In previous work we defined a Quillen model structure, determined by cycles, on the category Gph of directed graphs. In this paper we give a complete description of the homotopy category of graphs associated to our model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Z-sets, and to show that two finite directed graphs are almost-isospectral if and only if they are homotopy-equivalent in our sense.