Related papers: Frames generated by graphs
We introduce and study a generalization of conformal rigidity for graphs. A graph is $k$-conformally rigid if the uniform edge weights simultaneously maximize the sum of the $k$ smallest nontrivial Laplacian eigenvalues and minimize the sum…
Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density…
Network theory has proven to be a powerful tool in describing and analyzing systems by modelling the relations between their constituent objects. In recent years great progress has been made by augmenting `traditional' network theory.…
We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…
Graphs with diverse structural characteristics play a central role in modelling and optimization tasks. The ability to generate different types of graphs that exhibit shared properties is likewise essential for algorithm selection and…
A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.
The graph $G$ is said to be strongly regular with parameters $(n,k,\lambda,\mu)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $\lambda$ common neighbours; (3) any two…
The Random Geometric Graph (RGG) is a random graph model for network data with an underlying spatial representation. Geometry endows RGGs with a rich dependence structure and often leads to desirable properties of real-world networks such…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
This paper addresses the problem of constructing bearing rigid networks in arbitrary dimensions. We first show that the bearing rigidity of a network is a generic property that is critically determined by the underlying graph of the…
Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = \lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$. One can also consider the same graph…
This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
A finite-dimensional Hilbert space is usually described in terms of an orthonormal basis, but in certain approaches or applications a description in terms of a finite overcomplete system of vectors, called a finite tight frame, may offer…
Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…
A graph G is {\xi}-nearly planar if it can be embedded in the sphere so that each of its edges is crossed at most {\xi} times. The family of {\xi}-nearly planar graphs is widely extending the notion of planarity. We introduce an alternative…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
The prevalence of graph-based data has spurred the rapid development of graph neural networks (GNNs) and related machine learning algorithms. Yet, despite the many datasets naturally modeled as directed graphs, including citation, website,…