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We call an oriented odd cycle alternating if it has exactly one vertex whose in-degree and out-degree are both positive. In this paper, we investigate whether certain graphs admit an orientation that avoids alternating odd cycles as…
We continue our study of orderings on Bratteli diagrams started in previous work, joint with Jan Kwiatkowski, where Bratteli diagrams of finite rank were considered. We extend the notions of languages, permutations (called correspondences…
We modify the Pearce-Kelly algorithm that maintains a topological ordering for a directed acyclic graph in order to allow cycles to be tolerated. Cycles make topological orderings moot, of course, however tolerating them is useful in…
It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an…
In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics of the critical orbit with the exponents…
We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under the topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key…
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin,…
Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…
We report the observation of the Berezinskii-Kosterlitz-Thouless transition for a 2D gas of exciton-polaritons, and through the joint measurement of the first-order coherence both in space and time we bring compelling evidence of a…
We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion…
The simplest way to make a dynamical system out of a finite connected graph $G$ is to give it a polarization, that is to say a cyclic ordering of the edges incident to a vertex, for each vertex. The phase space $\mathcal{P}(G)$ then…
Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1…
We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological…
The chapter presents mathematical models intended for creating a topological drawing of a non-separable non-planar graph based on the methods of G. Ringel's vertex rotation theory. The induced system of cycles generates a topological…
Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…
We consider directed graph algorithms in a streaming setting, focusing on problems concerning orderings of the vertices. This includes such fundamental problems as topological sorting and acyclicity testing. We also study the related…
Let $(X,d,f)$ be a topological dynamical system, where $(X,d)$ is a compact metric space and $f:X\to X$ is a continuous map. We define $n$-ordered empirical measure of $x\in X$ by \begin{align*}…
A graph property is said to be elusive ( evasive) if every algorithm testing this property by asking questions of the form "is there an edge between vertices x and y" requires, in the worst case, to ask about all pairs of vertices. The…
We present an on-line algorithm for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created. Our algorithm takes O(m^{1/2}) amortized time per arc, where m is the total number…
Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the…