Related papers: The spectrum of the averaging operator on a networ…
We consider discrete Schr\"odinger operators with real periodic potentials on periodic graphs. The spectra of the operators consist of a finite number of bands. By "rolling up" a periodic graph along some appropriate directions we obtain…
A rooted network consists of a connected, locally finite graph G, equipped with edge conductances and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1 at o, and is harmonic at all…
Assortativity measures the tendency of a vertex in a network being connected by other vertexes with respect to some vertex-specific features. Classical assortativity coefficients are defined for unweighted and undirected networks with…
We obtain the estimate of the Lebesgue measure of the spectrum for the direct integral of matrix-valued functions. These estimates are applicable for a wide class of discrete periodic operators. For example: these results give new and sharp…
A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space…
Spectrum graph theory not only facilitate comprehensively reflect the topological structure and dynamic characteristics of networks, but also offer significant and noteworthy applications in theoretical chemistry, network science and other…
The spectral properties of traditional (dyadic) graphs, where an edge connects exactly two vertices, are widely studied in different applications. These spectral properties are closely connected to the structural properties of dyadic…
The spectrum of the adjacency matrix plays several important roles in the mathematical theory of networks and in network data analysis, for example in percolation theory, community detection, centrality measures, and the theory of dynamical…
The main goal of this work is to examine the structure of normal Hausdorff operators on $\mathbb{R}^n$. We show that normal Hausdorff operator in $L^2(\mathbb{R}^n)$ is unitary equivalent to the operator of multiplication by some…
Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical…
Understanding the mutual interdependence between the behavior of dynamical processes on networks and the underlying topologies promises new insight for a large class of empirical networks. We present a generic approach to investigate this…
We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by…
We consider discrete one-dimensional Schr\"odinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue…
We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schr\"odinger operator on a star-graph with a finite number of…
A finite dimensional operator that commutes with some symmetry group admits quotient operators, which are determined by the choice of associated representation. Taking the quotient isolates the part of the spectrum supporting the chosen…
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been…
Let $X$ be a Borel and Borel-regular metric measure space whose closed balls are of positive and finite measure. In this paper, we shall give equivalent conditions for averaging operators on non-reflexive Lebesgue spaces $L^1(X)$ and…
Degree heterogeneity and latent geometry, also referred to as popularity and similarity, are key explanatory components underlying the structure of real-world networks. The relationship between these components and the statistical…
We consider a Laplace operator on a random graph consisting of infinitely many loops joined symmetrically by intervals of unit length. The arc lengths of the loops are considered to be independent, identically distributed random variables.…
We derive a class of equations describing low Reynolds number steady flows of incompressible and viscous fluids in networks made of straight channels, with several sources and sinks, and adaptive conductivities. The flow is controlled by…