  We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a
noncommutative geometry setting framed in terms of Kasparov modules. We use a
mapping cone construction to relate odd index pairings to even index pairings
with APS boundary conditions in the setting of KK-theory, generalising the
commutative theory. We find that Cuntz-Kreiger systems provide a natural class
of examples for our construction and the index pairings coming from APS
boundary conditions yield complete K-theoretic information about certain graph
C*-algebras.
