scholarly journals Plünnecke inequalities for measure graphs with applications

2015 ◽  
Vol 37 (2) ◽  
pp. 418-439
Author(s):  
KAMIL BULINSKI ◽  
ALEXANDER FISH
Keyword(s):  
Recent Work ◽  
Measure Space ◽  
Correspondence Principle ◽  
Abelian Groups ◽  
Density Estimates ◽  
Vertex Set ◽  
Banach Density ◽  
Measure Preserving Actions

We generalize Petridis’s new proof of Plünnecke’s graph inequality to graphs whose vertex set is a measure space. Consequently, by a recent work of Björklund and Fish, this gives new Plünnecke inequalities for measure-preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that extend those given by Jin.

scholarly journals Plünnecke inequalities for countable abelian groups

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Michael Björklund ◽  
Alexander Fish
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Lower Bounds ◽  
Correspondence Principle ◽  
Additive Basis ◽  
Banach Density ◽  
New Form ◽  
Countable Abelian Group ◽  
Product Sets ◽  
Measure Preserving Actions

AbstractWe establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers.We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

On enhanced power graphs of finite groups

2018 ◽  
Vol 17 (08) ◽  
pp. 1850146 ◽  
Author(s):  
Sudip Bera ◽  
A. K. Bhuniya
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Cyclic Subgroups ◽  
Power Graph ◽  
Vertex Set ◽  
One To One

Given a group [Formula: see text], the enhanced power graph of [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are edge connected in [Formula: see text] if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. Here, we show that the graph [Formula: see text] is complete if and only if [Formula: see text] is cyclic; and [Formula: see text] is Eulerian if and only if [Formula: see text] is odd. We characterize all abelian groups and all non-abelian [Formula: see text]-groups [Formula: see text] such that [Formula: see text] is dominatable. Besides, we show that there is a one-to-one correspondence between the maximal cliques in [Formula: see text] and the maximal cyclic subgroups of [Formula: see text].

The intersection graph of a group

2015 ◽  
Vol 14 (05) ◽  
pp. 1550065 ◽  
Author(s):  
S. Akbari ◽  
F. Heydari ◽  
M. Maghasedi
Keyword(s):  
Disjoint Union ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Intersection Graph ◽  
Free Graph ◽  
Intersection Graphs ◽  
Cyclic Groups ◽  
Vertex Set

Let G be a group. The intersection graph of G, denoted by Γ(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if H ∩ K ≠ 1. In this paper, we show that the girth of Γ(G) is contained in the set {3, ∞}. We characterize all solvable groups whose intersection graphs are triangle-free. Moreover, we show that if G is finite and Γ(G) is triangle-free, then G is solvable. Also, we prove that if Γ(G) is a triangle-free graph, then it is a disjoint union of some stars. Among other results, we classify all abelian groups whose intersection graphs are complete. Finally, we study the intersection graphs of cyclic groups.

scholarly journals Representations of constant socle rank for the Kronecker algebra

2020 ◽  
Vol 32 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Daniel Bissinger
Keyword(s):  
Recent Work ◽  
Abelian Groups ◽  
Wild Type ◽  
Kronecker Algebra ◽  
Radical Property

AbstractInspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups {E_{r}} of rank r over a field of characteristic {p>0}, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra {\mathcal{K}_{r}=k\Gamma_{r}} with {r\geq 2} arrows and {1\leq d\leq r-1}. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in {\operatorname{mod}\mathcal{K}_{r}} are of wild type. Then we use a natural functor {\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}} to transfer our results to {\operatorname{mod}kE_{r}}.

scholarly journals Walrasian equilibrium theory with and without free-disposal: theorems and counterexamples in an infinite-agent context

2021 ◽  
Author(s):  
Robert M. Anderson ◽  
Haosui Duanmu ◽  
M. Ali Khan ◽  
Metin Uyanik
Keyword(s):  
Recent Work ◽  
Measure Space ◽  
Equilibrium Theory ◽  
Walrasian Equilibrium ◽  
Free Disposal ◽  
Trade Offs ◽  
Finite Set ◽  
Abstract Economies ◽  
Measure Space Of Agents

AbstractThis paper provides four theorems on the existence of a free-disposal equilibrium in a Walrasian economy: the first with an arbitrary set of agents with compact consumption sets, the next highlighting the trade-offs involved in the relaxation of the compactness assumption, and the last two with a countable set of agents endowed with a weighting structure. The results generalize theorems in the antecedent literature pioneered by Shafer–Sonnenschein in 1975, and currently in the form taken in He–Yannelis 2016. The paper also provides counterexamples to the existence of non-free-disposal equilibrium in cases of both a countable set of agents and an atomless measure space of agents. One of the examples is related to one Chiaki Hara presented in 2005. The examples are of interest because they satisfy all the hypotheses of Shafer’s 1976 result on the existence of a non-free-disposal equilibrium, except for the assumption of a finite set of agents. The work builds on recent work of the authors on abstract economies, and contributes to the ongoing discussion on the modelling of “large” societies.

scholarly journals Turán Problems on Non-Uniform Hypergraphs

10.37236/3901 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
J. Travis Johnston ◽  
Linyuan Lu
Keyword(s):  
Recent Work ◽  
Blow Up ◽  
Uniform Hypergraph ◽  
Expected Number ◽  
Uniform Hypergraphs ◽  
Turan Problems ◽  
Vertex Set ◽  
Lubell Function

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set of edge types. For a fixed hypergraph $H$, the Turán density $\pi(H)$ is defined to be $\lim_{n\to\infty}\max_{G_n}h_n(G_n)$, where the maximum is taken over all $H$-free hypergraphs $G_n$ on $n$ vertices satisfying $R(G_n)\subseteq R(H)$, and $h_n(G_n)$, the so called Lubell function, is the expected number of edges in $G_n$ hit by a random full chain. This concept, which generalizes  the Turán density of $k$-uniform hypergraphs, is motivated by recent work on extremal poset problems.  The details connecting these two areas will be revealed in the end of this paper.Several properties of Turán density, such as supersaturation, blow-up, and suspension, are generalized from uniform hypergraphs to non-uniform hypergraphs. Other questions such as "Which hypergraphs are degenerate?" are more complicated and don't appear to generalize well. In addition, we completely determine the Turán densities of $\{1,2\}$-hypergraphs.

scholarly journals Common neighborhood spectrum of commuting graphs of finite groups

2021 ◽  
Vol 32 (1) ◽  
pp. 33-48
Author(s):  
W. N. T. Fasfous ◽  
◽  
R. Sharafdini ◽  
R. K. Nath ◽  
◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Undirected Graph ◽  
Abelian Groups ◽  
Commuting Graph ◽  
Vertex Set ◽  
The Common

The commuting graph of a finite non-abelian group G with center Z(G), denoted by Γc(G), is a simple undirected graph whose vertex set is G∖Z(G), and two distinct vertices x and y are adjacent if and only if xy=yx. In this paper, we compute the common neighborhood spectrum of commuting graphs of several classes of finite non-abelian groups and conclude that these graphs are CN-integral.

scholarly journals Isomorphisms of Cayley digraphs of Abelian groups

1998 ◽  
Vol 57 (2) ◽  
pp. 181-188 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Cayley Graph ◽  
Prime Divisor ◽  
Open Problem ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Vertex Set ◽  
A Finite Group

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

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