递归树状小世界聚合物网络的Laplacian谱：解析解及其应用
Laplacian spectra of recursive treelike small-world polymer networks: Analytical solutions and applications
摘要：
高分子物理研究的一个中心问题是理解大分子的几何性质和各种动力学之间的关系，其中大部分动力学都编码在描述宏观结构的相关图的拉普拉斯谱中。本文介绍了一类带参数的树状聚合物网络，它的尺寸与描述规则超支化聚合物的Vicsek分形的尺寸相同。我们研究了网络的一些相关性质，证明了它们具有指数衰减的度分布和小世界行为。然后，我们研究的Laplacian特征值和相应的特征向量的网络中考虑，这两个数量被确定通过递归关系推导出的网络结构。利用所得到的递推关系，我们可以求出任意规模网络的所有特征值和特征向量。最后，作为应用，我们利用这些特征值解析或半解析地研究了网络中发生的三种动力学过程，包括无规行走、广义高斯结构框架下的弛豫动力学以及准共振能量转移下的荧光去极化.此外，我们将结果与Vicsek分形相对应的结果进行了比较，并表明两个网络家族的动力学差异很大，从而使我们能够区分它们。
英文摘要：
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.
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