Horofunctions of infinite Sierpinski polygon graphs

Generalizing works of D’Angeli and Donno, we describe, starting from an infinite sequence over $r$ letters with $r\mathrm{\ne }4i$ and $i\in \mathbb{N}$, a sequence of pointed finite graphs. We study the pointed Gromov-Hausdorff limit graphs giving a description of isomorphim classes in terms of dihedral groups and providing insights on the horofunction boundaries in terms of Busemann and non-Busemann points.