Symmetric Random Walks on Wheel-and-Spokes Graphs

We study symmetric random walks on the vertices of a wheel-and-spokes graph. We consider the following questions. How long does it take for the walk to go from one vertex to another? Starting from one vertex, how long does it take to visit all vertices? Having visited all vertices, how much additional time does it take to return to the starting vertex? The answers to these questions are random variables for which we desire the exact probability distributions, if possible; otherwise, we seek at least their means and standard deviations. We compare our results to those of symmetric random walks on the vertices of polygons.