Graph labelings obtainable by random walks

We initiate the study of what we refer to as random walk labelings of graphs. These are graph labelings that are obtainable by performing a random walk on the graph, such that the labeling occurs increasingly whenever an unlabeled vertex is encountered. Some of the results we obtain involve sums of inverses of binomial coefficients, for which we obtain new identities. In particular, we prove that $\sum_{k=0}^{n-1}2^{k}(2k+1)^{-1}\binom{2k}{k}^{-1}\binom{n+k}{k}=\binom{2n}{n}2^{-n}\sum_{k=0}^{n-1}2^{k}(2k+1)^{-1}\binom{2k}{k}^{-1}$, thus confirming a conjecture of Bala.