  The problem of inner vs outer conjugacy of subalgebras of certain graph
C*-algebras is investigated. For a large class of finite graphs E, we show that
whenever $\alpha$ is a vertex-fixing quasi-free automorphism of the
corresponding graph C*-algebra C*(E) such that \alpha(\D_E)\neq\D_E, where \D_E
is the canonical MASA in C*(E), then \alpha(\D_E)\neq w\D_E w^* for all
unitaries w\in C*(E). That is, the two MASAs \D_E and \alpha(\D_E) of C*(E) are
outer but not inner conjugate. Passing to an isomorphic C*-algebra by changing
the underlying graph makes this result applicable to certain non quasi-free
automorphisms as well. For the Cuntz algebras O_n, we find a criterion which
guarantees that a polynomial automorphism moves the canonical UHF subalgebra to
a non-inner conjugate UHF subalgebra. The criterion is phrased in terms of
rescaling of trace on diagonal projections.
