Orbital Graphs Research Articles

Strongly aperiodic subshifts on surface groups

We give strongly aperiodic subshifts of nite type on every hyperbolic surface group; more generally, for each pair of expansive primitive symbolic substitution systems with incommensurate growth rates, we construct strongly aperiodic subshifts of finite type on their orbit graphs.

О классификации дистанционно-транзитивных графов орбиталов надгрупп группы Джевонса

О классификации дистанционно-транзитивных графов орбиталов надгрупп группы Джевонса

Orbital diameters of the symmetric and alternating groups

For a primitive group G acting on a finite set Omega , we define the orbital diameter to be the maximum of the diameters of all orbital graphs of G. In this paper, we study the orbital diameters of symmetric and alternating groups. We give necessary numerical conditions for the orbital diameter to be bounded by some constant c and give precise descriptions of the actions for which the orbital diameter is bounded by 5. For each primitive action, we also either determine all orbital graphs of diameter 2 or give descriptions of infinite families of orbital graphs of diameter 2.

THE ORBIT GRAPH OF METACYCLIC 2-GROUPS OF NEGATIVE TYPE

In this paper, let G be a metacyclic 2-group of negative type of class two and class at least three. Let be the set of all subsets of all commuting elements of size two in the form of (a,b), where a and b commute and |a| = |b| = 2. The orbit graph is a graph whose vertices are non-central orbits under group action of G on . In this paper, the orbit graph of metacyclic 2-groups of negative type of nilpotency class two and class at least three is determined. Some graph properties are also provided.

Codimension one connectedness of the graph of associated varieties

Let $ \pi $ be an irreducible Harish-Chandra $ (\mathfrak{g}, K) $-module, and denote its associated variety by $ \mathcal{AV}(\pi) $. If $ \mathcal{AV}(\pi) $ is reducible, then each irreducible component must contain codimension one boundary component. Thus we are interested in the codimension one adjacency of nilpotent orbits for a symmetric pair $ (G, K) $. We define the notion of orbit graph and associated graph for $ \pi $, and study its structure for classical symmetric pairs; number of vertices, edges, connected components, etc. As a result, we prove that the orbit graph is connected for even nilpotent orbits. Finally, for indefinite unitary group $ U(p, q) $, we prove that for each connected component of the orbit graph $ \Gamma_K(\mathcal{O}^G_\lambda) $ thus defined, there is an irreducible Harish-Chandra module $ \pi $ whose associated graph is exactly equal to the connected component.

THE ORBIT GRAPH OF FINITE NON-ABELIAN GROUPS

Let G be a finite non-abelian group and let Ω be a set of elements of G. Let A be the set of commuting elements in Ω, i.e A = {v ∈ Ω : vg = gv, g ∈ G}. In this paper, we extend the work on conjugate graph by defining a new graph called the orbit graph, denoted as ΓGΩ. The vertices of ΓGΩ are non central elements in Ω but not in A in which two vertices of ΓGΩ are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined.

The Orbit Graph of Finite p-Groups and Groups of Order pq

In this paper G denotes a finite p-group and Γ denotes a simple undirected graph. The orbit graph is a graph whose vertices are non-central orbit under group action of G on a set. Two vertices v 1 and v 2 are adjacent in the graph if v 1 g = v 2 where v 1 ,v 2 ∈Ω and g∈G. In this paper, the orbit graph of some finite p-groups and group of order pq, where p and q are relatively prime, is found. The orbit graph is determined for the group in the case that a group acts regularly on itself, acts on itself by conjugation, and acts on a set. Besides, some graph properties are found.

Applications of graphs related to the probability that an element of finite metacyclic 2-group fixes a set

In this paper, G denotes a metacyclic 2-group of positive type of nilpotency class at least three and Ω is the set of all subsets of commuting elements of G of size two in the form of (a, b), where a and b commute and lcm(|a|, |b|) = 2. The probability that a group element of G fixes a set is one of the generalizations of the commutativity degree that has been recently introduced. In this paper, the probability that an element of fixes a set for metacyclic 2-groups of positive type of nilpotency class at least three is computed. The results obtained are then applied to graph theory, more precisely to the orbit graph and generalized conjugacy class graph.

Theoretical Study of N749 Dyes Anchoring on the (TiO2)28 Surface in DSSCs and Their Electronic Absorption Properties

We have performed calculations on the panchromatic N749 dyes adsorbed on the (TiO2)28 surface. N749 is a prototypical form of Ru(II) complexes for dye sensitized solar cells (DSSCs), which possesses a terpyridine tridentate ligand bearing four different protonation states (0, 1, 2, or 3 carboxylic protons). Depending on the type of proton bonding interaction (protonated and deprotonated), seven N749/(TiO2)28 surface models (N749-0H/(TiO2)28, N749-1H-P/(TiO2)28, N749-1H-DP/(TiO2)28, N749-2H-P/(TiO2)28, N749-2H-DP/(TiO2)28, N749-3H-P/(TiO2)28, and N749-3H-DP/(TiO2)28) have been applied in this study for the geometry optimization, frontier molecular orbital level diagrams, and calculated absorption spectra. The moderate surface area of the (TiO2)28 cluster is suitable for N749 dyes adsorbing behaviors so that all calculations can be performed using the Gaussian 09 program package. We have carefully examined these seven N749/(TiO2)28 assemblies that could influence the DSSC device performance. The calculated absorption spectra of these seven various N749/(TiO2)28 models are in good agreement with the experimental results by Hagfeldt et al. (J. Phys. Chem. B2002, 106, 12693–12704) with onset ranging from the visible to near-IR region. The combination of the adsorption energy onto TiO2 and calculated absorption spectra (cf. the experimental results) concludes that the deprotonated dyes constitute the most favorable and dominant structure in the DSSC devices. The frontier molecular orbital graphs indicate that the electron charge distributions of all HOMOs are located at the N749 dyes, while LUMOs are localized at the (TiO2)28 surface or delocalized at the interfacial regions of N749/(TiO2)28. The corresponding transitions are thus more like a type of optical electron transfer, injecting the electron to the interfacial TiO2.

A discussion on the electronic structure of arachno-borane B4H10

The electronic structure of the butterfly-shaped arachno-borane B4H10 is rationalized by the octet rule in comparison with its butterfly-shaped isoelectronic species C4H6. The computational results of geometry and molecular orbital (MO) graphs are compared between the two isoelectronic species. And the point group method is employed to describe the MO correlation between the hydrogen-bridged molecule B4H10 and the hypothetical molecule B4H6 4−. The results indicate that the electronic structure of arachno-borane B4H10 can be rationalized by the octet rule constrast with its isoelectronic hydrocarbon counterpart C4H6. And the MO description on the electronic structure of hydrogen-bridge bonds in B4H10 is provided through the point group treatment of a hypothetical reaction. All the computational work are performed at MP2/6-31++G(d, p) level.

Primitive permutation groups of bounded orbital diameter

We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to describing families of finite permutation groups such that every ultraproduct of the family is primitive. A key result is that, in the almost simple case with socle of fixed Lie rank, apart from very specific cases, there is such a diameter bound. This is proved using recent results on the model theory of pseudofinite fields and difference fields.

On the multiplicity-free actions of the sporadic simple groups

We introduce a database containing the character tables of the endomorphism rings of the multiplicity-free permutation modules of the sporadic simple groups, their automorphism groups, their Schur covers, and their bicyclic extensions. We describe the techniques used to compile the data, and present a couple of applications to orbital graphs.

On the Edge-forwarding Indices of Frobenius Graphs

A G-Frobenius graph Γ, as defined by Fang, Li, and Praeger, is a connected orbital graph of a Frobenius group G = K ⋊ H with Frobenius kernel K and Frobenius complement H. Γ is also shown to be a Cayley graph, Γ = Cay(K, S) for K and some subset S of the group K. On the other hand, a network N with a routing function R, written as (N,R), is an undirected graph N together with a routing R which consists of a collection of simple paths connecting every pair of vertices in the graph. The edge-forwarding index π(N) of a network (N,R), defined by Heydemann, Meyer, and Sotteau, is a parameter to describe the maximum load of edges of N. In this paper, we study the edge-forwarding indices of Frobenius graphs. In particular, we obtain the edge-forwarding index of a G-Frobenius graph Γ with rank(G) ≤ 50.

The Simple Connectedness of a Tame Weakly Shod Algebra#

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H1(A) with coefficients in A A A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H1(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸˜, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸˜.

On primitive permutation groups with small suborbits and their orbital graphs

In this paper, we study finite primitive permutation groups with a small suborbit. Based on the classification result of Quirin [Math. Z. 122 (1971) 267] and Wang [Comm. Algebra 20 (1992) 889], we first produce a precise list of primitive permutation groups with a suborbit of length 4. In particular, we show that there exist no examples of such groups with the point stabiliser of order 2 43 6, clarifying an uncertain question (since 1970s). Then we analyse the orbital graphs of primitive permutation groups with a suborbit of length 3 or of length 4. We obtain a complete classification of vertex-primitive arc-transitive graphs of valency 3 and valency 4, and we prove that there exist no vertex-primitive half-arc-transitive graphs of valency less than 10. Finally, we construct vertex-primitive half-arc-transitive graphs of valency 2 k for infinitely many integers k, with 14 as the smallest valency.

Symmetric group approach to spin-dependent CI

A general algorithm of configuration interaction (CI) method for spin-dependent Hamiltonians in a basis of spin-adapted N-electron configuration state functions is formulated. The method is derived from the symmetric group approach (SGA) to a many-electron problem and retains the simple structure of the spin-independent CI. The transition from an algorithm for a spin-independent Hamiltonian to a general spin-dependent case results in a replacement of the blocks of irreducible representation matrices of the symmetric group (representing permutation operators in the spin-adapted N-electron basis) by larger matrices representing the pertinent spin operators. The structure of the algorithm, in particular the set of loops in the orbital graph leading to non-zero contributions to the CI matrix elements, remains exactly the same as in the spin-independent case.

Novel point charge models: reliable instruments for­molecular electrostatic

AbstractA novel method to compute atomic charge distribution is proposed. The starting data for the computation are the topology of a molecule and electronegativities of either constituent atoms or atomic orbitals. The topology of a molecule is represented by a molecular graph or, in a more advanced model, by an orbital graph that reflects an orbital constitution of atoms. The distinctive feature of the method is that the starting atomic (orbital) electronegativities equilibrate in the same fashion that electrical potentials do in the nodes of a closed electrical network. Thus, the well‐developed formalism of the theory of electrical circuits was applied for computation of partial atomic charges. In addition, the problem of parametrization of halogens was solved using a specially developed procedures. Comparison of the obtained charges with those produced by means of various computational schemes [Partial Equalization of Orbital Electronegativities (PEOE), Mulliken population analysis, natural population analysis, Bader's AIM, method of generalized atomic polar tensors (GAPT) and an electrostatic potential‐based method, CHELPG] and with some molecular spectral characteristics proves that both models, but especially the orbital graph charge model, provide fast, convenient and reliable methods to calculate theoretically justifiable atomic charges for a variety of chemical species. The developed charge models are well suited for application in many areas of molecular modeling and QSAR/QSPR studies. Copyright © 2001 John Wiley & Sons, Ltd.

Prediction of heteroaromatic amine mutagenicity by means of correlation weighting of atomic orbital graphs of local invariants

Graphs of atomic orbitals (GAOs) have been used to represent molecular structures. Rules by which the labeled hydrogen-filled graphs (LHFGs) were converted into the GAOs are described. The GAO is an attempt at taking into account the structures of atoms (i.e. atomic orbitals, such as 1s 1, 2p 2, 3d 10) for QSPR/QSAR analyses. As a method of mutagenicity modeling, optimization of correlation weights of local invariants (OCWLI) of the LHFGs and the GAOs has been used. Statistical characteristics of such models based on the OCWLI of GAO are better than those based on the OCWLI of the LHFGs.

Characteristic polynomials of graphs having a semifree action

J.H. Kwak and J. Lee (Linear and Multilinear Algebra 32 (1992) 61–73) computed the characteristic polynomial of a finite graph G having an abelian automorphism group which acts freely on G. For a finite weighted symmetric pseudograph G having an abelian automorphism group which acts semifreely on G, K. Wang (Linear Algebra Appl. 51 (1983) 121–125) showed that the characteristic polynomial of G is factorized into a product of a polynomial associated to the orbit graph and a polynomial associated to the free part of the action. But he did not explicitly compute the characteristic polynomial of such a graph G. In this paper, we introduce a new method to construct a finite pseudograph G having an automorphism group which acts semifreely on G, and obtain an explicit formula to compute the characteristic polynomial of such a graph by using the construction method.

Applications of the molecular orbital graph theory (XI)

The tree graph method of evaluating the local molecular moment is proposed. By applying this method and the molecular moment formula of π-electron energy in the molecular orbital graph theory, a topological method of using the molecular moment to judge the relatively reactive point of even AH is achieved.