  We construct the representations of Cayley graphs of wreath products using
finite automata, pushdown automata and nested stack automata. These
representations are in accordance with the notion of Cayley automatic groups
introduced by Kharlampovich, Khoussainov and Miasnikov and its extensions
introduced by Elder and Taback. We obtain the upper and lower bounds for a
length of an element of a wreath product in terms of the representations
constructed.
