  Unique expansions in non-integer bases $q$ have been investigated in many
papers during the last thirty years. They are often conveniently generated by
labeled directed graphs. In the first part of this paper we give a precise
description of the set of sequences generated by these graphs.
  Using the description of univoque graphs, the second part of the paper is
devoted to the study of multiple expansions. Contrary to the unique expansions,
we prove for each $j\ge 2$ that the set $U_q^j$ of numbers having exactly $j$
expansions is closed only if it is empty.
  Furthermore, generalizing an important example of Sidorov, we prove for a
large class of bases that the Hausdorff dimension of $U_q^j$ is independent of
$j$.
  In the last two sections our results are illustrated by many examples.
