The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

Cryptocurrency Market Consolidation in 2020–2021

Time series of price returns for 80 of the most liquid cryptocurrencies listed on Binance are investigated for the presence of detrended cross-correlations. A spectral analysis of the detrended correlation matrix and a topological analysis of the minimal spanning trees calculated based on this matrix are applied for different positions of a moving window. The cryptocurrencies become more strongly cross-correlated among themselves than they used to be before. The average cross-correlations increase with time on a specific time scale in a way that resembles the Epps effect amplification when going from past to present. The minimal spanning trees also change their topology and, for the short time scales, they become more centralized with increasing maximum node degrees, while for the long time scales they become more distributed, but also more correlated at the same time. Apart from the inter-market dependencies, the detrended cross-correlations between the cryptocurrency market and some traditional markets, like the stock markets, commodity markets, and Forex, are also analyzed. The cryptocurrency market shows higher levels of cross-correlations with the other markets during the same turbulent periods, in which it is strongly cross-correlated itself.

Resting-State Electroencephalography Functional Connectivity Networks Relate to Pre- and Postoperative Language Functioning in Low-Grade Glioma and Meningioma Patients

Introduction: Preservation of language functioning in patients undergoing brain tumor surgery is essential because language impairments negatively impact the quality of life. Brain tumor patients have alterations in functional connectivity (FC), the extent to which brain areas functionally interact. We studied FC networks in relation to language functioning in glioma and meningioma patients.Method: Patients with a low-grade glioma (N = 15) or meningioma (N = 10) infiltrating into/pressing on the language-dominant hemisphere underwent extensive language testing before and 1 year after surgery. Resting-state EEG was registered preoperatively, postoperatively (glioma patients only), and once in healthy individuals. After analyzing FC in theta and alpha frequency bands, weighted networks and Minimum Spanning Trees were quantified by various network measures.Results: Pre-operative FC network characteristics did not differ between glioma patients and healthy individuals. However, hub presence and higher local and global FC are associated with poorer language functioning before surgery in glioma patients and predict worse language performance at 1 year after surgery. For meningioma patients, a greater small worldness was related to worse language performance and hub presence; better average clustering and global integration were predictive of worse outcome on language function 1 year after surgery. The average eccentricity, diameter and tree hierarchy seem to be the network metrics with the more pronounced relation to language performance.Discussion: In this exploratory study, we demonstrated that preoperative FC networks are informative for pre- and postoperative language functioning in glioma patients and to a lesser extent in meningioma patients.