scholarly journals Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1418
Author(s):  
Serge Lawrencenko ◽  
Abdulkarim M. Magomedov
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Simplicial Complexes ◽  
Fixed Number ◽  
Type Ii ◽  
Computing Technology ◽  
Quaternion Group ◽  
Labeled Graph ◽  
First Time ◽  
Abstract Simplicial Complexes

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P′(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G=K2,2,2,2, in which specific case P(x)=x31. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q8 with the three imaginary quaternions i, j, k as generators.

scholarly journals Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K_{2,2,2,2}

2021 ◽  
Author(s):  
Serge Lawrencenko ◽  
Abdulkarim Magomedov
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Simplicial Complexes ◽  
Fixed Number ◽  
Type Ii ◽  
Computing Technology ◽  
Quaternion Group ◽  
Labeled Graph ◽  
First Time ◽  
Abstract Simplicial Complexes

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P'(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G = K_{2,2,2,2}, in which specific case P(x) = x^{31}. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (twelve in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q_8 with the three imaginary quaternions i, j, k as generators.

scholarly journals Generating the Triangulations of the Torus with the Vertex-Labeled Graph K_{2,2,2,2}

2021 ◽  
Author(s):  
Serge Lawrencenko ◽  
Abdulkarim M. Magomedov
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Simplicial Complexes ◽  
Fixed Number ◽  
Type Ii ◽  
Computing Technology ◽  
Quaternion Group ◽  
Labeled Graph ◽  
First Time ◽  
Abstract Simplicial Complexes

Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following useful properties: (I) P(1) is equal to the number of unlabeled complexes (of a given type), (II) the derivative P'(1) is equal to the number of non-trivial automorphisms over all unlabeled complexes, (III) the integral of P(x) from 0 to 1 is equal to the number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees, and then is applied to triangulations of the torus with the vertex-labeled complete four-partite graph G = K_{2,2,2,2}, in which specific case P(x) = x^{31}. The graph G embeds on the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (twelve in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q_8 with three quaternions, i, j, k, as generators.

Ionized AGN outflows are less powerful than assumed: A multi-wavelength census of outflows in type II AGN

2019 ◽  
Vol 15 (S356) ◽  
pp. 225-225
Author(s):  
Dalya Baron
Keyword(s):  
Optical Emission ◽  
Infrared Emission ◽  
Type Ii ◽  
Ionized Gas ◽  
Kinetic Coupling ◽  
Mass Outflow ◽  
Multi Wavelength ◽  
Novel Method ◽  
First Time ◽  
Ionization Parameter

AbstractIn this talk I will show that multi-wavelength observations can provide novel constraints on the properties of ionized gas outflows in AGN. I will present evidence that the infrared emission in active galaxies includes a contribution from dust which is mixed with the outflow and is heated by the AGN. We detect this infrared component in thousands of AGN for the first time, and use it to constrain the outflow location. By combining this with optical emission lines, we constrain the mass outflow rates and energetics in a sample of 234 type II AGN, the largest such sample to date. The key ingredient of our new outflow measurements is a novel method to estimate the electron density using the ionization parameter and location of the flow. The inferred electron densities, ∼104.5 cm−3, are two orders of magnitude larger than found in most other cases of ionized outflows. We argue that the discrepancy is due to the fact that the commonly-used [SII]-based method underestimates the true density by a large factor. As a result, the inferred mass outflow rates and kinetic coupling efficiencies are 1–2 orders of magnitude lower than previous estimates, and 3–4 orders of magnitude lower than the typical requirement in hydrodynamic cosmological simulations. These results have significant implications for the relative importance of ionized outflows feedback in this population.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

scholarly journals Cubic arc-transitive Cayley graphs on Frobenius groups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850126 ◽  
Author(s):  
Hailin Liu ◽  
Lei Wang
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Frobenius Groups

A Cayley graph [Formula: see text] is called arc-transitive if its automorphism group [Formula: see text] is transitive on the set of arcs in [Formula: see text]. In this paper, we give a characterization of cubic arc-transitive Cayley graphs on a class of Frobenius groups.

Transport Properties of Type II Sodium-Silicon Clathrates

2005 ◽  
Vol 886 ◽  
Author(s):  
Matt Beekman ◽  
Jan Grkyo ◽  
George S. Nolas
Keyword(s):  
Thermal Conductivity ◽  
Sodium Content ◽  
Type I ◽  
Type Ii ◽  
Conductivity Measurements ◽  
Scanning Calorimetry ◽  
Resistivity Measurements ◽  
Silicon Clathrates ◽  
Covalently Bonded ◽  
First Time

ABSTRACTWe have synthesized the type II silicon clathrates Na1Si136 and Na8Si136, and report on the electrical and thermal transport in these materials. The crystal structure consists of a covalently bonded silicon framework in which sodium guest atoms are encapsulated inside the silicon host framework. Differential scanning calorimetry measurements show the compounds decompose above 600°C to diamond-structure silicon. Temperature dependant electrical resistivity measurements show the specimens to have an insulating character, with magnitudes that decrease with increasing sodium content. For the first time, thermal conductivity measurements on type II sodium-silicon clathrates are presented. The thermal conductivity is very low for both specimens, and for Na8Si136 exhibits a clear dip in the range from 50 to 70 K. These data suggest that the “rattling” behavior observed in type I clathrates may also be present in type II clathrates.

scholarly journals Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer

2019 ◽  
Vol 17 (1) ◽  
pp. 513-518
Author(s):  
Hailin Liu
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Frobenius Groups ◽  
Full Automorphism Group

Abstract A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.

RF-Sputtered MoS2 Film Morphology and the Imperfection Nucleation Model

2003 ◽  
Vol 10 (02n03) ◽  
pp. 443-448 ◽  
Author(s):  
G. Jakovidis ◽  
I. M. Jamieson ◽  
A. Singh
Keyword(s):  
Substrate Temperature ◽  
Film Formation ◽  
High Substrate Temperature ◽  
Film Morphology ◽  
Type Ii ◽  
Rf Power ◽  
Oriented Films ◽  
Nucleation Model ◽  
Mos2 Film ◽  
First Time

RF-sputtered MoS2 films revealing the characteristics of bulk type II orientation on GaAs are reported for the first time. It is found that RF power and temperature have a pronounced effect on film morphology. Type II bulk-oriented films are obtained with a combination of low RF power and high substrate temperature. The results on GaAs are successfully interpreted within the context of an extension to the imperfection nucleation model of film formation. Films deposited on glass display an unusual morphology consisting of two distinct phases. Such phases may be related to the presence of sodium in the glass that leads to chemical texturing via a sodium thio-molybdate phase.

A 2-ARC TRANSITIVE PENTAVALENT CAYLEY GRAPH OF 

2016 ◽  
Vol 93 (3) ◽  
pp. 441-446 ◽  
Author(s):  
BO LING ◽  
BEN GONG LOU
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Discrete Math ◽  
Simple Groups ◽  
Alternating Group ◽  
Full Automorphism Group ◽  
Symmetric Graphs

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

scholarly journals Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

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