Related papers: Characteristic Power Series of Graph Limits
The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a…
We continue our study of matrix models of dually weighted graphs. Among the attractive features of these models is the possibility to interpolate between ensembles of regular and random two-dimensional lattices, relevant for the study of…
We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not…
Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties…
The power graph $P(\Omega)$ of a group $\Omega$ is a graph with the vertex set $\Omega$ such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph…
We collect some general results on graph limits associated to hereditary classes of graphs. As examples, we consider some classes defined by forbidden subgraphs and some classes of intersection graphs, including triangle-free graphs,…
Uncertainty principles present an important theoretical tool in signal processing, as they provide limits on the time-frequency concentration of a signal. In many real-world applications the signal domain has a complicated irregular…
In this short note, we introduce cospectral graphons, paralleling the notion of cospectral graphs. As in the graph case, we give three equivalent definitions: by equality of spectra, by equality of cycle densities, and by a unitary…
The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have…
For each graph and each positive integer $n$, we define a chain complex whose graded Euler characteristic is equal to an appropriate $n$-specialization of the dichromatic polynomial. This also gives a categorification of $n$-specializations…
Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix $A$, one may look at the spectrum of $\psi(A)$ for a properly chosen…
Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on…
A method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices. The topological classification of Feynman diagrams, which was actively used in last years for matrix…
In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. In this paper some of this research is extended to more general matrix norms, like the Schatten p-norms and the Ky Fan k-norms.…
In two recent papers by Veitch and Roy and by Borgs, Chayes, Cohn, and Holden, a new class of sparse random graph processes based on the concept of graphexes over $\sigma$-finite measure spaces has been introduced. In this paper, we…
In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain…
Using appropriate power series evaluations, we determine all moments of arbitrary positive powers of the arcsine. As consequences we evaluate several doubly infinite classes of power series involving central binomial coefficients and…
We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function $W(x,y)$ on the unit square, with $x$ and $y$ uniform on the interval $(0,1)$.…
We define train track maps for graphs-of-groups $\cal G$ and exhibit the precise conditions under which the fundamental finiteness properties known for classical train track maps extend to this generalization. These finiteness properties…
Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios the information of interest resides…