Vertex Symmetry Research Articles

A group theoretic approach to model comparison with simplicial representations

The complexity of biological systems, and the increasingly large amount of associated experimental data, necessitates that we develop mathematical models to further our understanding of these systems. Because biological systems are generally not well understood, most mathematical models of these systems are based on experimental data, resulting in a seemingly heterogeneous collection of models that ostensibly represent the same system. To understand the system we therefore need to understand how the different models are related to each other, with a view to obtaining a unified mathematical description. This goal is complicated by the fact that a number of distinct mathematical formalisms may be employed to represent the same system, making direct comparison of the models very difficult. A methodology for comparing mathematical models based on their underlying conceptual structure is therefore required. In previous work we developed an appropriate framework for model comparison where we represent models, specifically the conceptual structure of the models, as labelled simplicial complexes and compare them with the two general methodologies of comparison by distance and comparison by equivalence. In this article we continue the development of our model comparison methodology in two directions. First, we present a rigorous and automatable methodology for the core process of comparison by equivalence, namely determining the vertices in a simplicial representation, corresponding to model components, that are conceptually related and the identification of these vertices via simplicial operations. Our methodology is based on considerations of vertex symmetry in the simplicial representation, for which we develop the required mathematical theory of group actions on simplicial complexes. This methodology greatly simplifies and expedites the process of determining model equivalence. Second, we provide an alternative mathematical framework for our model-comparison methodology by representing models as groups, which allows for the direct application of group-theoretic techniques within our model-comparison methodology.

Linear time algorithms for finding independent spanning trees on pyramid networks

The use of independent spanning trees (ISTs) has scientific applications in fault-tolerant requirement in network protocols and secure message distributions. Most of the designs of ISTs are for those interconnection networks with vertex symmetric property, implying that one can find ISTs rooted on a designated vertex, and, by the vertex symmetry property of the given network, hence have solved the ISTs problem on any arbitrary vertex. The existence of asymmetry makes the ISTs problem even harder than its symmetric counterpart. Cheriyan and Maheshwari (J Algorithms 9:507–537, 1988) showed that, for any 3-connected graph, 3-ISTs rooted at any vertex can be found in O(|V||E|) time. In this paper, we propose linear time algorithms that solved 3-ISTs rooted at an arbitrary vertex of pyramid networks.

Supramolecular Tessellations at Surfaces by Vertex Design

Assembly and tessellation of organic species at surfaces are important for the design of advanced materials, particularly for the development of spontaneous self-assemblies of supramolecular systems of increasing complexity. However, there are few examples where the ability to steer the system between supramolecular tessellations has been achieved. Here, we demonstrate a series of steps to reduce and then restore molecular symmetry; those variations impact vertex symmetry and thus generate a series of tessellations that reflect the molecular symmetry. We deposit 4,4'-dihydroxybiphenyl on the Ag(111) surface, then anneal at specific temperatures to achieve stepwise dehydrogenation of the terminal hydroxyls. The symmetry of tessellation vertices in the self-assembled structure also changes, as characterized by scanning tunneling microscopy and synchrotron radiation photoemission spectroscopy. This control over vertex geometry and spontaneous tessellation structure extends our understanding of supramolecular design control and advances architectural complexity for the development of functional surfaces.

Bayesian precision and covariance matrix estimation for graphical Gaussian models with edge and vertex symmetries

SUMMARYGraphical Gaussian models with edge and vertex symmetries were introduced by Hojsgaard & Lauritzen (2008), who gave an algorithm for computing the maximum likelihood estimate of the precision matrix for such models. In this paper, we take a Bayesian approach to its estimation. We consider only models with symmetry constraints and which thus form a natural exponential family with the precision matrix as the canonical parameter. We identify the Diaconis–Ylvisaker conjugate prior for these models, develop a scheme to sample from the prior and posterior distributions, and thus obtain estimates of the posterior mean of the precision and covariance matrices. Such a sampling scheme is essential for model selection in coloured graphical Gaussian models. In order to verify the precision of our estimates, we derive an analytic expression for the expected value of the precision matrix when the graph underlying our model is a tree, a complete graph on three vertices, or a decomposable graph on four vertices with various symmetries, and we compare our estimates with the posterior mean of the precision matrix and the expected mean of the coloured graphical Gaussian model, that is, of the covariance matrix. We also verify the accuracy of our estimates on simulated data.

Benchmark Applications of Variations of Multireference Equation of Motion Coupled-Cluster Theory.

In this work, several variations of the multireference equation of motion (MR-EOM) methodology are investigated for the calculation of excitation spectra. These variants of MR-EOM are characterized by the following aspects: (1) the operators included in the sequence of similarity transformations of the molecular electronic Hamiltonian, (2) whether permutational symmetries (i.e., hermitization, vertex symmetry) are imposed on the final elements of the similarity-transformed Hamiltonian, (3) the size of the manifold over which the similarity-transformed Hamiltonian is diagonalized, (4) whether the two-body cumulant is included in the expressions defining the amplitudes and the elements of the transformed Hamiltonian. The MR-EOM methods are benchmarked for the calculation of the excitation energies of a test set of organic molecules. With the availability of reliable benchmark data for this test set, it is possible to gauge the relative accuracy of these approaches. We also further examine a subset of the MR-EOM methods for the calculation of the excitation energies of some transition-metal complexes. These systems prove to be particularly difficult for single-reference coupled-cluster methods.

Broken vertex symmetry and finite zero-point entropy in the artificial square ice ground state

We study degeneracy and entropy in the ground state of artificial square ice. In theoretical models, individual nanomagnets are typically treated as single spins with only two degrees of freedom, leading to a twofold degenerate ground state with intensive entropy and thus no zero-point entropy. Here, we show that the internal degrees of freedom of the nanostructures can result, through edge bending of the magnetization and breaking of local magnetic symmetry at the vertices, in a transition to a highly degenerate ground state with finite zero-point entropy, similar to that of the pyrochlore spin ices. We find that these additional degrees of freedom have observable consequences in the resonant spectrum of the lattice, and predict the occurrence of edge ``melting'' above a critical temperature at which the magnetic symmetry is restored.

Two-band model of Raman scattering on iron pnictide [formula omitted] superconductors

Based on a two-band model, we theoretically study the electronic Raman scattering spectra in both normal and superconducting states of iron-pnictide Ba(Fe1-xCox)2As2 superconductors. In the normal state, due to the match or mismatch of band hybridization and Raman vertex symmetries, overall B2g Raman intensity is found to be much stronger than that of the B1g channel. Besides, in the non-resonant limit, there could exhibit an interband excitation peak at high frequency ω≃7.3t1(6.8t1) in the B1g (B2g) channel with t1 the nearest-neighbor hopping. In the superconducting state, due to the composite effect of Raman vertex, gap symmetry, and Fermi surface topology, both B1g and B2g Raman intensities are dominated by α- (β-) band contribution for the extended s-wave (dx2-y2-wave) pairing, whereas both bands are equally important for the s±-wave pairing. It is shown that both extended s- and s±-wave pairings can lead to a good fitting for the reported B1g data Muschler et al. (2009) [11], while the dx2-y2-wave pairing seems not favorable.

Chiral nature of magnetic monopoles in artificial spin ice

Micromagnetic properties of monopoles in artificial kagome spin ice systems are investigated using numerical simulations. We show that micromagnetics brings additional complexity into the physics of these monopoles that is, by essence, absent in spin models: in addition to a fractionalized classical magnetic charge, monopoles in the artificial kagome ice are chiral at remanence. Our simulations predict that the chirality of these monopoles can be controlled without altering their charge state. This chirality breaks the vertex symmetry and triggers a directional motion of the monopole under an applied magnetic field. Our results also show that the choice of the geometrical features of the lattice can be used to turn on and off this chirality, thus allowing the investigation of chiral and achiral monopoles.

New Approach Construction for Wireless ZigBee Sensor Based on Embedding Pancake Graphs

Wireless Sensor Networks (WSN) based on the IEEE 802.15.4 standard are constantly expanding. Applications like production control, building control are more and more based on WSN because of their energy efficiency, self organizing capacity and protocol flexibility. However, the construction of Cluster-Tree networks based on the beacon mode Pancake graphs is still undefined by the IEEE 802.15.4 standard. In order to enable the construction of such topology, i.e., Beacon Cluster-Tree based on Pancake graphs, we present, in this paper, a new topology construction approach. The Pancake is one of the Cayley graphs that were proposed as alternative to the Hypercube for interconnecting processors in parallel computers. This network offers attractive and desirable properties: Vertex symmetry, small degree and diameter, extensibility, high connectivity, easy routing, regularity of topology, fault-tolerance, and embed ability of other topologies. We present in this work the many-to-one embedding of Multiply-Twisted Hypercube into the Pancake networks with dilation 5 as a new approach for wireless networks. The presented approach is based on the exploitation of RF front-end capabilities in treating multipath signals and, thus, avoiding the introduction of beacon or Super Frames scheduling algorithms. Avoiding the introduction of scheduling algorithms ensures a simple solution that could be easily implemented and executed by ZigBee sensor nodes.

Embedding n-Dimensional Crossed Hypercube into Pancake Graphs

Among Cayley graphs on the symmetric group, the pancake graph is one as a viable interconnection scheme for parallel computers, which has been examined by a number of researchers. The pancake was proposed as alternatives to the hypercube for interconnecting processors in parallel computers. Some good and attractive properties of this interconnection network include: vertex symmetry, small degree, a sub-logarithmic diameter, extendability, and high connectivity (robustness), easy routing, and regularity of topology, fault tolerance, extensibility and embeddability of other topologies. In this paper, we present the many-to-one dilation 5 embedding of n-dimensional crossed hypercube into n-dimensional pancake patients. These predictors, however, need further work to validate reliability.

Hamiltonian connectivity and globally 3∗-connectivity of dual-cube extensive networks

In 2000, Li et al. introduced dual-cube networks, denoted by DC n for n ⩾ 1 , using the hypercube family Q n and showed the vertex symmetry and some fault-tolerant hamiltonian properties of DC n . In this article, we introduce a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN ( G ) . Given any arbitrary graph G , DCEN ( G ) is generated from G using the similar structure of DC n . We show that if G is a nonbipartite and hamiltonian connected graph, then DCEN ( G ) is hamiltonian connected. In addition, if G has the property that for any two distinct vertices u , v of G , there exist three disjoint paths between u and v such that these three paths span the graph G , then DCEN ( G ) preserves the same property. Furthermore, we prove that the similar results hold when G is a bipartite graph.

Graphical Gaussian Models With Edge and Vertex Symmetries

SummaryWe introduce new types of graphical Gaussian models by placing symmetry restrictions on the concentration or correlation matrix. The models can be represented by coloured graphs, where parameters that are associated with edges or vertices of the same colour are restricted to being identical. We study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates. We identify conditions for restrictions on the concentration and correlation matrices being equivalent. This is for example the case when symmetries are generated by permutation of variable labels. For such models a particularly simple maximization of the likelihood function is available.

A Spectral Study of the Manhattan Networks

The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity.

Inference in Graphical Gaussian Models with Edge and Vertex Symmetries with thegRcPackage forR

In this paper we present the R package gRc for statistical inference in graphical Gaussian models in which symmetry restrictions have been imposed on the concentration or partial correlation matrix. The models are represented by coloured graphs where parameters associated with edges or vertices of same colour are restricted to being identical. We describe algorithms for maximum likelihood estimation and discuss model selection issues. The paper illustrates the practical use of the gRc package.

Cover Picture: Toward Large-Scale Alignment of Electrohydrodynamic Patterning of Thin Polymer Films (Adv. Funct. Mater. 15/2006)

By integrating lithography and self-assembly via electrohydrodynamic instabilities, Russel and co-workers are able to guide initially flat polymer films to evolve into periodic arrays of pillars over regions much greater in extent than their natural domain sizes, as detailed on p. 1992. Novel structures that involve a combination of linear ridges and pillars are also produced, mainly as as result of the dynamic merging among preformed pillars. To pattern thin polymer films via electrohydrodynamic instabilities, we design and utilize two different kinds of mask patterns to guide pillars into alignment over regions much greater in extent than their natural domain sizes. First, narrow protruding ridges that intersect to form regular patterns on the mask trigger the growth of pillars beneath. Later, square and triangular packings of pillars develop in the regions enclosed by those ridges, preserving the registry from one domain to the next over a much larger area than within individual domains in unpatterned portions of the mask. Second, small square protrusions that are prealigned into a large regular array on the mask guide the formation of square packings of pillars in domains that conform to the mask, forming a large array of pillars. Novel structures involving a combination of linear ridges and pillars are also produced mainly due to the dynamic merging among preformed pillars. Finally, we find vertex symmetry of the mask pattern is necessary for generating and preserving ordered patterns on the polymer film.

Toward Large-Scale Alignment of Electrohydrodynamic Patterning of Thin Polymer Films

To pattern thin polymer films via electrohydrodynamic instabilities, we design and utilize two different kinds of mask patterns to guide pillars into alignment over regions much greater in extent than their natural domain sizes. First, narrow protruding ridges that intersect to form regular patterns on the mask trigger the growth of pillars beneath. Later, square and triangular packings of pillars develop in the regions enclosed by those ridges, preserving the registry from one domain to the next over a much larger area than within individual domains in unpatterned portions of the mask. Second, small square protrusions that are prealigned into a large regular array on the mask guide the formation of square packings of pillars in domains that conform to the mask, forming a large array of pillars. Novel structures involving a combination of linear ridges and pillars are also produced mainly due to the dynamic merging among preformed pillars. Finally, we find vertex symmetry of the mask pattern is necessary for generating and preserving ordered patterns on the polymer film.

A 6-Regular Torus Graph Family with Applications to Cellular and Interconnection Networks

This paper introduces a special graph family, together with an extensive characterization of some of its properties and two of its immediate applications. The graph denoted by Ti,j, is a regular hexavalent toroidal graph. The topological features of Ti,j include vertex symmetry, Hamiltonian decomposition, translation and rotation isomorphism. Topologically, the graph is a torus, while algebraically, it can also be expressed as a Cayley graph, defined on the cyclic group , where k can be determined from the (i,j) parameters defining graph T i,j . As a direct consequence, the proposed graph, which has j

The Hyperstar Interconnection Network

In this paper, a multiprocessor interconnection topology, the hyperstar, based on the Cartesian product of star graphs is studied. The basic properties of the hyperstar are discussed and proved. This includes reduced degree and diameter, hierarchical structure, vertex symmetry, optimal routing, and shortest path characterization. The hyperstar is shown to be a member of the Cayley class of symmetric graphs. Embeddings of hypercubes, star graphs, and meshes are discussed. An optimal one-to-all broadcasting algorithm is obtained and analyzed. Some results on fault tolerance, parallel paths, Hamiltonian cycles, and VLSI layouts are obtained. Furthermore, a comparative study between the hyperstar and seven related networks is conducted. The comparison is based on scalability, broadcasting cost, link requirements, cost/performance ratio, and other static parameters such as degree, diameter, and average diameter.

Vertex symmetry and the asymptotic frequency dependence of the self-energy

We show that the exact 1/${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{n}}$ dependence of the self-energy for the Hubbard model for large ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{n}}$ is obtained entirely from the second-order skeleton diagram evaluated with the exact propagator. Explicit calculation demonstrates that contributions from higher-order skeleton diagrams cancel exactly as a consequence of the crossing symmetry of the renormalized four-point vertex function. From an analysis of the skeleton diagram expansion and direct numerical calculation, we show that an imperfect cancellation of contributions from higher-order skeleton diagrams appears at third and fourth order in U for the shielded potential approximation and the fluctuation exchange approximation, respectively. By itself, this indication of crossing symmetry violation of the vertex function says little about the accuracy of the resulting self-energy and propagator, particularly at low frequency.

Hamiltonicity, vertex symmetry, and broadcasting of uni-directional hypercubes

In the paper, the authors show that the two uni-directional hypercubes, namely UHC1n and UHC2n, proposed by Chou and Du as interconnection schemes, are Hamiltonian. In addition, the authors show that if n is even, both architectures are vertex symmetric, and that if n is odd, both architectures have exactly two vertex-symmetric components. Furthermore, the study of the symmetry leads to an effective analysis of the maximum delay of one-port one-to-all broadcasting for either architecture which is at most ⌈1.5n⌉.