Related papers: Graph and backward asymptotics of the tent map

We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: t\^ete-\`a-t\^ete graphs and twists. We completely characterize mapping…

Confirming a conjecture of Ne\v{s}et\v{r}il, we show that up to isomorphism there is only a finite number of finite minimal asymmetric undirected graphs. In fact, there are exactly 18 such graphs. We also show that these graphs are exactly…

Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…

A finite graph $\Gamma$ is called $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. We study a family of symmetric graphs, called the unitary…

Proximity networks are time-varying graphs representing the closeness among humans moving in a physical space. Their properties have been extensively studied in the past decade as they critically affect the behavior of spreading phenomena…

We prove the Ingram Conjecture, i.e., we show that the inverse limit spaces of every two tent maps with different slopes in the interval [1, 2] are non-homeomorphic. Based on the structure obtained from the proof, we also show that every…

A continuum $X$ is a dendrite if it is locally connected and contains no simple closed curve, a self mapping $f$ of $X$ is called monotone if the preimage of any connected subset of $X$ is connected. If $X$ is a dendrite and $f:X\to X$ is a…

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every…

A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortest-path, and the resistance distances at the limiting values of the family…

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking…

The dynamics of one dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a…

We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as…

Circulant graphs are a widely studied family of graphs whose members possess varying amounts of symmetry. Although considerable progress has been made in finding the automorphism groups of circulant graphs under certain restrictions, a…

In this note we consider a collection $\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\{\mu f\}$ where $\mu\in [0,1].$ Denoting the kneading sequence of $\mu f$ by $K(\mu f)$, we…

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the…

We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs. We extend various theorems concerning…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…

A Toeplitz graph $T_n \langle t_1,t_2,\ldots,t_k\rangle$ is a simple graph with the vertex set $[n]$ such that two vertices $v$ and $w$ are adjacent if and only if $|v-w| = t_i$ for some $i \in [k]$. In this paper, we investigate line…

In a recent paper we constructed a family of foliated 2-complexes of thin type whose typical leaves have two topological ends. Here we present simpler examples of such complexes that are, in addition, symmetric with respect to an involution…

Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$.…