Hitting Time on the Petersen Graph

The Petersen graph has $10$ vertices and $15$ edges; it is $3$-regular, vertex-transitive, and has diameter $2$ (every pair of non-adjacent vertices has exactly one common neighbor, and the graph has girth $5$). A random walk at each step moves to one of the $3$ neighbors uniformly at random. Starting from a vertex $u$, what is the expected number of steps to reach a specified non-adjacent vertex $v$ for the first time?