Homology, Homotopy and Applications

Volume 11 (2009)

Number 1

A homotopical algebra of graphs related to zeta series

Pages: 171 – 184

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n1.a8


Terrence Bisson (Department of Mathematics and Statistics, Canisius College, Buffalo, New York, U.S.A.)

Aristide Tsemo (Department of Mathematics, College Boreal, Toronto, Ontario, Canada)


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.


category of directed graphs, topos, Quillen model structure, weak factorization system, cycles, zeta function

2010 Mathematics Subject Classification

05C20, 18G55, 55U35

Published 1 January 2009