Homotopy equivalence of isospectral graphs

TL;DR

This paper explores the homotopy theory of directed graphs, establishing equivalences between their homotopy categories and categories of periodic Z-sets, and linking spectral properties to homotopy equivalence.

Contribution

It provides a detailed description of the homotopy category of directed graphs and connects isospectrality with homotopy equivalence.

Findings

Homotopy category of graphs is equivalent to the category of periodic Z-sets.

Two finite directed graphs are almost-isospectral iff they are homotopy-equivalent.

Homotopy equivalence captures spectral similarity in directed graphs.

Abstract

In this paper, we investigate the Quillen model structure defined by Bisson and Tsemo in the category of directed graphs Gph. In particular, we give a precise description of the homotopy category of graphs associated 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.