Two-Arc Transitive Graphs and Twisted Wreath Products

The paper addresses a part of the problem of classifying all 2-arc transitive graphs: namely, that of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices. A construction is given for such groups, together with the associated graphs, in terms of the following ingredients: a nonabelian simple group T, a permutation group P acting 2-transitively on a set , and a map F : Tsuch that x = x ^–1 for all x F() and such that Tis generated by F(). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.     

Stereogenicity/Astereogenicity as Global/Local Permutation-Group Symmetry and Relevant Concepts for Restructuring Stereochemistry

Molecules of ligancy 4 that have been derived from an allene, an ethylene, a tetrahedral, and a square-planar skeleton have been investigated to show that their symmetries are dually and distinctly controlled by point groups and permutation groups. Insomuch as the point-group symmetry was exhibited to control the chirality/achirality of a molecule, sphericity in a molecule, and enantiomeric relationship between molecules [S. Fujita, J. Am. Chem. Soc. 112 (1990) 3390], the permutation-group symmetry has been now clarified to control the stereogenicity of a molecule, tropicity in a molecule, and diastereomeric relationship between molecules. To characterize permutation groups, proper and improper permutations have been defined by comparing proper and improper rotations. Thereby, such permutation groups are classified into stereogenic and astereogenic ones. After a coset representation (CR) of a permutation group has been ascribed to an orbit (equivalence class), the tropicity of the orbit has been defined in term of the global stereogenicity and the local stereogenicity of the CR. As a result, the conventional stereogenicity has now been replaced by the concept local stereogenicity of the present investigation. The terms homotropic, enantiotropic, and hemitropic are coined and used to characterize prostereogenicity. Thus, a molecule is defined as being prostereogenic if it has at least one enantiotropic orbit. Since this definition has been found to be parallel with the definition of prochirality, relevant concepts have been discussed with respect to the parallelism between stereogenicity and chirality in order to restructure the theoretical foundation of stereochemistry and stereoisomerism. The derivation of the skeletons has been characterized by desymmetrization due to the subduction of CRs. The Cahn–Ingold–Prelog (CIP) system has been discussed from the permutational point of view to show that it specifies diastereomeric relationships only. The apparent specification of enantiomeric relationships by the CIP system has been shown to stem from the fact that diastereomeric relationships and enantiomeric ones overlap occasionally in case of tetrahedral molecules.     

Estimations of subgraph positions in molecular graphs and their common subgraph peculiarities

Some questions emerged from electronic data processing of molecular structures (graphs) and its fragments have been considered in this work. Quantitative estimations of subgraph positions in molecular graphs are presented and some properties of their maximal common subgraphs are described.     

Computer generation of chemical structures from known fragments

The interactive generation of chemical structures from given fragments is described and discussed. It is implemented as a part of our expert system CARBON, based on C-13 NMR spectra. As it is designed, this program can also be a useful tool in the structure elucidation process when information on parts of the structure is obtained by other means (IR, mass and other spectrometries, chemical analysis, other relevant information). The topological characteristics of candidate fragments are first chosen interactively and then the elements are connected in all topologically possible ways. In the following step, the topological building blocks are substituted by chemical structural fragments resulting in a set of all chemical structures consistent with the input information.     

Three-dimensional aromaticity in deltahedral boranes and carboranes

Methods derived from topology and graph theory indicate that the deltahedral boranes B _n H _n ^2– and the corresponding carboranes C₂B_n–2H _n (6 n 12) may be regarded as three-dimensional delocalized aromatic systems in which surface bonding and core bonding correspond to -bonding and -bonding, respectively, in planar polygonal two-dimensional hydrocarbons CinnH _n ^(n–6)+ (n=5/7). The two extreme types of topologies which may be used to model core bonding in deltahedral boranes and carboranes are the deltahedral (D _n ) topology based on the skeleton of the underlying deltahedron and the complete (K _n ) topology based on the corresponding complete graph. Analyses of the Hoffmann-Lipscomb LCAO extended Hückel computations, the Armstrong—Perkins—Stewart self-consistent molecular orbital computations, and SCF MOab initio GAUSSIAN-82 computations on B₆H₆ ^2– indicate that the approximation of the atomic orbitals by the sum of the molecular orbitals, as is typical in modernab initio computations, leads to significantly weaker apparent core bonding approximated more closely by deltahedral (D _n ) topology than by complete (K _n ) topology.This work was presented at the Workshop The Modern Problems of Heteroorganic Chemistry sponsored by the A. N. Nesmeyanov Institute of Organoelement Compounds of the Russian Academy of Sciences (May 8–13, 1993).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1353–1360, August, 1993.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

A Family of Fitness Landscapes Modeled through Gene Regulatory Networks

Fitness landscapes are a powerful metaphor for understanding the evolution of biological systems. These landscapes describe how genotypes are connected to each other through mutation and related through fitness. Empirical studies of fitness landscapes have increasingly revealed conserved topographical features across diverse taxa, e.g., the accessibility of genotypes and “ruggedness”. As a result, theoretical studies are needed to investigate how evolution proceeds on fitness landscapes with such conserved features. Here, we develop and study a model of evolution on fitness landscapes using the lens of Gene Regulatory Networks (GRNs), where the regulatory products are computed from multiple genes and collectively treated as phenotypes. With the assumption that regulation is a binary process, we prove the existence of empirically observed, topographical features such as accessibility and connectivity. We further show that these results hold across arbitrary fitness functions and that a trade-off between accessibility and ruggedness need not exist. Then, using graph theory and a coarse-graining approach, we deduce a mesoscopic structure underlying GRN fitness landscapes where the information necessary to predict a population’s evolutionary trajectory is retained with minimal complexity. Using this coarse-graining, we develop a bottom-up algorithm to construct such mesoscopic backbones, which does not require computing the genotype network and is therefore far more efficient than brute-force approaches. Altogether, this work provides mathematical results of high-dimensional fitness landscapes and a path toward connecting theory to empirical studies.     

Picture Fuzzy Threshold Graphs with Application in Medicine Replenishment

In this study, a novel concept of picture fuzzy threshold graph (PFTG) is introduced. It has been shown that PFTGs are free from alternating 4-cycle and it can be constructed by repeatedly adding a dominating or an isolated node. Several properties about PFTGs are discussed and obtained the results that every picture fuzzy graph (PFG) is equivalent to a PFTG under certain conditions. Also, the underlying crisp graph (UCG) of PFTG is a split graph (SG), and conversely, a given SG can be applied to constitute a PFTG. A PFTG can be decomposed in a unique way and it generates three distinct fuzzy threshold graphs (FTGs). Furthermore, two important parameters i.e., picture fuzzy (PF) threshold dimension (TD) and PF partition number (PN) of PFGs are defined. Several properties on TD and PN have also been discussed. Lastly, an application of these developed results are presented in controlling medicine resources.     

Space-Air-Ground Integrated Mobile Crowdsensing for Partially Observable Data Collection by Multi-Scale Convolutional Graph Reinforcement Learning

Mobile crowdsensing (MCS) is attracting considerable attention in the past few years as a new paradigm for large-scale information sensing. Unmanned aerial vehicles (UAVs) have played a significant role in MCS tasks and served as crucial nodes in the newly-proposed space-air-ground integrated network (SAGIN). In this paper, we incorporate SAGIN into MCS task and present a Space-Air-Ground integrated Mobile CrowdSensing (SAG-MCS) problem. Based on multi-source observations from embedded sensors and satellites, an aerial UAV swarm is required to carry out energy-efficient data collection and recharging tasks. Up to date, few studies have explored such multi-task MCS problem with the cooperation of UAV swarm and satellites. To address this multi-agent problem, we propose a novel deep reinforcement learning (DRL) based method called Multi-Scale Soft Deep Recurrent Graph Network (ms-SDRGN). Our ms-SDRGN approach incorporates a multi-scale convolutional encoder to process multi-source raw observations for better feature exploitation. We also use a graph attention mechanism to model inter-UAV communications and aggregate extra neighboring information, and utilize a gated recurrent unit for long-term performance. In addition, a stochastic policy can be learned through a maximum-entropy method with an adjustable temperature parameter. Specifically, we design a heuristic reward function to encourage the agents to achieve global cooperation under partial observability. We train the model to convergence and conduct a series of case studies. Evaluation results show statistical significance and that ms-SDRGN outperforms three state-of-the-art DRL baselines in SAG-MCS. Compared with the best-performing baseline, ms-SDRGN improves 29.0% reward and 3.8% CFE score. We also investigate the scalability and robustness of ms-SDRGN towards DRL environments with diverse observation scales or demanding communication conditions.