Dipartimento di Matematica, Sapienza Università di Roma, P. le A. Moro 5, 00185, Roma, Italy.
Dipartimento SBAI (Ingegneria), Sapienza Università di Roma, via A. Scarpa 16, 00161, Roma, Italy.
Sci Rep. 2017 Jan 9;7:39962. doi: 10.1038/srep39962.
We consider the Dyson hierarchical graph , that is a weighted fully-connected graph, where the pattern of weights is ruled by the parameter σ ∈ (1/2, 1]. Exploiting the deterministic recursivity through which is built, we are able to derive explicitly the whole set of the eigenvalues and the eigenvectors for its Laplacian matrix. Given that the Laplacian operator is intrinsically implied in the analysis of dynamic processes (e.g., random walks) occurring on the graph, as well as in the investigation of the dynamical properties of connected structures themselves (e.g., vibrational structures and relaxation modes), this result allows addressing analytically a large class of problems. In particular, as examples of applications, we study the random walk and the continuous-time quantum walk embedded in , the relaxation times of a polymer whose structure is described by , and the community structure of in terms of modularity measures.
我们考虑戴森等级图，它是一个加权完全连通图，其中权重的模式由参数 σ ∈ (1/2, 1] 决定。利用其确定性递归性，我们能够显式地推导出其拉普拉斯矩阵的所有特征值和特征向量。由于拉普拉斯算子本质上隐含在图上发生的动态过程（例如随机游走）的分析中，以及在连接结构本身的动态特性（例如振动结构和弛豫模式）的研究中，因此该结果允许对一大类问题进行分析。特别地，作为应用示例，我们研究了嵌入在其中的随机游走和连续时间量子游走、由 描述的聚合物的弛豫时间，以及基于模块度度量的 中的社区结构。
