A homotopical algebra of graphs related to zeta series

Abstract

The purpose of this paper is to develop a homotopical algebra for graphs, relevant to the zeta series and the 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.