The independence polynomial on recursive sequences of graphs

TL;DR

This paper investigates the zero sets of the independence polynomial on recursive graph sequences, showing boundedness under certain conditions and revealing universal dynamical behaviors.

Contribution

It introduces a framework linking recursive graph sequences, rational dynamical systems, and zero set properties of the independence polynomial, highlighting universal features.

Findings

Zeros of the independence polynomial are uniformly bounded for specific recursive sequences.

The dynamics of the associated rational systems exhibit universal qualitative features.

The approach connects graph polynomials with dynamical systems theory.

Abstract

We study the zero sets of the independence polynomial on recursive sequences of graphs. We prove that for a maximally independent starting graph and a stable and expanding recursion algorithm, the zeros of the independence polynomial are uniformly bounded. Each of the recursion algorithms leads to a rational dynamical system whose formula, degree and the dimension of the space it acts upon depend on the specific algorithm. Nevertheless, we demonstrate that the qualitative behavior of the dynamics exhibit universal features that can be exploited to draw conclusions about the zero sets.