  The wreath product of graphs is a graph composition inspired by the notion of
wreath product of groups, with interesting connections with Geometric Group
Theory and Probability. This paper is devoted to the description of some degree
and distance-based invariants, of large interest in Chemical Graph Theory, for
a wreath product of graphs. An explicit formula is obtained for the Zagreb
indices, in terms of the Zagreb indices of the factor graphs. A detailed
analysis of distances in a wreath product is performed, allowing to describe
the antipodal graph and to provide a formula for the Wiener index. Finally, a
formula for the Szeged index is obtained. Several explicit examples are given.
