# Probability of random walk traversal

Consider a random walk on an connected, non-bipartite, undirected graph G. Show that, in the long run, the walk will traverse each edge with equal probability.

Note: The walk can traverse each edge in two directions.

If I understand the model correctly, at any vertex the particle chooses one of the outgoing edges with equal probability. If that is the case, then the transition probabilities of the discrete-time Markov chain are given by: $$P_{i,j} = \frac{1}{E_i},$$ where $E_i$ is number of edges connected to vertex $i$.
One can they verify that the stationary measure of this chain (assuming finite state space) is : $$\pi_i = \frac{E_i}{\sum_k E_k}.$$
Since, any outgoing edge will be selected uniformly at random, the fraction of the times the chain will traverse an edge $e_{i,j}$ is then $$\frac{1}{\sum_k E_k},$$ which is the same for any edge $e_{i,j}$.