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In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph G oH in terms of the total distances in H and of some distance-based indices of G. We explicitly compute these indices for the star graph Sn, providing a closed formula for the total distances in Sn o H when H is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of Sn o Sm in terms of tail conditional expectations of an associated binomial distribution. Mathematics Subject Classifications: 05C12, 05C57, 05C76

Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs