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The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…
Large real-life complex networks are often modeled by various random graph constructions and hundreds of further references therein. In many cases it is not at all clear how the modeling strength of differently generated random graph model…
Epsilon-nets and approximate unitary $t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. The former constitute subsets of unitary…
We construct all the unitary cubic curvature gravity theories built on the contractions of the Riemann tensor in D -dimensional (anti)-de Sitter spacetimes. Our construction is based on finding the equivalent quadratic action for the…
We present an explicit deep neural network construction that transforms uniformly distributed one-dimensional noise into an arbitrarily close approximation of any two-dimensional Lipschitz-continuous target distribution. The key ingredient…
In this survey we overview known results on the strong subgraph $k$-connectivity and strong subgraph $k$-arc-connectivity of digraphs. After an introductory section, the paper is divided into four sections: basic results, algorithms and…
Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. We present here an efficient Fourier-space based…
We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…
We investigate a graph-theoretic problem motivated by questions in quantum computing concerning the propagation of information in quantum circuits. A graph $G$ is said to be a bounded extension of its subgraph $L$ if they share the same…
Geometric structures and dualities arise naturally in quantum field theories and string theory. In fact, these tools become very useful when studying strong coupling effects, where standard perturbative techniques can no longer be used. In…
The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this…
Designing well-connected graphs is a fundamental problem that frequently arises in various contexts across science and engineering. The weighted number of spanning trees, as a connectivity measure, emerges in numerous problems and plays a…
A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find…
We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus is close to the minimum genus of the primal graph. When the number of vertices is congruent to 5 modulo 12, we further guarantee that…
The graphical notion of effective resistance has found wide-ranging applications in many areas of pure mathematics, applied mathematics and control theory. By the nature of its construction, effective resistance can only be computed in…
We study the problem of cooperative localization of a large network of nodes in integer-coordinated unit disk graphs, a simplified but useful version of general random graph. Exploiting the property that the radius $r$ sets clear cut on the…
We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let $P$ be a set of $n$ points in the plane and let $S$ be a set of…
Nearest neighbor search and k-nearest neighbor graph construction are two fundamental issues arise from many disciplines such as multimedia information retrieval, data-mining and machine learning. They become more and more imminent given…
The total effective resistance, also called the Kirchhoff index, provides a robustness measure for a graph $G$. We consider two optimization problems of adding $k$ new edges to $G$ such that the resulting graph has minimal total effective…
Robust optimization is concerned with constructing solutions that remain feasible also when a limited number of resources is removed from the solution. Most studies of robust combinatorial optimization to date made the assumption that every…