Special cases and equivalent forms of Katznelson’s problem on recurrence

Unitary Cayley graphs of Dedekind domain quotients

Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ <i>disperses</i> if $\lim_

A kind of conditional connectivity of Cayley graphs generated by wheel graphs

Minimum face-spanning subgraphs of plane graphs

A discrete norm on an Abelian groupA is a non-negative function ‖ · ‖A → ℝ which satisfies the triangle inequality, is homogenous with respect to scaling ofA by ℤ and is bounded away from 0 onA/{0}. A countable Abelian group is discretely normed if and only if the group is free.

Let K⊂ℝ d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in KΛ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(j−i) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ.

Let G G be a locally compact SIN group, and w w a weight on G G which is diagonally bounded with bound K K on a dispersed set. We show that the topological centre of L U C ( w − 1 ) ∗ {LUC}(w^{-1})^* equals M ( w ) M(w) , and there are ⌊ K ⌋ + 1 \lfloor K \rfloor + 1 many points in the spectrum of L U C ( w − 1 ) {LUC}(w^{-1}) which are determining for the topological centre (DTC). As corollaries, we obtain DTC results as well for L 1 ( w ) ∗ ∗ L_1(w)^{**} , C 0 ( w − 1 ) ⊥ C_0(w^{-1})^\perp and L ∞ , 0 ( w − 1 ) ⊥ L_{\infty , 0}(w^{-1})^\perp . We also give a short proof of the first result in the unweighted case for all locally compact second countable abelian groups G G .

Nous établissons que, pour tout groupe dénombrable abélien Γ et tout sous-groupe finiment engendré Δ de Γ, l’ensemble des actions de Δ sur un espace de probabilités standard (X,μ) qui peuvent être étendues en une action libre de Γ sur (X,μ) est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où Δ est un groupe cyclique infini.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

0. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\chi(G)$ is the chromatic number of $G$, and $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. In this paper, via an original speech and simple results, we rigorously simplify the understanding of the Hadwiger conjecture. It will appear that to solve the famous Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs.\textbf{1.} The Hadwiger conjecture (see [3] or [4] or [6] or [7] or [8] or [9] or [10]) and the Berge problem (see [2] or [4] or [5] or [6] or [8]) are well known. Recall in a graph $G=[V(G),E(G),\chi(G),\omega(G), \bar{G}]$, $V(G)$ is the set of vertices, $E(G)$ is the set of edges, $\chi(G)$ is the chromatic number, $\omega(G)$ is the clique number and $\bar{G}$ is the complementary graph of $G$. The Hadwiger conjecture [recall that the famous four-color problem is a special case of the Hadwiger conjecture] states that every graph $G$ satisfies $\chi(G) \leq \eta(G)$ [where $\eta(G)$ is the hadwiger number of $G$ (i.e.\ the maximum of $p$ such that $G$ is contractible to the complete graph $K_{p}$)]. We say that a graph $B$ is berge if every $B'\in \lbrace B, \bar{B}\rbrace$ does not contain an induced cycle of odd length $\geq 5$. A graph $G$ is perfect if every induced subgraph $G'$ of $G$ satisfies $\chi(G')=\omega(G')$. Indeed, the Berge problem (see [4] or [5] or [6]) consists to show that $\chi(B)=\omega(B)$ for every berge graph $B$ [we recall (see {\bf{[0]), that the Berge problem was solved in a paper of $146$ pages long by Chudnovsky, Robertson, Seymour and Thomas]. In [4], it is presented an original investigation around the Berge problem and the Hadwiger conjecture, and, via two simple Theorems, it is shown that the Berge problem and the Hadwiger conjecture are curiously resembling, so resembling that they seem identical [indeed, they can be restated in ways that resemble each other (see [4])]. Now, in this paper, via only original speech and results, we rigorously simplify the understanding of the Hadwiger conjecture. Moreover, it will appear that to solve the Hadwiger conjecture is equivalent to solve an analytic conjecture stated on a very small class of graphs

Let G and H be connected graphs. The tensor product G + H is a graph with vertex set V(G+H) = V (G) X V(H) and edge set E(G + H) ={(a , b)(x , y)| ax ∈ E(G) & by ∈ E(H)}. The graph H is called the strongly triangular if for every vertex u and v there exists a vertex w adjacent to both of them. In this article the tensor product of G + H under some distancebased topological indices are investigated, when H is a strongly triangular graph. As a special case most of results given by Hoji, Luob and Vumara in [Wiener and vertex PI indices of Kronecker products of graphs, Discrete Appl. Math., 158 (2010), 1848-1855] will be deduced.

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

An actual topic in the graph drawing is the question how to draw two edge sets on the same vertex set, the so-called simultaneous drawing of graphs. The goal is to simultaneously find a nice drawing for both of the sets. It has been found out that only restricted classes of planar graphs can be drawn simultaneously using straight lines and without crossings within the same edge set. In this paper, we negatively answer one of the most often posted open questions namely whether any two trees with the same vertex set can be drawn simultaneously crossing-free in a straight line way.

Cayley graphs have been well-studied as a model for interconnection networks due to their low diameter, optimal fault tolerance, and algorithmic efficiency, among other properties. A problem of practical and theoretical interest is to determine or estimate the diameter of Cayley graphs. Let Γ be a Cayley graph on n! vertices generated by a transposition tree on vertex set {1,2,…,n}. In an oft-cited paper [1], it was shown that the diameter of Γ is bounded as: $$\mathop{\mathrm{diam}}\nolimits (\Gamma)~ \le ~\max_{\pi \in S_n} \left\{ c(\pi)-n+\sum_{i=1}^n \mathop{\mathrm{dist}}\nolimits _T(i,\pi(i)) \right\},$$ where the maximization is over all permutations π in the symmetric group, c(π) denotes the number of cycles in π, and \(\mathop{\mathrm{dist}}\nolimits _T\) is the distance function in T. It is of interest to determine how far away this upper bound can be from the true diameter value in the worst case and for which families of graphs this bound should be utilized or not utilized. In this work, we investigate the worst case performance of this upper bound. We show that for every n, there exists a transposition tree on n vertices such that the maximum possible difference Δ n between the upper bound and the true diameter value is at least n − 4. The lower bound we provide for Δ n is seen to be best possible, and an open problem is to determine an upper bound for Δ n .