Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes

2017 ◽  
Vol 73 (3) ◽  
pp. 238-245
Author(s):  
Jean-Guillaume Eon
Keyword(s):  
Topological Analysis ◽  
Modular Structure ◽  
Quotient Graph ◽  
Quotient Graphs ◽  
Partial Symmetry ◽  
Symmetry Operations

The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral.(2016),28, 189–203], has been performed on the respective labelled quotient graphs (LQGs). It is shown that the structure and maximum symmetry of the module,i.e.its layer group, can be determined directly from the LQG. Partial symmetry operations between different modules have been associated with automorphisms of the quotient graph that may not be consistent with net voltages over the respective cycles. These operations have been shown to generate the pyroxene groupoid structure.

Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings

2016 ◽  
Vol 72 (3) ◽  
pp. 268-293 ◽  
Author(s):  
Jean-Guillaume Eon
Keyword(s):  
Crystal Structures ◽  
Direct Analysis ◽  
Building Blocks ◽  
Quotient Graph ◽  
Quotient Graphs ◽  
Edge Contraction ◽  
Topological Features ◽  
Structure Decomposition ◽  
Bond Topology ◽  
Ring Analysis

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is thecombinatorialorbond topologyof the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal. Since periodic nets can be faithfully represented by their labelled quotient graphs, it may be inferred that their topological features can be recovered by a direct analysis of the labelled quotient graph. Evidence is given for ring analysis and structure decomposition into building units and building networks. An algebraic treatment is developed for ring analysis and thoroughly applied to a description of coesite. Building units can be finite or infinite, corresponding to 1-, 2- or even 3-periodic subnets. The list of infinite units includes linear chains or sheets of corner- or edge-sharing polyhedra. Decomposing periodic nets into their building units relies on graph-theoretical methods classified assurgery techniques. The most relevant operations are edge subdivision, vertex identification, edge contraction and decoration. Instead, these operations can be performed on labelled quotient graphs, evidencing in almost a mechanical way the nature and connection mode of building units in the derived net. Various examples are discussed, ranging from finite building blocks to 3-periodic subnets. Among others, the structures of strontium oxychloride, spinel, lithiophilite and garnet are addressed.

Topological transformations performed on nets via their quotient graphs

2006 ◽  
Vol 221 (1) ◽  
Author(s):  
Jean-Guillaume Eon
Keyword(s):  
Quotient Graph ◽  
Quotient Graphs ◽  
Edge Transitive ◽  
Topological Transformations

AbstractTopological transformations in nets resulting from the insertion or deletion of edges or vertices are analyzed through the analogous operations performed on their quotient graphs. The role of strong rings and cages of the net is emphasized. It is shown that closed trails of the oriented quotient graph define the topology of 3-periodic nets derived from regular, vertex and edge transitive, 4-periodic minimal nets.

scholarly journals STUDY OF CONTINUOUS-TIME QUANTUM WALKS ON QUOTIENT GRAPHS VIA QUANTUM PROBABILITY THEORY

2008 ◽  
Vol 06 (04) ◽  
pp. 945-957 ◽  
Author(s):  
S. SALIMI
Keyword(s):  
Probability Theory ◽  
Adjacency Matrix ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Original Graph ◽  
Quotient Graph ◽  
Quotient Graphs ◽  
Time Quantum

In the present paper, we study the continuous-time quantum walk on quotient graphs. On such graphs, there is a straightforward reduction of the problem to a subspace that can be considerably smaller than the original one. Along the lines of reductions, by using the idea of calculation of the probability amplitudes for continuous-time quantum walk in terms of the spectral distribution associated with the adjacency matrix of graphs [Jafarizadeh and Salimi (Ann. Phys.322 (2007))], we show that the continuous-time quantum walk on original graph Γ induces a continuous-time quantum walk on quotient graph ΓH. Finally, for example, we investigate the continuous-time quantum walk on some quotient Cayley graphs.

scholarly journals Bounds on the Distance Energy and the Distance Estrada Index of Strongly Quotient Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Ş. Burcu Bozkurt ◽  
Chandrashekara Adiga ◽  
Durmuş Bozkurt
Keyword(s):  
Quotient Graph ◽  
Quotient Graphs ◽  
Estrada Index

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Güngör and Bozkurt (2009) and Zaferani (2008).

scholarly journals Isotopy classes for 3-periodic net embeddings

2020 ◽  
Vol 76 (3) ◽  
pp. 275-301 ◽  
Author(s):  
Stephen C. Power ◽  
Igor A. Baburin ◽  
Davide M. Proserpio
Keyword(s):  
Repeat Unit ◽  
Connected Components ◽  
Quotient Graph ◽  
Parallel Orientation ◽  
Maximal Symmetry ◽  
Single Vertex ◽  
Quotient Graphs ◽  
Linear Graph ◽  
Dimension Type

Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. Periodic isotopy classifications are obtained for various families of embedded nets with small quotient graphs. The 25 periodic isotopy classes of depth-1 embedded nets with a single-vertex quotient graph are enumerated. Additionally, a classification is given of embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, as well as demonstrations of their maximal symmetry periodic isotopes. The methodology of linear graph knots on the flat 3-torus [0,1)3 is introduced. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.

Net topologies, space groups, and crystal phases

2006 ◽  
Vol 221 (12) ◽  
Author(s):  
Georg Thimm ◽  
Björn Winkler
Keyword(s):  
Crystal Structures ◽  
Space Group ◽  
Space Groups ◽  
Quotient Graphs ◽  
Crystal Phases ◽  
Symmetry Operations

Quotient graphs and nets — the graph theore tical correspondences of cells and crystal structures — are reintroduced independent from crystal structures. Based on this, the issue of iso- and automorphism of nets, the graph theoretical equivalent of symmetry operations, is closely examined. A result, it is shown that the topology of a net (that is the bonds in a crystal) constrains severely the symmetry of the embedding (that is the crystal), and in the case of connected nets the space group except for the setting. Several examples are studied and conclusions on phases are drawn (pseudo-cubic FeS

DIMENSIONING OF THE FRONTAL SUSPENSION SYSTEM OF THE DOUBLE-A TYPE THROUGH TOPOLOGICAL ANALYSIS.

2017 ◽  
Author(s):  
Tiago Ferreira ◽  
Thiago Moreira ◽  
Gustavo Melchiades ◽  
Lucas Ferreira ◽  
Diógenes Sena de França e Silva
Keyword(s):  
Topological Analysis ◽  
Suspension System
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