Orbital Graphs Research Articles

On a polynomial bound for the orbital diameter of primitive affine groups

Let VG be a finite primitive affine permutation group, where V is a vector space of dimension d over the prime field F p and G is an irreducible linear group on V. We prove that if p divides |G|, then the diameters of all nondiagonal orbital graphs of VG are at most 9 d 3 . This improves an earlier exponential bound by A. Maróti and the author.

A review on the applications of graph neural networks in materials science at the atomic scale

AbstractIn recent years, interdisciplinary research has become increasingly popular within the scientific community. The fields of materials science and chemistry have also gradually begun to apply the machine learning technology developed by scientists from computer science. Graph neural networks (GNNs) are new machine learning models with powerful feature extraction, relationship inference, and compositional generalization capabilities. These advantages drive researchers to design computational models to accelerate material property prediction and new materials design, dramatically reducing the cost of traditional experimental methods. This review focuses on the principles and applications of the GNNs. The basic concepts and advantages of the GNNs are first introduced and compared to the traditional machine learning and neural networks. Then, the principles and highlights of seven classic GNN models, namely crystal graph convolutional neural networks, iCGCNN, Orbital Graph Convolutional Neural Network, MatErials Graph Network, Global Attention mechanism with Graph Neural Network, Atomistic Line Graph Neural Network, and BonDNet are discussed. Their connections and differences are also summarized. Finally, insights and prospects are provided for the rapid development of GNNs in materials science at the atomic scale.

On the Laplacian Energy of an Orbit Graph of Finite Groups

Let β H denote the orbit graph of a finite group H . Let ζ be the set of commuting elements in H with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in ζ , and two vertices in ζ are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order 2 w .

On Some Topological Indices for the Orbit Graph of Dihedral Groups

Let ΓG be the orbit graph of G, with non-central orbits in the subset of order two commuting elements in G, and the vertices of ΓG connected if they are conjugate. The main objective of this study is to compute several topological indices for the orbit graph of a dihedral group, including the Wiener index, the Zagreb index, the Schultz index, and others. We also develop a relationship between the Wiener index and the other indices for the dihedral group’s orbit graph. Furthermore, their polynomial has been computed as well.

Bounds for the Diameters of Orbital Graphs of Affine Groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer H ≤GL(V ), where the vector space V has dimension d over the prime field, can be bounded in terms of d and log |V|/log |H| only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

On the orbital diameter of groups of diagonal type

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of simple diagonal type. As a consequence we obtain a classification of simple diagonal groups with orbital diameter less than or equal to 4. As part of this, we classify all finite simple groups with covering number and conjugacy width at most 3. We also prove some general bounds on the covering number and conjugacy width of groups of Lie type.

The Orbital Graph of Primitive Group with Socle <i>A</i><sub>7</sub> × <i>A</i><sub>7</sub>

In this paper, we mainly study the orbital graphs of primitive groups with the socle A7 x A7 which acts by diagonal action. Firstly, we calculate the element conjugate classes of A7, then we discuss the stabilizer of two points in A7. Finally, according to the relation between suborbit and orbital, we obtain the orbitals, and determine the orbital graphs.

Schreier graphs of spinal groups

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on [Formula: see text]-regular rooted trees and on their boundaries, viewed as topological spaces or as spaces with measure. They form interesting families of finitely ramified graphs, and we study their combinatorics, their isomorphism classes and their geometric properties, such as growth and the number of ends.

Confined subgroups of Thompson's group $F$ and its embeddings into wobbling groups

We obtain a characterisation of confined subgroups of Thompson's group F . As a result, we deduce that the orbital graph of a point under an action of F has uniformly subexponential growth if and only if this point is fixed by the commutator subgroup. This allows us to prove non-embeddability of F into wobbling groups of graphs with uniformly subexponential growth.

Kepler’s shaping of Mars’ orbit or astronomy on a tabletop

We describe a method that could have been used by Kepler to determine the shape and dimensions of Mars’ orbit around the Sun, starting from observations of its path across the night sky. Though we use Kepler’s work as a motivation, our approach does not intend to suggest that this method was actually used by him. Rather, it is only intended to show beginning undergraduate students what may be discovered about the planets using basic observational data together with simple geometric methods. We employ publicly available Harvard sky patrol (HSP) photographs of Mars at different moments in the Earth’s year to get its ecliptic coordinates, locate them on a Sun-centered coordinate system, and finally plot them on a sheet of paper. From this orbital graph, we obtain the shape and dimensions of Mars’ orbit around the Sun, using the radius of the Earth’s orbit to set the scale. By generalizing this orbital shape to the other planets in the Solar System, students may be led to conclude Kepler’s first and second laws. Other features of the scaled-down Martian orbit may be directly measured from the paper orbit. To obtain the actual values for such features, we only need to scale them up. The quantities thus determined agree well (within a surprising 5%) with those reported by NASA.

Topological and Electronic Properties of Chlorine-Substituents on the α-Diimine Ni-based Catalysts

The effect of substituent nature and position on a series of late transition metal catalysts based on nickel (A–D) was investigated practically and theoretically. Catalyst A bearing isopropyl groups on the ortho position showed the highest activity while for the ortho and para chlorine substituted catalysts B–D insignificant activities were observed. These experimental results were confirmed by theoretical study on the (pre)catalysts. Based on that, thermodynamics, atomic/molecular effective parameters such as bond distances, bond angles, band gap or chemical hardness, charge of Mulliken on Ni center, electronic chemical potential, global electrophilicity index and also activation energy of the pre-catalyst through the alkylation were calculated. To pursue the steric and electronic effects of substituent on the axial and equatorial sites of active center, electronic density, electron location function, localized orbital locator graphs as topological properties were studied. As a result, complex A with lower activation energy for alkylation, higher stability, greater Mulliken charge on Ni center and high electronic density around the active center was found as a highly active catalyst. In addition, polymerization conditions such as polymerization temperature and ethylene pressure showed significant effect on the catalysts productivity and branching density of the produced polyethylene.

Orbital graph convolutional neural network for material property prediction

Material representations that are compatible with machine learning models play a key role in developing models that exhibit high accuracy for property prediction. Atomic orbital interactions are one of the important factors that govern the properties of crystalline materials, from which the local chemical environments of atoms is inferred. Therefore, to develop robust machine learningmodels for material properties prediction, it is imperative to include features representing such chemical attributes. Here, we propose the Orbital Graph Convolutional Neural Network (OGCNN), a crystal graph convolutional neural network framework that includes atomic orbital interaction features that learns material properties in a robust way. In addition, we embedded an encoder-decoder network into the OGCNN enabling it to learn important features among basic atomic (elemental features), orbital-orbital interactions, and topological features. We examined the performance of this model on a broad range of crystalline material data to predict different properties. We benchmarked the performance of the OGCNN model with that of: 1) the crystal graph convolutional neural network (CGCNN), 2) other state-of-the-art descriptors for material representations including Many-body Tensor Representation (MBTR) and the Smooth Overlap of Atomic Positions (SOAP), and 3) other conventional regression machine learning algorithms where different crystal featurization methods have been used. We find that OGCNN significantly outperforms them. The OGCNN model with high predictive accuracy can be used to discover new materials among the immense phase and compound spaces of materials

Stabilizer Graph and Orbit Graph of Finite Groups

Stabilizer Graph and Orbit Graph of Finite Groups

Cores of cubelike graphs

A graph is cubelike if it is a Cayley graph for some elementary abelian 2-group Z2n. The core of a graph is its smallest subgraph to which it admits a homomorphism. More than ten years ago, Nešetřil and Šámal (2008) asked whether the core of a cubelike graph is cubelike, but since then very little progress has been made towards resolving the question. Here we investigate the structure of the core of a cubelike graph, deducing a variety of structural, spectral and group-theoretical properties that the core ”inherits” from the host cubelike graph. These properties constrain the structure of the core quite severely — even if the core of a cubelike graph is not actually cubelike, it must bear a very close resemblance to a cubelike graph. Moreover we prove the much stronger result that not only are these properties inherited by the core of a cubelike graph, but also by the orbital graphs of the core. Even though the core and its orbital graphs look very much like cubelike graphs, we are unable to show that this is sufficient to characterize cubelike graphs. However, our results are strong enough to eliminate all non-cubelike graphs on up to 32 vertices as potential cores of cubelike graphs (of any size). Thus, if one exists at all, a cubelike graph with a non-cubelike core has at least 128 vertices and its core has at least 64 vertices.

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

AbstractThe Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper.

Related graphs of the conjugacy classes of a 3-generator 5-group

The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of which is of size two where and commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.

Using geometric morphometrics for the genetics analysis of shape and size of lactation curves in Israeli first-parity Holstein cattle

Our objective was to combine the methods of geometric morphometrics and multivariate quantitative genetics to genetic evaluation of the size and shape of lactation curves of milk of 3,492 Israeli first-parity Holstein cattle. Lactation records were treated as morphological data, for which 2 different lactation shape functions were evaluated, one depicted by a line graph and the other by an orbital graph. The lactation curves from both shape functions were represented by 2-dimensional Cartesian landmark coordinates. The 2 sets of landmarks were then analyzed individually for each shape function with geometric morphometrics to separate variation into components of size and shape. The analysis yielded 2 size measures and 2 sets of shape variables, and they were the inputs to estimate variance components using the MTC REML individual animal model program. Variance components were also estimated for the 305-d lactation production as a reference. Shape variables showed negligible correlation with 305-d production, providing evidence of size and shape of lactation curve as separate characters. The size measure derived from the orbital-depicted lactation curve had equal heritability (0.39 ± 0.01; ± standard error) and complete genetic and environmental correlations with 305-d production, whereas the size measure derived from the line-depicted lactation curve showed low heritability (0.09 ± 0.01) and environmental correlation (0.02 ± 0.004) and relative high genetic correlation with 305-d production (0.48 ± 0.04). This may validate both the orbital graph to depict lactation records and the use of geometric morphometrics to split variation of lactation curve into size and shape components. The maximal heritability for shape of lactation curve was 0.55 for orbital- and 0.56 for line-depicted lactation curves. The respective patterns of variations were visualized as shape changes from the mean shape in the data set. Geometric morphometrics are well grounded within the theory of shape analysis and can be paired with conventional methods in the field to characterize the patterns of phenotypic and genetic variation of shape and size of lactation curve in dairy cattle.

New refiners for permutation group search

Partition backtrack is the current generic state of the art algorithm to search for subgroups of a given permutation group. We describe an improvement of partition backtrack for set stabilizers and intersections of subgroups by using orbital graphs. With extensive experiments we demonstrate that our methods improve performance of partition backtrack – in some cases by several orders of magnitude.

Классификация дистанционно транзитивных графов орбиталов надгрупп группы Джевонса

Группа Джевонса является группой изометрий метрики Хемминга на $n$-мерном векторном пространстве $V_{n}$ над полем $GF(2)$. Она порождена группой сдвигов пространства $V_{n}$ и группой подстановочных $(n \times n)$-матриц над полем $GF(2)$. Ранее авторы провели классификацию подметрик метрики Хемминга на ${V_n}$ для $n \geqslant 4$, а также всех надгрупп $G$ группы Джевонса, являющихся группами изометрий данных надметрик. В свою очередь, каждая надгруппа $G$ задаeт графы орбиталов, «естественными» метриками которых являются подметрики метрики Хемминга. Среди графов орбиталов надгрупп группы Джевонса авторами были выявлены все дистанционно транзитивные графы. В данной работе проводится классификация дистанционно транзитивных графов орбиталов надгрупп группы Джевонса. В том числе показано, что некоторые графы изоморфны следующим графам: полному графу ${K_{{2^n}}}$, полному двудольному графу ${K_{{2^{n - 1}}{{,2}^{n - 1}}}}$, половинному $(n + 1)$-кубу, сложенному $(n + 1)$-кубу, графам знакопеременных форм, графу Тейлора, графу Адамара, графам инцидентности блок-схем.