  The purpose of this paper is to develop a homotopical algebra for graphs,
relevant to zeta series and spectra of finite graphs. More precisely, we define
a Quillen model structure in a category of graphs (directed and possibly
infinite, with loops and multiple arcs allowed). The weak equivalences for this
model structure are the Acyclics (graph morphisms which preserve cycles). The
cofibrations and fibrations for the model are determined from the class of
Whiskerings (graph morphisms produced by grafting trees). Our model structure
seems to fit well with the importance of acyclic directed graphs in many
applications.
  In addition to the weak factorization systems which form this model
structure, we also describe two Freyd-Kelly factorization systems based on
Folding, Injecting, and Covering graph morphisms.
