Homotopy equivalence of isospectral graphs

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.

💡 Research Summary

This paper investigates the Quillen model structure on the category Gph of directed graphs, originally introduced by Bisson and Tsemo, and provides a complete description of the associated homotopy category Ho(Gph). The authors first recall the definition of the model structure: weak equivalences are graph morphisms that induce isomorphisms on the underlying path‑space (equivalently, they preserve the non‑zero part of the adjacency‑matrix spectrum); fibrations are morphisms satisfying a right‑lifting property with respect to a set of generating trivial cofibrations, and cofibrations are defined dually. By exhibiting explicit generating (trivial) cofibrations, they prove that the model structure is proper, cofibrantly generated, and both left and right proper, which guarantees the existence of a well‑behaved homotopy category.

Next, the authors construct parallel model structures on the categories of N‑sets (sets equipped with a natural‑number action) and Z‑sets (sets equipped with an integer action). They define adjoint functors that “linearise’’ a directed graph into an N‑set (by forgetting all but a single infinite forward chain) and “circulise’’ a graph into a Z‑set (by collapsing each strongly connected component into a bi‑infinite orbit). These functors preserve cofibrations, fibrations, and weak equivalences, establishing Quillen equivalences \