Differential transcendence and walks on self-similar graphs

Abstract Symmetrically self-similar graphs are an important type of fractal graph. Their Green’s functions satisfy order one iterative functional equations. We show that when the branching number of a generating cell is two, either the graph is a star consisting of finitely many one-sided lines meeting at an origin vertex, in which case the Green’s function is algebraic, or the Green’s function is differentially transcendental over $\mathbb {C}(z)$ . The proof strategy relies on a result in a recent preprint of Di Vizio, Fernandes, and Mishna. The result adds evidence to a conjecture of Krön and Teufl about the spectra of the difference Laplacian of this family of graphs.