Non-crystallographic nets: characterization and first steps towards a classification

2014 ◽  
Vol 70 (3) ◽  
pp. 217-228 ◽  
Author(s):  
Montauban Moreira de Oliveira ◽  
Jean-Guillaume Eon
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Crystal Structures ◽  
Finite Order ◽  
Quotient Graphs ◽  
Order Form ◽  
Crystallographic Symmetry ◽  
Vertex Set ◽  
Barycentric Representation

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroupF(N) of the automorphism group of NC nets (N,T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroupF(N).

Automorphism Groups of Bouwer Graphs

2018 ◽  
Vol 25 (03) ◽  
pp. 493-508 ◽  
Author(s):  
Mimi Zhang ◽  
Jinxin Zhou
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Transitive Graph ◽  
Full Automorphism Group ◽  
Vertex Set ◽  
Edge Transitive ◽  
Positive Integers

Let k, m and n be three positive integers such that 2m ≡ 1 (mod n) and k ≥ 2. The Bouwer graph, which is denoted by B(k, m, n), is the graph with vertex set [Formula: see text] and two vertices being adjacent if they can be written as (a, b) and (a + 1, c), where either c = b or [Formula: see text] differs from [Formula: see text] in exactly one position, say the jth position, where [Formula: see text]. Every B(k, m, n) is a vertex- and edge-transitive graph, and Bouwer proved that B(k, 6, 9) is half-arc-transitive for every k ≥ 2. In 2016, Conder and Žitnik gave the classification of half-arc-transitive Bouwer graphs. In this paper, the full automorphism group of every B(k, m, n) is determined.

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.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

scholarly journals Nets with collisions (unstable nets) and crystal chemistry

2013 ◽  
Vol 69 (6) ◽  
pp. 535-542 ◽  
Author(s):  
Olaf Delgado-Friedrichs ◽  
Stephen T. Hyde ◽  
Shin-Won Mun ◽  
Michael O'Keeffe ◽  
Davide M. Proserpio
Keyword(s):  
Crystal Structures ◽  
Crystal Chemistry ◽  
Barycentric Coordinates ◽  
Real Crystal ◽  
Crystallographic Symmetry

Nets in which different vertices have identical barycentric coordinates (i.e.have collisions) are called unstable. Some such nets have automorphisms that do not correspond to crystallographic symmetries and are called non-crystallographic. Examples are given of nets taken from real crystal structures which have embeddings with crystallographic symmetry in which colliding nodes either are, or are not, topological neighbors (linked) and in which some links coincide. An example is also given of a crystallographic net of exceptional girth (16), which has collisions in barycentric coordinates but which also has embeddings without collisions with the same symmetry. In this last case the collisions are termedunforced.

Classification of the minerals of the graftonite group

2018 ◽  
Vol 82 (6) ◽  
pp. 1301-1306 ◽  
Author(s):  
Frank C. Hawthorne ◽  
Adam Pieczka
Keyword(s):  
Crystal Structures ◽  
Distinct Species ◽  
International Mineralogical Association ◽  
Coordination Numbers ◽  
Mineralogical Association ◽  
Strong Order

ABSTRACTA classification and nomenclature scheme has been approved by the International Mineralogical Association Commission on New Minerals, Nomenclature and Classification for the minerals of the graftonite group. The crystal structures of these minerals have three distinct sites that are occupied by Fe2+, Mn2+and Ca2+. These sites have coordination numbers [8], [5] and [6], and these differences lead to very strong order of Fe2+, Mn2+and Ca2+over these sites. As a result of this strong order, the following compositions have been identified as distinct species: graftonite: FeFe2(PO4)2; graftonite-(Ca): CaFe2(PO4)2; graftonite-(Mn): MnFe2(PO4)2; beusite: MnMn2(PO4)2; and beusite-(Ca): CaMn2(PO4)2.

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Tournaments With a Given Automorphism Group

1964 ◽  
Vol 16 ◽  
pp. 485-489 ◽  
Author(s):  
J. W. Moon
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Oriented Graph ◽  
Not Given ◽  
Oriented Graphs ◽  
Vertex Set ◽  
A Finite Group ◽  
Graph Form

The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.

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