Independence Attractors in Graph Dynamics

• Independence attractors are compact subsets of the complex plane defined by the limit of roots of iterated independence polynomials, linking graph theory and complex dynamics.
• They emerge from the iteration of a reduced independence polynomial, where preimages of -1 under successive lexicographic graph products reveal intricate fractal structures.
• The framework delineates various topologies—ranging from real line segments to fractal sets—with connectivity properties governed by the dynamics of critical orbits.

An independence attractor is a compact subset of the complex plane characterizing the asymptotic root set of iterated independence polynomials associated to the lexicographic powers of a finite simple graph. The concept builds a bridge between algebraic graph invariants and Julia-set theory from complex dynamics, providing a new class of graph-theoretic attractors with rich geometric and topological properties.

1. Definition and Formal Construction

Let $G$ be a finite simple graph with independence number $\alpha(G)$, and independence polynomial

$I_G(z) = \sum_{j=0}^{\alpha(G)} a_j z^j,$

where $a_j$ counts the independent sets of size $j$. The $m$-fold lexicographic power $G^m$ is formed by repeated lexicographic products. The roots of the independence polynomial of $G^m$, denoted $\mathrm{Roots}(I_{G^m})$, are studied in the limit $m \to \infty$ with respect to the Hausdorff metric on compact subsets of the plane.

The independence attractor of $G$, denoted $\mathcal{A}(G)$, is

$$\mathcal{A}(G) = \lim_{m \to \infty} \{ z : I_{G^m}(z) = 0 \}.$$

This limit is well-defined, yielding a compact set that is invariant under the polynomial dynamics derived from $G$'s structure. A reduced independence polynomial $P_G(z)\equiv I_G(z)-1$ encodes the iteration: $I_{G^m}(z) = P_G^{\circ m}(z) + 1$, so the roots at stage $m$ are the preimages of $-1$ under the $m$-fold iteration of $P_G$ (Khetawat et al., 27 May 2025, Manna et al., 6 Aug 2025).

2. Julia Set Correspondence and Independence Fractals

Classic results from complex dynamics show that, except for a discrete set associated to "super-attracting" multiple roots at $z=-1$, the independence attractor $\mathcal{A}(G)$ equals the Julia set $J(P_G)$ of the polynomial $P_G$:

$\mathcal{A}(G) = J(P_G)$

in all cases except where $-1$ is a multiple root of $I_G(z)$ (Khetawat et al., 27 May 2025, Manna et al., 6 Aug 2025). The independence fractal $F(G)=J(P_G)$. The topology of $\mathcal{A}(G)$, therefore, is dictated by the critical orbit structure of $P_G$ and falls into regimes familiar from the Fatou-Julia theory (connected, totally disconnected, disconnected but not totally, etc.).

3. Topological Classification: Circles and Segments

Research has precisely characterized the possible simple topologies realizable as independence attractors:

• No Circular Attractors: For any nontrivial finite graph $G$, $\mathcal{A}(G)$ can never be a geometric circle. If $-1$ is not a multiple root, a polynomial with a circular Julia set must be conjugate to $z \mapsto Bz^n$, $|B|=1$, but positivity of $P_G$'s coefficients and $P_G(0)=0$ forces $P_G(z) = (1+z)^n-1$, corresponding only to the empty graph. If $-1$ is a multiple root, invariance is broken by the basin of $-1$ (Khetawat et al., 27 May 2025).
• Line Segment Attractors: If $\mathcal{A}(G)$ is a segment, then

$\mathcal{A}(G) = [-4/k, 0]$

for some $k \in \{1,2,3,4\}$. The segment is always real, classified using affine conjugation to Chebyshev polynomials, and constraints on positivity and binomial divisibility restrict $k$ to these integer values. For independence number 4, explicit graph constructions realize all four options (Khetawat et al., 27 May 2025).

| Attractor | Independence polynomial $I_G(z)$ (for $\alpha=4$) | Example type | |----------------|-----------------------------------------------------------|-------------------| | $[-4,0]$ | $1+16z+20z^2+8z^3+z^4$ | Connected | | $[-2,0]$ | $1+16z+40z^2+32z^3+8z^4$ | Connected | | $[-4/3,0]$ | $1+16z+60z^2+72z^3+27z^4$ | Disconnected | | $[-1,0]$ | $1+16z+80z^2+128z^3+64z^4$ | Disconnected |

Outside these cases, independence attractors exhibit more intricate (fractal) topology, faithfully reflecting Julia sets of $P_G$.

4. Connectivity and Critical Orbit Dichotomy

The connectivity properties of independence attractors are governed by the dynamics of critical points of $P_G(z)$, with precise discriminant-based conditions:

• Connected attractor: Every critical orbit of $P_G$ remains bounded.
• Totally disconnected: All critical orbits escape to infinity.
• Disconnected but not totally: Some, but not all, critical orbits escape.

For independence polynomials of degree three, these conditions are refined by explicit discriminants $\Delta_1 = a_2^2 - 3a_1a_3$ and $\Delta_2 = a_2^2 - 4a_3(a_1-1)$. For special cases such as $a_1=3$, the attractor forms $\{-1\} \cup \{z : |z+1|=1\}$ (a circle plus isolated point), otherwise the regime is determined algorithmically from the coefficients (Manna et al., 6 Aug 2025).

Condition	Connectivity type
$a_2^2 < 3a_1a_3$	Totally disconnected
$a_2^2 = 3a_1a_3$ ($a_1=3$)	Circle plus point
$a_2^2 = 4a_3(a_1-1)$, $a_1=5$	Connected
$a_2^2 = 4a_3(a_1-1)$, $a_1\neq5$	Disconnected but not totally
(others, see (Manna et al., 6 Aug 2025))	Varies (see text for details)

5. Explicit Examples and Realizability

Constructive results demonstrate that for independence number 4, both connected and disconnected graphs realize all four possible segment attractors. The independence polynomial factorization and composition correspond to various configurations (single graph, product of smaller graphs). Infinitely many non-isomorphic graphs can realize a given segment, with explicit component factor polynomials provided for disconnected examples (Khetawat et al., 27 May 2025).

For independence number 3, the paper provides examples for every connectivity type—complete graphs with an edge removed (totally disconnected), unions of isolated vertices (circle plus point), and unions of cliques or components (parabolic-disconnected or connected, as per critical discriminant) (Manna et al., 6 Aug 2025).

6. Open Problems and Research Directions

Key unresolved questions include:

• Which other Jordan curves (beyond real segments) can occur as independence attractors?
• Can the topological classification be extended for graphs with larger independence numbers $(\alpha>4)$?
• Which reduced independence polynomials $P_G$ are conformally conjugate to monic unicritical polynomials $z^n+c$—potentially leading to new classes of attractor sets via classic complex dynamics results?
• Systematic determination of graph families whose attractors possess fractal or other higher-genus topology.

A plausible implication is that further exploration of the connection between independence polynomial iteration and polynomial dynamics will yield new invariants of interest in both graph theory and holomorphic dynamical systems (Khetawat et al., 27 May 2025, Manna et al., 6 Aug 2025).

7. Significance and Broader Context

The independence attractor framework incorporates algebraic graph invariants, functional iteration, and complex dynamics, providing a precise and calculable invariant capturing the global asymptotics of root loci for graph polynomials under lexicographic powering. The interaction with Julia set theory permits the importation of powerful techniques from holomorphic dynamics, establishing attractor-based dichotomies and trichotomies analogous to classic results on the Mandelbrot and Julia sets. Unlike classical attractor notions in dissipative ODEs, independence attractors may manifest as topologically connected sets (segments), disconnected but non-totally disconnected, fractal, or even totally disconnected, depending on both discrete graph structure and the resultant polynomial dynamics.

These results also highlight the constraints imposed by graph-theoretic positivity on the palette of attainable dynamical behaviors, in contrast with general polynomial dynamics where Julia sets can be circles, Sierpiński gaskets, or other exotica. Further classification, especially in higher independence number regimes, promises to elucidate the intersection of discrete, algebraic, and dynamical systems phenomena.