On the spectrality of a class of Moran measures

In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{\mathcal{D}_{n}\}_{n=1}^{\infty}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#\mathcal{D}_{n}\in \{2,3,N_{n}\}$. We find a countable set $\Lambda\subset\mathbb{R}$ such that the set $\{e^{-2\pi i \lambda x}|\lambda\in\Lambda\}$ is a orthonormal basis of $L^{2}(\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$

Graph Concatenations to Derive Weighted Fractal Networks

Given an initial weighted graph G0, an integer m>1, and m scaling factors f1,…,fm∈0,1, we define a sequence of weighted graphs Gkk=0∞ iteratively. Provided that Gk−1 is given for k≥1, we let Gk−11,…,Gk−1m be m copies of Gk−1, whose weighted edges have been scaled by f1,…,fm, respectively. Then, Gk is constructed by concatenating G0 with all the m copies. The proposed framework shares several properties with fractal sets, and the similarity dimension dfract has a great impact on the topology of the graphs Gk (e.g., node strength distribution). Moreover, the average geodesic distance of Gk increases logarithmically with the system size; thus, this framework also generates the small-world property