  Building on a construction of J-P. Serre, we associate to any graph of
C*-algebras a maximal and a reduced fundamental C*-algebra and use this theory
to construct the fundamental quantum group of a graph of discrete quantum
groups. This construction naturally gives rise to a quantum Bass-Serre tree
which can be used to study the K-theory of the fundamental quantum group. To
illustrate the properties of this construction, we prove that if all the vertex
qantum groups are amenable, then the fundamental quantum group is K-amenable.
This generalizes previous results of P. Julg, A. Valette, R. Vergnioux and the
first author.
