scholarly journals Explicit Cayley Covers of Kautz Digraphs

10.37236/592 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Josep M. Brunat
Keyword(s):  
Group Action ◽  
Explicit Construction ◽  
Cayley Digraph ◽  
Kautz Digraphs ◽  
Finite Set ◽  
Vertex Set

Given a finite set $V$ and a set $S$ of permutations of $V$, the group action graph $\mathrm{GAG}(V,S)$ is the digraph with vertex set $V$ and arcs $(v,v^\sigma)$ for all $v\in V$ and $\sigma\in S$. Let $\langle S\rangle$ be the group generated by $S$. The Cayley digraph $\textrm{Cay}(\langle S\rangle, S)$ is called a Cayley cover of $\mathrm{GAG}(V,S)$. We define the Kautz digraphs as group action graphs and give an explicit construction of the corresponding Cayley cover. This is an answer to a problem posed by Heydemann in 1996.

Sufficient conditions for expansive group action

2019 ◽  
Vol 20 (03) ◽  
pp. 2050022 ◽  
Author(s):  
Ali Barzanouni
Keyword(s):  
Solvable Group ◽  
Group Action ◽  
Uniform Space ◽  
Sufficient Conditions ◽  
Open Cover ◽  
Probability Measures ◽  
Continuous Action ◽  
Paracompact Space ◽  
Finite Set ◽  
Borel Probability

Existence of expansivity for group action [Formula: see text] depends on algebraic properties of [Formula: see text] and the topology of [Formula: see text]. We give an expansive action of a solvable group on [Formula: see text] while there is no expansive action of a solvable group on a dendrite [Formula: see text]. We prove that a continuous action [Formula: see text] on a compact metric space [Formula: see text] is expansive if and only if there exists an open cover [Formula: see text] such that for any other open cover [Formula: see text], [Formula: see text] for some finite set [Formula: see text]. In this paper, we introduce the notion of topological expansivity of a group action [Formula: see text] on a [Formula: see text]-paracompact space [Formula: see text]. If a [Formula: see text]-paracompact space [Formula: see text] admits topologically expansive action, then [Formula: see text] is Hausdorff space. We also show that a continuous action [Formula: see text] of a finitely generated group [Formula: see text] on a compact Hausdorff uniform space [Formula: see text] is expansive with an expansive neighborhood [Formula: see text] if and only if for every [Formula: see text] there is an entourage [Formula: see text] such that for every two [Formula: see text]-pseudo orbit [Formula: see text] if [Formula: see text] for all [Formula: see text], then [Formula: see text] for all [Formula: see text]. Finally, we introduce measure [Formula: see text]-expansive actions on a uniform space. The set of all [Formula: see text]-expansive measures with common expansive neighborhood for a group action [Formula: see text] is a convex, closed and [Formula: see text]-invariant subset of the set of all Borel probability measures on [Formula: see text]. Also, we show that a group action [Formula: see text] is expansive if all Borel probability measures are [Formula: see text]-expansive or all Dirac measures [Formula: see text], [Formula: see text], have a common expansive neighborhood.

Local conditions for the stochastic comparison of particle systems

2004 ◽  
Vol 36 (4) ◽  
pp. 1252-1277 ◽  
Author(s):  
Rosario Delgado ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Queueing Networks ◽  
Interacting Particle Systems ◽  
Sufficient Conditions ◽  
Explicit Construction ◽  
Particle Systems ◽  
Stochastic Comparison ◽  
Local Conditions ◽  
Jackson Networks ◽  
Finite Set ◽  
Necessary And Sufficient

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

scholarly journals Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture

10.37236/773 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jeremy F. Alm ◽  
Roger D. Maddux ◽  
Jacob Manske
Keyword(s):  
Complete Graph ◽  
Ramsey Theory ◽  
Projective Planes ◽  
Relation Algebras ◽  
Ramsey Numbers ◽  
Edge Colorings ◽  
Finite Set ◽  
Vertex Set ◽  
Special Case

Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y \in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${\cal M} \subseteq L^{3}$. Let $c : E \rightarrow L$. Let $[n]$ denote the integer set $\{1, 2, \ldots, n\}$. For $x,y,z \in V$, let $c(xyz)$ denote the ordered triple $\big(c(xy)$, $c(yz), c(xz)\big)$. We say that $c$ is good with respect to ${\cal M}$ if the following conditions obtain: 1. $\forall x,y \in V$ and $\forall (c(xy),j,k) \in {\cal M}$, $\exists z \in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $\forall x,y,z \in V$, $c(xyz) \in {\cal M}$; and 3. $\forall x \in V \ \forall \ell\in L \ \exists \, y\in V$ such that $ c(xy)=\ell $. We investigate particular subsets ${\cal M}\subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${\cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.

Extrema of a Class of Functions on a Finite Set

1974 ◽  
Vol 26 (4) ◽  
pp. 806-819
Author(s):  
Kenneth W. Lebensold
Keyword(s):  
Positive Integer ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Polygonal Path ◽  
Finite Set ◽  
Vertex Set ◽  
The Traveling Salesman Problem ◽  
The Cost ◽  
Special Case ◽  
Class Of Functions

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

An inverse for the Gohberg-Krupnik symbol map

1980 ◽  
Vol 87 (1-2) ◽  
pp. 153-165 ◽  
Author(s):  
Martin Costabel
Keyword(s):  
Singular Integral ◽  
Mellin Transform ◽  
Integral Operators ◽  
Closed Operator ◽  
Explicit Construction ◽  
One Dimensional ◽  
Finite Set ◽  
Piecewise Continuous ◽  
Set Of Points ◽  
Points Of Discontinuity

SynopsisIt is shown that the elements of the closed operator algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients with a fixed finite set of points of discontinuity can be written as the sum of a singular integral operator, a compact operator, and generalized Mellin convolutions. Their Gohberg-Krupnik symbol is given in terms of the Mellin transform. This gives an explicit construction of an operator with prescribed Gohberg—Krupnik symbol.

scholarly journals Graph Labelings and Continuous Factors of Dynamical Systems

10.37236/2213 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Stephen M. Shea
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Shift Space ◽  
Finite Set ◽  
Vertex Set ◽  
Definition Of

A labeling of a graph is a function from the vertex set of the graph to some finite set.  Certain dynamical systems (such as topological Markov shifts) can be defined by directed graphs.  In these instances, a labeling of the graph defines a continuous, shift-commuting factor of the dynamical system.  We find sufficient conditions on the labeling to imply classification results for the factor dynamical system.  We define the topological entropy of a (directed or undirected) graph and its labelings in a way that is analogous to the definition of topological entropy for a shift space in symbolic dynamics.  We show, for example, if $G$ is a perfect graph, all proper $\chi(G)$-colorings of $G$ have the same entropy, where $\chi(G)$ is the chromatic number of $G$.

scholarly journals Order Distances and Split Systems

Order ◽  
2021 ◽  
Author(s):  
Vincent Moulton ◽  
Andreas Spillner
Keyword(s):  
Shortest Path ◽  
Classification Theory ◽  
Pairwise Distance ◽  
Split Systems ◽  
Negative Edge ◽  
Finite Set ◽  
Vertex Set ◽  
Total Preorders ◽  
Total Preorder

AbstractGiven a pairwise distance D on the elements in a finite set X, the order distanceΔ(D) on X is defined by first associating a total preorder ≼x on X to each x ∈X based on D, and then quantifying the pairwise disagreement between these total preorders. The order distance can be useful in relational analyses because using Δ(D) instead of D may make such analyses less sensitive to small variations in D. Relatively little is known about properties of Δ(D) for general distances D. Indeed, nearly all previous work has focused on understanding the order distance of a treelike distance, that is, a distance that arises as the shortest path distances in a tree with non-negative edge weights and X mapped into its vertex set. In this paper we study the order distance Δ(D) for distances D that can be decomposed into sums of simpler distances called split-distances. Such distances D generalize treelike distances, and have applications in areas such as classification theory and phylogenetics.

A Combinatorial Determinant Dual to the Group Determinant

2015 ◽  
Vol 29 ◽  
pp. 17-29
Author(s):  
Murali Srinivasan ◽  
Ashish Mishra
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Group Action ◽  
Dimensional Vector ◽  
Finite Group Action ◽  
Irreducible Polynomials ◽  
Dimensional Vector Space ◽  
Finite Set ◽  
A Finite Group

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We give the following combinatorial example of a commuting algebra determinant. Let $\Bq(n)$ denote the set of all subspaces of an $n$-dimensional vector space over $\Fq$. The {\em type} of an ordered pair $(U,V)$ of subspaces, where $U,V\in \Bq(n)$, is the ordered triple $(\mbox{dim }U, \mbox{dim }V, \mbox{dim }U\cap V)$ of nonnegative integers. Assume that there are independent indeterminates corresponding to each type. Let $X_q(n)$ be the $\Bq(n)\times \Bq(n)$ matrix whose entry in row $U$, column $V$ is the indeterminate corresponding to the type of $(U,V)$. We factorize the determinant of $X_q(n)$ into irreducible polynomials.

scholarly journals Automorphisms and Enumeration of Switching Classes of Tournaments.

10.37236/1516 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
L. Babai ◽  
P. J. Cameron
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Probability Distributions ◽  
Explicit Construction ◽  
Linear Orders ◽  
Full Automorphism Group ◽  
Odd Order ◽  
Vertex Set ◽  
Countably Infinite ◽  
Switching Class

Two tournaments $T_1$ and $T_2$ on the same vertex set $X$ are said to be switching equivalent if $X$ has a subset $Y$ such that $T_2$ arises from $T_1$ by switching all arcs between $Y$ and its complement $X\setminus Y$. The main result of this paper is a characterisation of the abstract finite groups which are full automorphism groups of switching classes of tournaments: they are those whose Sylow 2-subgroups are cyclic or dihedral. Moreover, if $G$ is such a group, then there is a switching class $C$, with Aut$(C)\cong G$, such that every subgroup of $G$ of odd order is the full automorphism group of some tournament in $C$. Unlike previous results of this type, we do not give an explicit construction, but only an existence proof. The proof follows as a special case of a result on the full automorphism group of random $G$-invariant digraphs selected from a certain class of probability distributions. We also show that a permutation group $G$, acting on a set $X$, is contained in the automorphism group of some switching class of tournaments with vertex set $X$ if and only if the Sylow 2-subgroups of $G$ are cyclic or dihedral and act semiregularly on $X$. Applying this result to individual permutations leads to an enumeration of switching classes, of switching classes admitting odd permutations, and of tournaments in a switching class. We conclude by remarking that both the class of switching classes of finite tournaments, and the class of "local orders" (that is, tournaments switching-equivalent to linear orders), give rise to countably infinite structures with interesting automorphism groups (by a theorem of Fraïssé).

scholarly journals On the Turán Density of $\{1, 3\}$-Hypergraphs

10.37236/8072 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Shuliang Bai ◽  
Linyuan Lu
Keyword(s):  
Turan Problems ◽  
Finite Set ◽  
Vertex Set ◽  
Positive Integers

In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs.  We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where  $H^{\{1, 3\}}_5$ is a hypergraph with vertex set  $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\},\{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$,  there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs. 

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