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"... to this 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 sh ..."

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to this 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. ยง0. Introduction. Mathematicians often study complicated categories by means of invariants (which are equal for isomorphic objects in the category). Sometimes a complicated category can be replaced by a (perhaps simpler) homotopy category which is better related to the various invariants used to study it. In topology, this was first achieved by declaring two continuous functions to be equivalent when one could be deformed into the other. But it