Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index H. First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension $\langle D(0)\rangle $ of HVGs approximately satisfies the prominent linear formula $\langle D(0)\rangle =2-H$ ; while the average information dimension $\langle D(1)\rangle $ and average correlation dimension $\langle D(2)\rangle $ are all approximately bi-linear functions of H when $H\geqslant 0.15$ . Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue $\langle \ln \left({{u}_{2}}\right)\rangle $ , the average logarithm of third-smallest eigenvalue $\langle \ln \left({{u}_{3}}\right)\rangle $ , and the average logarithm of maximum eigenvalue $\langle \ln \left({{u}_{n}}\right)\rangle $ of these HVGs are approximately linear functions of H; while the average Laplacian energy $\langle {{E}_{\text{nL}}}\rangle $ is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, $\langle \ln \left({{u}_{2}}\right)\rangle $ and $\langle \ln \left({{u}_{3}}\right)\rangle $ of these HVGs approximately satisfy linear functions of H; while $\langle \ln \left({{u}_{n}}\right)\rangle $ and $\langle {{E}_{\text{nL}}}\rangle $ are approximately a 4th and cubic polynomial function of H respectively.

We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

This paper proves the local Lipschitz property for harmonic (or positive subharmonic) functions on graphs. An example is also obtained to show that the global Lipschitz property of harmonic function on graphs does not hold.

A self-similar measure on R^n is defined to be a probability measure satisfying \mu = \sum_{j=1}^N p_j\mu\circ S_j^{-1} + \sum_{j=1}^M q_j(\mu*\mu)\circ T_j^{-1}, where S_j x = \rho_j R_j x + b_j, T_j x = \eta_j Q_j x + c_j are contractive similarities, 0 < \rho_j < 1, 0 < \eta_j < 1/2, 0 < p_j < 1, 0 < q_j < 1; \sum_{j=1}^N p_j + \sum_{j=1}^M q_j = 1, R_j, Q_j are orthogonal matrix and \mu*\mu is the convolution of two measures. When M = 0, \mu is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of \mu, such as \int_{|x|\le R} |(\frac{\partial}{\partial x})^\alpha \hat\mu(x)|^2 dx = O(R^{n-\beta}) for any order derivation of \hat\mu(x) as R\to\infty under certain additional hypotheses. When M > 0, \mu is a nonlinear self-similar measure, we get some results of L^p boundedness for maximal operators of l, from the pointwise asymptotic estimate of the Fourier transform of \mu made by Strichartz.