Séminaire Lotharingien de Combinatoire, 93B.63 (2025), 12 pp.
Yakob Kahane and Marni Mishna
Differential Transcendence and Walks on Self-Similar 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 C(z). The proof strategy relies on a recent work of Di Vizio, Fernandes and Mishna. The result adds evidence to a conjecture of Krön and Teufl about the spectrum of this family of graphs. A long version of this abstract with complete proofs is available
[A HREF="https://arxiv.org/abs/2411.19316">arχiv:2411.19316].
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
The following versions are available: