The suspension of a graph, and associated C^⁎-algebras

Given a directed graph E, we construct for each real number l a quiver whose vertex space is the topological realisation of E, and whose edges are directed paths of length l in the vertex space. These quivers are not topological graphs in the sense of Katsura, nor topological quivers in the sense of Muhly and Tomforde. We prove that when l=1 and E is finite, the infinite-path space of the associated quiver is homeomorphic to the suspension of the one-sided shift of E. We call this quiver the suspension of E. We associate both a Toeplitz algebra and a Cuntz-Krieger algebra to each of the quivers we have constructed, and show that when l=1 the Cuntz-Krieger algebra admits a natural faithful representation on the ℓ 2 -space of the suspension of the one-sided shift of E. For graphs E in which sufficiently many vertices both emit and receive at least two edges, and for rational values of l, we show that the Toeplitz algebra and the Cuntz-Krieger algebra of the associated quiver are homotopy equivalent to the Toeplitz algebra and Cuntz-Krieger algebra respectively of a graph that can be regarded as encoding the lth higher shift associated to the one-sided shift space of E.