Zeros of the independence polynomial on recursive sequences of graphs
Abstract
We study the hard-core model on recursively defined sequences (Gn)n≥0 of graphs with a fixed number k≥ 1 of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number m≥ 2 of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing k labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs Gn are uniformly bounded and the distances between the labeled vertices in Gn diverge, the complex zeros of the univariate independence polynomials ZGn(λ) avoid a neighborhood of the non-negative real axis. By the Lee--Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph G0. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps Fλ on the (2k−1)-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.
Related
- Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences
- Geometric parameter sequences for Q-polynomial distance-regular graphs
- ON THE ZEROS OF THE RAMIFICATION POLYNOMIAL
- Linear Independence of Polynomial Compositions with Bounded Input Degree
- Convergence of a Family of Accelerated Schröder-like Methods for the Simultaneous Extraction of Simple and Multiple Polynomial Zeros
BibTeX
@misc{hlushchanka2026zeros,
title = {{Zeros of the independence polynomial on recursive sequences of graphs}},
author = {Mikhail Hlushchanka and Han Peters},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35102},
eprint = {2609.35102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35102},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zeros of the independence polynomial on recursive sequences of graphs
AU - Mikhail Hlushchanka
AU - Han Peters
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35102
UR - https://arxiv.org/abs/2609.35102
AB - We study the hard-core model on recursively defined sequences (G_n)_n≥0 of graphs with a fixed number k≥ 1 of labeled vertices in each graph. The next graph in the sequence is constructed by taking a fixed number m≥ 2 of copies of the previous graph, connecting these copies by identifying some labeled vertices according to a fixed rule, and afterward choosing k labeled vertices in the resulting graph, again in accordance with a fixed rule. Examples of such sequences include the Sierpiński gasket graphs, hierarchical lattices, and many more. We prove that, when the vertex degrees of the graphs G_n are uniformly bounded and the distances between the labeled vertices in G_n diverge, the complex zeros of the univariate independence polynomials Z_G_n(λ) avoid a neighborhood of the non-negative real axis. By the Lee--Yang theory this implies that no phase transitions occur for the hard-core model on these recursive sequences of graphs, independently of the starting graph G_0. The proof relies on the study of the dynamical properties of a one-parameter family of rational maps F_λ on the (2^k−1)-dimensional complex projective space induced by the graph recursion operator. The dynamical framework developed in this paper can be naturally extended to other classical models in statistical mechanics (such as the Ising or Potts models) and to more general notions of graph recursions.
ER -