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.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

“…In recent years, extensive attention has been focused on the study of spectra for various matrices of complex networks [14][15][16], such as the adjacency matrix [17][18][19], the Laplacian matrix [20][21][22][23][24], the modularity matrix [25,26], and the nonback-tracking matrix [27]. In the context of the transition matrix, a conscientious effort has been devoted to determine the eigenvalues for some classic fractals [28][29][30][31] and treelike networks [32][33][34][35].…”

Exploring the applications of fractional calculus: Hierarchically built semiflexible polymers

“…Now, for several classes of polymers the role of their topology on the dynamics in external fields has been studied using the standard GGS formalism. This classes include stars and dendritic polymers [3], scale-free polymer networks [4,5], hyperbranched polymers modeled by Vicsek fractals [6,7], Sierpinski gaskets [8], and also multihierarchical fractals [9]. In this respect regular fractal structures are of much interest since their dynamical properties may display scaling.…”

Maximal entropy random walk improves efficiency of trapping in dendrimers

“…Seemingly, there are k +1 multipliers in both the denominator and numerator in Eq. (30). In fact, for any k we can decrease the number of multipliers from k + 1 to two by simplifying Eq.…”

“…A primary quantity related to trapping is average trapping time (ATT), which is the average of mean first-passage time (MFPT) [18][19][20][21][22][23] to the target over all starting nodes, where MFPT from a node to the trap is the expected time steps needed for a walker starting off from this node to visit the trap for the first time. ATT is a quantitative indicator measuring the trapping efficiency, which has been much studied for diverse complex systems [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] . Because of the theoretical and practical relevance, trapping in dendrimers has also been devoted to concerted efforts.…”