(v\) - called the degree of \(v\). The random walk can jump to each one of the neighbors with probability \(1 / d(v)\) and can be described by a Markov chain. Show that, if the chain is aperiodic, the limiting probability that the chain is in state \(v\) is equal to \(d(v) / \sum v^{\prime} \in \mathscr{V} d\left(u^{\prime}\right)\).