Circles and line segments as independence attractors of graphs

TL;DR

This paper investigates the geometric structure of independence attractors derived from graph independence polynomials, proving they can be line segments with specific bounds or circles are impossible, and provides examples with particular attractor shapes.

Contribution

It characterizes the topologically simple independence attractors of graphs, showing circles cannot occur and identifying line segments as attractors with specific bounds, supported by graph examples.

Findings

Independence attractors can never be circles.

If an attractor is a line segment, it must be of the form [-4/k, 0] for k in {1,2,3,4}.

Examples exist with independence number four whose attractors are line segments.

Abstract

By an independent set in a simple graph $G$, we mean a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$ is defined as $I_G(z)=a_0 + a_1 z + a_2 z^2+\cdots+a_\alpha z^{\alpha}$, where $a_i$ is the number of independent sets in $G$ with cardinality $i$ and $\alpha$ is the cardinality of a largest independent set in $G$, known as the independence number of $G$. Let $G^m$ denote the $m$-times lexicographic product of $G$ with itself. The independence attractor of $G$, denoted by $\mathcal{A}(G)$, is defined as $\mathcal{A}(G) = \lim_{m\rightarrow \infty} \{z: I_{G^m}(z)=0\}$, where the limit is taken with respect to the Hausdorff metric on the space of all compact subsets of the plane. This paper deals with independence attractors that are topologically simple. It is shown that $\mathcal{A}(G)$ can never be a circle. If $\mathcal{A}(G)$ is a line segment then it is proved that the line segment is $[-\frac{4}{k}, 0]$ for some $k \in \{1, 2, 3, 4 \}$. Examples of graphs with independence number four are provided whose independence attractors are line segments.