Applications of closed models defined by counting to graph theory and topology

TL;DR

This paper introduces a new framework of closed models based on counting, explores their homotopy categories in graph theory and topology, and demonstrates their limitations and applications to Galoisian complexes and dessins d'enfant.

Contribution

It defines closed models by counting, computes their homotopy categories, and applies these to graphs and topological structures, revealing limitations in characterizing certain graph invariants.

Findings

No closed model in undirected graphs characterizes the Ihara Zeta function.

The construction applies to Galoisian complexes and dessins d'enfant.

Homotopy categories are computed for these models.

Abstract

In this paper, we define the notion of closed models defined by counting, and we compute their homotopy categories. We apply this construction to various categories of graphs. We show that there does not exist a closed model in the category of undirected graphs which characterizes the Ihara Zeta function in the sense that, a morphism $f:X\rightarrow Y$ is a weak equivalence for this model if and only if it induces a bijection between the sets of non degenerated cycles of $X$ and $Y$. Finally, we apply our construction to Galoisian complexes and dessins d'enfant.