  We show that the Hilbert bimodule associated to a compact topological graph
can be recovered from the C*-algebraic triple consisting of the Toeplitz
algebra of the graph, its gauge action and the commutative subalgebra of
functions on the vertex space of the graph. We discuss connections with work of
Davidson-Katsoulis and of Davidson-Roydor on local conjugacy of topological
graphs and isomorphism of their tensor algebras. In particular, we give a
direct proof that a compact topological graph can be recovered up to local
conjugacy from its Hilbert bimodule, present an example of nonisomorphic
locally conjugate compact topological graphs with isomorphic Hilbert bimodules.
We also give an elementary proof that for compact topological graphs with
totally disconnected vertex space the notions of local conjugacy, Hilbert
bimodule isomorphism, isomorphism of C*-algebraic triples, and isomorphism all
coincide.
