Differential transcendence and walks on self-similar graphs

TL;DR

This paper investigates the nature of Green functions on self-similar fractal graphs, showing they are either algebraic or differentially transcendental depending on the graph's structure, with implications for spectral theory.

Contribution

It establishes a clear criterion for when Green functions are algebraic or differentially transcendental on self-similar graphs with branching number two.

Findings

Green functions are algebraic for star graphs with finitely many lines.

Green functions are differentially transcendental for other self-similar graphs.

Supports a conjecture about the spectral properties of these graphs.

Abstract

Symmetrically self-similar graphs are an important type of fractal graph. Their Green functions satisfy order one iterative functional equations. We show 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 function is algebraic, or the Green function is differentially transcendental over $\mathbb{C}(z)$. The proof strategy relies on a recent work of Di Vizio, Fernandes and Mishna. The result adds evidence to a conjecture of Kr\"on and Teufl about the spectrum of this family of graphs.