Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications

TL;DR

This paper introduces a family of treelike small-world polymer networks, derives their Laplacian spectra analytically, and applies these results to study various dynamical processes, highlighting differences from Vicsek fractals.

Contribution

It provides recursive solutions for the Laplacian spectra of these networks and demonstrates their application in analyzing network dynamics.

Findings

Networks have exponential degree distribution and small-world properties.

All eigenvalues and eigenvectors can be obtained recursively for any network size.

Differences in dynamics between the new networks and Vicsek fractals are significant.

Abstract

A central issue in the study of polymer physics is to understand the relation between the geometrical properties of macromolecules and various dynamics, most of which are encoded in the Laplacian spectra of a related graph describing the macrostructural structure. In this paper, we introduce a family of treelike polymer networks with a parameter, which has the same size as the Vicsek fractals modeling regular hyperbranched polymers. We study some relevant properties of the networks and show that they have an exponentially decaying degree distribution and exhibit the small-world behavior. We then study the Laplacian eigenvalues and their corresponding eigenvectors of the networks under consideration, with both quantities being determined through the recursive relations deduced from the network structure. Using the obtained recursive relations we can find all the eigenvalues and eigenvectors for the networks with any size. Finally, as some applications, we use the eigenvalues to study analytically or semi-analytically three dynamical processes occurring in the networks, including random walks, relaxation dynamics in the framework of generalized Gaussian structure, as well as the fluorescence depolarization under quasiresonant energy transfer. Moreover, we compare the results with those corresponding to Vicsek fractals, and show that the dynamics differ greatly for the two network families, which thus enables us to distinguish between them.