We construct a locally compact Hausdorff topology on the path space of a directed graph E and identify its boundary-path space ∂E as the spectrum of a commutative C*-subalgebra DE of C*(E). We then show that ∂E is homeomorphic to a subset of the infinite-path space of any desingularisation F of E. Drinen and Tomforde showed that we can realise C*(E) as a full corner of C*(F), and we deduce that DE is isomorphic to a corner of DF. Lastly, we show that this isomorphism implements the homeomorphism between the boundary-path spaces.
Webster, S. B. G. (2014). The path space of a directed graph. Proceedings of the American Mathematical Society, 142 (1), 213-225.