Gromov Hyperbolicity of Substitution graphs

In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J: edges among vertices with the same parent follow G, while edges between vertices whose parents are horizontally linked follow J. The substitution graph is defined as its underlying graph. Substitution graphs provide a purely combinatorial model of self-similar structures, independent of any underlying geometric structure. Furthermore, we establish a necessary and sufficient condition for substitution graphs to be hyperbolic, formulated in terms of the vanishing of path matrices associated with sufficiently long shortest horizontal paths. Based on this characterization, we further derive several conditions that are either necessary or sufficient for hyperbolicity, depending only on the generators G and J.