A homotopical algebra of graphs related to zeta series

TL;DR

This paper develops a homotopical algebra framework for graphs, introducing a Quillen model structure that captures acyclic graph morphisms and relates to zeta series and spectral properties of finite graphs.

Contribution

It introduces a novel homotopical algebraic structure for graphs, including a Quillen model structure with specific weak equivalences, cofibrations, and fibrations, connecting graph theory with algebraic topology.

Findings

Defined a Quillen model structure on graphs with acyclic morphisms as weak equivalences

Established factorization systems based on folding, injecting, and covering morphisms

The model structure aligns with applications involving acyclic directed graphs

Abstract

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.