Starting from exact relations for finite Husimi cacti we determine their complete spectra to very high accuracy. The Husimi cacti are dual structures to the dendrimers but, distinct from these, contain loops. Our solution makes use of a judicious analysis of the normal modes. Although close to those of dendrimers, the spectra of Husimi cacti differ. From the wealth of applications for measurable quantities which depend only on the spectra, we display for Husimi cacti the behavior of the fluorescence depolarization under quasiresonant Forster energy transfer.

We focus on polymer networks with a scale-free topology. In the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix, we determined numerically the averaged monomer displacement under external forces and the mechanical relaxation moduli (storage and loss modulus). First, we monitor these physical quantities and additionally the eigenvalue spectrum for structures of different sizes, but with the same γ, which is a parameter that measures the connectivity of the structure. Second, we vary the parameter γ, and we keep constant the size of the structures. This allows us to study in detail the crossover behavior from a simple linear chain to a starlike structure. As expected we encounter a more chainlike behavior for high values of γ, while for small values of γ a more starlike behavior is observed. In the intermediate time (frequency) domain, we encounter regions of constant slope for some intermediate values of γ.

We study the transport efficiency of excitations on complex quantum networks with loops. For this we consider sequentially growing networks with different topologies of the sequential subgraphs. This can lead either to a universal complete breakdown of transport for complete-graph-like sequential subgraphs or to optimal transport for ring-like sequential subgraphs. The transition to optimal transport can be triggered by systematically reducing the number of loops of complete-graph-like sequential subgraphs in a small-world procedure. These effects are explained on the basis of the spectral properties of the network's Hamiltonian. Our theoretical considerations are supported by numerical Monte-Carlo simulations for complex quantum networks with a scale-free size distribution of sequential subgraphs and a small-world-type transition to optimal transport. [5]. Only recently, network theory has been combined with quantum theory, in order to study, say, the quantum dynamic properties on complex structures [6][7][8][9][10][11][12].The majority of networks will have (some) loops, which -for classical networks-influence the dynamics. For instance, the target search on looped DNA is of superdiffusive type [13]. In the cell, DNA appears as supercoils (plectonemes), which also influences the dynamics [14]. It is not clear, if and how the presence of loops influences the quantum dynamics. For the subclass of quantum networks without loops, we have recently demonstrated that there are universal features when the complexity of the network leads to a complete breakdown of the quantum transport properties [15].

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We consider continuous-time quantum walks (CTQWs) on multilayer dendrimer networks (MDs) and their application to quantum transport. A detailed study of properties of CTQWs is presented and transport efficiency is determined in terms of the exact and average return probabilities. The latter depends only on the eigenvalues of the connectivity matrix, which even for very large structures allows a complete analytical solution for this particular choice of network. In the case of MDs we observe an interplay between strong localization effects, due to the dendrimer topology, and good efficiency from the linear segments. We show that quantum transport is enhanced by interconnecting more layers of dendrimers.

We focus on the generalized Husimi cacti, which are dual structures to the dendrimers but, distinct from the latter, contain loops. We determine their complete spectra by making use of the normal mode analysis. These spectra have been used in computing some physical quantities, such as the averaged monomer displacement and the mechanical relaxation moduli with its two components: the storage and the loss modulus. We also study the dynamics of Husimi cacti in solutions, introducing the hydrodynamic interactions in a preaveraged Oseen fashion, the so-called Zimm model. We observe that the relaxation quantities mentioned above do not scale, in the presence or in the absence of the hydrodynamic interactions. Our results show that all the relaxation forms depend on the number of monomers in the networks in the absence of the hydrodynamic interactions (Rouse model), while by taking into account the hydrodynamic interactions the results do not vary too much.

We focus on perturbed regular hyperbranched fractals (pRHF), which are RHF whose f coordinated centers (f CC) are traps. We compute the mechanical properties (storage and loss modulus) and the average displacement in the framework of generalized Gaussian structures, by making use of the eigenvalue spectrum of the connectivity matrix. We generalize the analysis to the case of a connectivity matrix perturbed by a diagonal and pure imaginary operator. Although the above-cited observables in this new situation lose their original meaning, they still give important information about the underlying structures and they could help to analyze other phenomena where complex operators are involved. We obtain analytically the eigenvalue spectrum for pRHF. A drastic change was observed in the behavior of the studied quantities even for a very small perturbation strength. However, it is still possible to depict the scaling of the fractals in the intermediate time (frequency) domains.

In this paper, we focus on the relaxation dynamics of Sierpinski hexagon fractal polymer. The relaxation dynamics of this fractal polymer is investigated in the framework of the generalized Gaussian structure model using both Rouse and Zimm approaches. In the Rouse-type approach, by performing real-space renormalization transformations, we determine analytically the complete eigenvalue spectrum of the connectivity matrix. Based on the eigenvalues obtained through iterative algebraic relations we calculate the averaged monomer displacement and the mechanical relaxation moduli (storage modulus and loss modulus). The evaluation of the dynamical properties in the Rouse-type approach reveals that they obey scaling in the intermediate time/frequency domain. In the Zimm-type approach, which includes the hydrodynamic interactions, the relaxation quantities do not show scaling. The theoretical findings with respect to scaling in the intermediate domain of the relaxation quantities are well supported by experimental results.