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An exact formula for the contraction factor of a subdivided Gaussian topological polymer

Jason Cantarella, Tetsuo Deguchi, Clayton Shonkwiler, Erica Uehara

TL;DR

The paper derives an exact, subdivision-aware formula for the contraction factor of Gaussian topological polymers on subdivision graphs. It expresses the mean squared radius of gyration and the contraction factor in terms of a weighted graph Laplacian $\mathcal{L}_{\Omega}$ and its pseudoinverse, via the weighted Kirchhoff index $\operatorname{Kf}(\mathbf{G},\Omega)=\omega\operatorname{tr} \mathcal{L}_{\Omega}^+$, with a precise dependence on the subdivision parameter $n$ through a carefully chosen weight vector $\Omega'$. This unifies and extends prior asymptotic results by giving exact formulas for arbitrary subdivision counts and base graphs, and it is validated through multiple concrete examples (cycle, star, theta, bipyramid, etc.). The results have implications for predicting the size and shape of subdivided Gaussian topological polymers and clarify the role of topology via the weighted Laplacian spectrum in determining contraction behavior.

Abstract

We consider the radius of gyration of a Gaussian topological polymer $G$ formed by subdividing a graph $G'$ of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of $G$ in terms of the number of subdivisions of each edge of $G'$ and a new weighted Kirchhoff index for $G'$. The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.

An exact formula for the contraction factor of a subdivided Gaussian topological polymer

TL;DR

The paper derives an exact, subdivision-aware formula for the contraction factor of Gaussian topological polymers on subdivision graphs. It expresses the mean squared radius of gyration and the contraction factor in terms of a weighted graph Laplacian and its pseudoinverse, via the weighted Kirchhoff index , with a precise dependence on the subdivision parameter through a carefully chosen weight vector . This unifies and extends prior asymptotic results by giving exact formulas for arbitrary subdivision counts and base graphs, and it is validated through multiple concrete examples (cycle, star, theta, bipyramid, etc.). The results have implications for predicting the size and shape of subdivided Gaussian topological polymers and clarify the role of topology via the weighted Laplacian spectrum in determining contraction behavior.

Abstract

We consider the radius of gyration of a Gaussian topological polymer formed by subdividing a graph of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of in terms of the number of subdivisions of each edge of and a new weighted Kirchhoff index for . The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.

Paper Structure

This paper contains 5 sections, 16 theorems, 58 equations, 5 figures.

Table of Contents

  1. Notation and review of theory
  2. Gaussian topological polymers based on subdivision graphs
  3. Weighted Radius of Gyration and Weighted Graph Laplacians
  4. The expected radius of gyration of subdivision graphs
  5. Example computations

Key Result

Theorem 1

For any connected multigraph $\mathbf{G}$ (including loop and multiple edges), if $\mathbf{G}_n$ is the $n$-part edge subdivision of $\mathbf{G}$, then the contraction factor $g(\mathbf{G}_n)$ of a Gaussian topological polymer with underlying graph $\mathbf{G}_n$ embedded in $\mathbb{R}^d$ obeys where $\operatorname{Loops}(\mathbf{G}) = \mathbf{e}(G) - \mathbf{v}(G) + 1$ is the cycle rank of $\ma

Figures (5)

  • Figure 1: The graph $\mathbf{G}$ (at left) is a subdivision of the graph $\mathbf{G}'$ at right. The vertices $v_{i,j}$ and edges $e_{i,j}$ of $\mathbf{G}$ are numbered to correspond with vertices and edges of $\mathbf{G}'$; each $v_{0,j}$ corresponds to a vertex $v'_j$ of $\mathbf{G}'$, while vertices $v_{i,1}, \dotsc, v_{i,n-1}$ are those created by subdividing edge $e'_i$ of $\mathbf{G}'$ into $n$ new edges. The edges $e_{i,j}$ of $\mathbf{G}$ aren't labeled in the picture, but are constructed so that $e_{i,1}, \dotsc, e_{i,n}$ are the edges created by subdividing $e'_i$. Here $n=4$, $\mathbf{v} = 22$, $\mathbf{e} = 24$, $\mathbf{v}' = 4$ and $\mathbf{e}'= 6$ so we can check that $\mathbf{e} = n \mathbf{e}'$ and $\mathbf{v} = \mathbf{v}' + (n-1) \mathbf{e}'$. The map $f_0(v'_i) = v_{0,i}$ while $f_1(e'_i) = \sum_{j=1}^{n} e_{i,j}$.
  • Figure 2: The cycle graph $\mathbf{G}$ (at left) is a subdivision of the multigraph $\mathbf{G}'$ with one vertex and one edge (at right).
  • Figure 3: The star polymer graph $\mathbf{G}$ (at left) is a subdivision of the star graph with four vertices $\mathbf{G}'$ (at right).
  • Figure 4: The graph $\mathbf{G}$ (at left) is a subdivision of the multitheta graph with 4 edges $\mathbf{G}'$ (at right).
  • Figure 5: The graph $\mathbf{G}$ (at left) is a subdivision of the bipyramid graph $\mathbf{G}'$ (at right).

Theorems & Definitions (31)

  • Theorem 1: Cantarella2022ROG, Theorem 5
  • Theorem 2
  • Corollary 1
  • Proposition 1: cf. MasseyBasicCourse
  • Definition 1
  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 21 more