  We initiate a systematic investigation of endomorphisms of graph C*-algebras
C*(E), extending several known results on endomorphisms of the Cuntz algebras
O_n. Most but not all of this study is focused on endomorphisms which permute
the vertex projections and globally preserve the diagonal MASA D_E of C*(E).
Our results pertain both automorphisms and proper endomorphisms. Firstly, the
Weyl group and the restricted Weyl group of a graph C*-algebra are introduced
and investigated. In particular, criteria of outerness for automorphisms in the
restricted Weyl group are found. We also show that the restriction to the
diagonal MASA of an automorphism which globally preserves both the diagonal and
the core AF-subalgebra eventually commutes with the corresponding one-sided
shift. Secondly, we exhibit several properties of proper endomorphisms,
investigate invertibility of localized endomorphisms both on C*(E) and in
restriction to D_E, and develop a combinatorial approach to analysis of
permutative endomorphisms.
