Related papers: Using a new zero forcing process to guarantee the …
In this paper, we revisit spectral sparsification for sums of arbitrary positive semidefinite (PSD) matrices. Concretely, for any collection of PSD matrices $\mathcal{A} = \{A_1, A_2, \ldots, A_r\} \subset \mathbb{R}^{n \times n}$, given…
A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{ 1,2,\ldots ,n\right\} $ to the vertices of $G$, where each $uv\in E\left( G\right) $ is labeled $f\left( u\right) +f\left( v\right)…
Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is…
If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some…
A global forcing set for maximal matchings of a graph $G=(V(G), E(G))$ is a set $S \subseteq E(G)$ such that $M_1\cap S \neq M_2 \cap S$ for each pair of maximal matchings $M_1$ and $M_2$ of $G$. The smallest such set is called a minimum…
We obtain the sharp arithmetic Gordon's theorem: that is, absence of eigenvalues on the set of energies with Lyapunov exponent bounded by the exponential rate of approximation of frequency by the rationals, for a large class of…
We consider the {\em Deligne-Simpson problem}: {\em Give necessary and sufficient conditions for the choice of the conjugacy classes $c_j\subset gl(n,{\bf C})$ or $C_j\subset GL(n,{\bf C})$, $j=1,..., p+1$, so that there exist irreducible…
In this paper the limit probabilities of first-order properties are studied. The random graph $G(n,p)$ {\it obeys Zero-One $k$-Law} if for each first-order property with quantifier depth not greater than $k$ its probability tends to 0 or…
We consider bounds on maximum nullity of a graph via transversal numbers of compatible collections of forts. Results include generalizations of theorems from symmetric to combinatorially symmetric matrices, special bases of matrix…
Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions…
A graph $G$ contains a strong parity factor $F$ if for every subset $X\subseteq V(G)$ with $|X|$ even, $G$ has a spanning subgraph $F$ satisfying $\delta(F)\geq1$, $d_F(u)\equiv1$ (mod 2) for any $u\in X$, and $d_F(v)\equiv0$ (mod 2) for…
We present a different way to obtain generators of metric spaces having the property that the ``position'' of every element of the space is uniquely determined by the distances from the elements of the generators. Specifically we introduce…
The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Let $G$ be embeddable on a surface whose Euler characteristic $\chi$ is as large as possible,…
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable…
We prove a robust version of a graph embedding theorem of Sauer and Spencer. To state this sparser analogue, we define $G(p)$ to be a random subgraph of $G$ obtained by retaining each edge of $G$ independently with probability $p \in…
A vertex $w$ of a connected graph $G$ strongly resolves two vertices $u,v\in V(G)$, if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric generator for…
There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the…
We study the probabilistic zero forcing process, a probabilistic variant of the classical zero forcing process. We show that for every connected graph $G$ on $n$ vertices, there exists an initial set consisting of a single vertex such that…
A graph is $\mathcal{R}_d$-independent (resp. $\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\mathcal{R}_d$-independent graph satisfies…
We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset…