scholarly journals Hyperbolicity of Direct Products of Graphs

Symmetry ◽  
2018 ◽  
Vol 10 (7) ◽  
pp. 279 ◽  
Author(s):  
Walter Carballosa ◽  
Amauris de la Cruz ◽  
Alvaro Martínez-Pérez ◽  
José Rodríguez
Keyword(s):  
Direct Product ◽  
Bipartite Graphs ◽  
The Other ◽  
Necessary Condition ◽  
Direct Products ◽  
Product Graphs ◽  
Hyperbolicity Constant

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

2015 ◽  
Vol 230 (12) ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Kushan Ratnayake ◽  
Sajdeh Sajjadi
Keyword(s):  
Direct Product ◽  
Free Space ◽  
Special Class ◽  
The Other ◽  
Pure Point ◽  
Direct Products ◽  
Space Groups ◽  
Point Transformations ◽  
Point Groups ◽  
Torsion Free

AbstractPoint groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

scholarly journals Gromov Hyperbolicity in Strong Product Graphs

10.37236/3271 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Walter Carballosa ◽  
Rocío M. Casablanca ◽  
Amauris De la Cruz ◽  
José M. Rodríguez
Keyword(s):  
Metric Space ◽  
The Other ◽  
Gromov Hyperbolicity ◽  
Geodesic Triangle ◽  
Product Graph ◽  
Strong Product ◽  
Geodesic Metric Space ◽  
Product Graphs ◽  
Geodesic Metric ◽  
Hyperbolicity Constant

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta (X)=\inf\{\delta\geq 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}\,.$ In this paper we characterize the strong product of two graphs $G_1\boxtimes G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the strong product graph $G_1\boxtimes G_2$ is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between $\delta (G_1\boxtimes G_2)$, $\delta (G_1)$, $\delta (G_2)$ and the diameters of $G_1$ and $G_2$ (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the exact values of the hyperbolicity constant for many strong product graphs.

The general position problem and strong resolving graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1126-1135 ◽  
Author(s):  
Sandi Klavžar ◽  
Ismael G. Yero
Keyword(s):  
Connected Graph ◽  
General Position ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Direct Products ◽  
Strong Product ◽  
Product Graphs ◽  
Complete Bipartite Graphs ◽  
Position Problem

Abstract The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(GSR), where GSR is the strong resolving graph of G, and ω(GSR) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp(G ⊠ H) ≥ gp(G)gp(H), and asked whether the equality holds for arbitrary connected graphs G and H. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

scholarly journals ON NEAR-RING IDEMPOTENTS AND POLYNOMIALS ON DIRECT PRODUCTS OF Ω-GROUPS

2001 ◽  
Vol 44 (2) ◽  
pp. 379-388 ◽  
Author(s):  
Erhard Aichinger
Keyword(s):  
Direct Product ◽  
Subject Classification ◽  
Idempotent Element ◽  
Direct Products ◽  
Primary 16Y30 ◽  
Secondary 08A40

AbstractLet $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

scholarly journals ALGEBRA* VIOLENCE ALGEBRA CRATY

2020 ◽  
pp. 12-23
Author(s):  
Vadym Chuiko ◽  
Valerii Atamanchuk-Angel
Keyword(s):  
Sustainable Development ◽  
Direct Product ◽  
The State ◽  
State System ◽  
The Other ◽  
Philosophical Thought ◽  
Human Culture ◽  
Full Responsibility ◽  
Abstract Notion ◽  
Almost All

Almost all philosophy about the state system has concentrated on the authorities. Any function of the state can be represented as a superposition of the functions of violence / coercion. Ultimately, the state appears to be a kind of plurality of subjects with a definite crater power / coercion / violence operation on it. The algebra of trust on the multiplicity of owners of themselves, endowed with free future, is each of them is only a part of nature, еру carrier of the part of the general human culture, and for their completeness, they have and understand the need for the Other. This is the philosophy of solving political, environmental, and climate challenges not through violent / voluntaristic methods, but by the recognition of sovereign rights and the search for ways to achieve sustainable development. Any cracy / power / coercion / violence must be separated from the models of society, the state. Public agreement is not an agreement with the abstract notion of the state, but an agreement with definite elected people who have gained the trust of those to whom they temporarily render their services. Contract is temporary, limited by period, with obligatory full responsibility of the parties. Scientific novelty. For more than two thousand years, long before Aristotle and Plato, European philosophical thought, reflecting on the structure of society, wanders in the labyrinths of kratia. Modern achievements of mathematics provide an opportunity to build ideal political objects, and a direct product of material and ideal government building. (Example of a trust algebra [4].)

scholarly journals Vertex-Transitive Direct Products of Graphs

10.37236/6999 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Wilfried Imrich
Keyword(s):  
Direct Product ◽  
Direct Products ◽  
Odd Cycle ◽  
Vertex Transitive ◽  
Odd Cycles

It is known that for graphs $A$ and $B$ with odd cycles, the direct product $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. But this is not necessarily true if one of $A$ or $B$ is bipartite, and until now there has been no characterization of such vertex-transitive direct products. We prove that if $A$ and $B$ are both bipartite, or both non-bipartite, then $A\times B$ is vertex-transitive if and only if both $A$ and $B$ are vertex-transitive. Also, if $A$ has an odd cycle and $B$ is bipartite, then $A\times B$ is vertex-transitive if and only if both $A\times K_2$ and $B$ are vertex-transitive.

scholarly journals Deferred On-Line Bipartite Matching

10.37236/5756 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Competitive Ratio ◽  
Bipartite Graphs ◽  
Deterministic Algorithm ◽  
The Other ◽  
Bipartite Matching ◽  
Classical Version ◽  
New Model ◽  
On Line

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$.

Monetary and Financial Stability

2013 ◽  
Author(s):  
Brigitte Granville
Keyword(s):  
Financial Stability ◽  
The Other ◽  
Necessary Condition ◽  
Financial Instability ◽  
Information Flows ◽  
Asset Bubbles ◽  
Root Cause ◽  
Monetary Stability ◽  
Long Run ◽  
The Mean

This chapter examines the relation between monetary and financial stability, looking at possible chains of cause and effect running in both directions between them—from the possibility that an unexpected tightening of monetary policy increases the mean probability of financial system distress, to the general risk of monetary stability being undermined by financial instability. The idea that monetary stability encourages financial instability is controversial. Inflation is often the root cause of financial instability by distorting information flows between lenders and borrowers, leading to asset bubbles and over investment. Most empirical evidence tends to support the view that monetary stability should promote financial stability in the long run and not the other way around. But while monetary stability is a necessary condition for financial stability, it is not a sufficient one. In other words, financial instability can still occur even with the inflation rate under control.

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