Differential-transcendence classification for symmetrically self-similar graphs

Let $X$ be a symmetrically self-similar graph with bounded geometry, and let its Green's function be the generating function for walks on $X$. A star is a graph consisting of finitely many one-sided lines with exactly one origin vertex in common. Differential-transcendence classification conjecture. The Green's function of $X$ is algebraic if and only if $X$ is a star. Otherwise, it is differentially transcendental.

This conjecture connects the analytic nature of Green's functions with the combinatorial structure of self-similar graphs. The paper develops results in this direction, including differential transcendence outside the star case for branching number $2$, but the full classification remains open.