Circles and line segments as independence attractors of graphs

TL;DR

The paper studies how the independence attractor $\\mathcal{A}(G)$, defined via the independence polynomial $I_G(z)$ and lexicographic powers $G^m$, behaves as a geometric limit in the plane and connects it to complex dynamics through the reduced polynomial $P_G(z)=I_G(z)-1$. By exploiting the relation $I_{G^m}(z)=P_G^m(z)+1$ and the dynamics of $P_G$, it shows that $\\mathcal{A}(G)$ cannot be a circle and, when it is a line segment, must be $[-\\frac{4}{k},0]$ with $k\\in\\{1,2,3,4\\}$; these segments are realized by conjugacies to Chebyshev polynomials. The paper also provides explicit graphs with independence number four whose independence attractors are these line segments, and analyzes the corresponding independence polynomials, including constructions with multiple components. Overall, the work links graph-theoretic polynomials with Julia-set dynamics to classify simple attractors and to construct graphs realizing them, enriching the interaction between combinatorics and complex dynamics.

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 $I_G(z)=a_0 + a_1 z + a_2 z^2+\cdots+a_αz^α$, where $a_i$ is the number of independent sets in $G$ with cardinality $i$ and $α$ 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.

Figures (3)

• Figure 4: Twenty five possibilities for the complement of $G_3$ where $I_{G_3}(z)= 1+8z +8z^2$.
• Figure 5: Complement of $G'$
• Figure 6: Complement of $G"$

Theorems & Definitions (46)

• Definition 1.1
• Theorem A
• Theorem B
• Theorem C
• Theorem 2.1: Theorem 3.2.5, beardon
• Theorem 2.2: Theorem 7.1.2, beardon
• Theorem 2.3: Theorem 4.2.8, beardon
• Theorem 2.4: Theorem 6.9.4, beardon
• Theorem 2.5: Theorem 4.2.7, beardon
• Theorem 2.6: Appendix A, brownetal2003