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

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.

arXiv commentDefinitive version accepted for publication in The Journal of Chemical Physics

References in corpus (16)

• The structure and function of complex networks
• Assortative mixing in networks
• Classes of behavior of small-world networks
• Quantum random walks - an introductory overview
• Theory of resistor networks: The two-point resistance
• Continuous-Time Quantum Walks: Models for Coherent Transport on Complex Networks
• Quantum walk approach to search on fractal structures
• Determining global mean-first-passage time of random walks on Vicsek fractals using eigenvalues of Laplacian matrices
• Determining mean first-passage time on a class of treelike regular fractals
• Exact solution of mean geodesic distance for Vicsek fractals
• Trapping in dendrimers and regular hyperbranched polymers
• Small-World Rouse Networks as models of cross-linked polymers
• Dynamics of continuous-time quantum walks in restricted geometries
• Influence of trap location on the efficiency of trapping in dendrimers and regular hyperbranched polymers
• Asymmetries in symmetric quantum walks on two-dimensional networks
• Constrained spin dynamics description of random walks on hierarchical scale-free networks

Cited by in corpus (7)

• Random walks in weighted networks with a perfect trap: An application of Laplacian spectra
• Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications
• Efficiency analysis of diffusion on T-fractals in the sense of random walks
• Maximal entropy random walk improves efficiency of trapping in dendrimers
• Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices
• Anomalous behavior of trapping in extended dendrimers with a perfect trap
• Generalized Flory Theory for Rotational Symmetry Breaking of Complex Macromolecules