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Many studies have shown that additional information can be gained on timeseries by investigating their associated complex networks. In this work, weinvestigate the multifractal property and Laplace spectrum of the horizontalvisibility graphs (HVGs) constructed from fractional Brownian motions. We aimto identify via simulation and curve fitting the form of these properties interms of the Hurst index $H$. First, we use the sandbox algorithm to study themultifractality of these HVGs. It is found that multifractality exists in theseHVGs. We find that the average fractal dimension $\langle D(0)\rangle$ of HVGsapproximately satisfies the prominent linear formula $\langle D(0)\rangle = 2 -H$; while the average information dimension $\langle D(1)\rangle$ and averagecorrelation dimension $\langle D(2)\rangle$ are all approximately bi-linearfunctions of $H$ when $H\ge 0.15$. Then, we calculate the spectrum and energyfor the general Laplacian operator and normalized Laplacian operator of theseHVGs. We find that, for the general Laplacian operator, the average logarithmof second-smallest eigenvalue $\langle \ln (u_2) \rangle$, the averagelogarithm of third-smallest eigenvalue $\langle \ln (u_3) \rangle$, and theaverage logarithm of maximum eigenvalue $\langle \ln (u_n) \rangle$ of theseHVGs are approximately linear functions of $H$; while the average Laplacianenergy $\langle E_{nL} \rangle$ is approximately a quadratic polynomialfunction of $H$. For the normalized Laplacian operator, $\langle \ln (u_2)\rangle$ and $\langle \ln (u_3) \rangle$ of these HVGs approximately satisfylinear functions of $H$; while $\langle \ln (u_n) \rangle$ and $\langle E_{nL}\rangle$ are approximately a 4th and cubic polynomial function of $H$respectively.

Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions