Ring Graph Research Articles

Engineering of Hyperentangled Complex Quantum Networks

Abstract We propose a novel scheme to engineer the atomic hyperentangled cluster and ring graph state invoking cavity-QED technique for applicative relevance to quantum biology and quantum communications utilizing the complex quantum networks. These states are engineered using both external quantized momenta states and energy levels of neutral atoms under off-resonant and resonant Atomic Bragg Diffraction (ABD) technique. The study of dynamical capacity and potential efficiency have certainly enhanced the range of usefulness of these states. In order to assess the operational behavior of such states when subjected to a realistic noise environment has also been simulated, demonstrating long enough sustainability of the proposed states. Moreover, experimental feasibility of the proposed scheme has also been elucidated under the prevailing cavity-QED research scenario.

The number of spanning trees in K n -chain and ring graphs

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K 5-chain network K5ℓ constructed by connecting ℓcopys of complete graphs K 5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K n -chain graphs and K n -ring graphs. The results show that there is a simple relation between the number of spanning trees of the K n -chain graph Lnℓ and the K n -ring graph Cnℓ . We also calculate the corresponding tree entropy (or so called ‘the asymptotic growth constant’) and find that the tree entropy of the corresponding K n -chain graphs and K n -ring graphs are totally the same.

Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for r≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when r=1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.

Geodetic Domination Integrity of Thorny Graphs

The concept of geodetic domination integrity is a crucial parameter when examining the potential damage to a network. It has been observed that the removal of a geodetic set from the network can increase its vulnerability. This study explores the geodetic domination integrity parameter and presents general results on the geodetic domination integrity values of thorn ring graphs, $n$-sunlet graphs, thorn path graphs, thorn rod graphs, thorn star graphs, helm graphs, $E_p^t$ tree graphs, dendrimer graphs, spider graphs, and bispider graphs, which are the frequently used graph classes in the literature.

Machine Learning-Based Traffic Flow Prediction and Intelligent Traffic Management

With the rapid development of information technology, multiple time series forecasting, which is typical of traffic flow forecasting, has become increasingly important in big data analysis. As the cornerstone of intelligent transportation system, traffic flow forecasting has important scientific research value and practical application value for urban traffic operation scheduling, quality and efficiency improvement of logistics transportation industry and public travel planning. Traffic flow prediction is always an important task of intelligent transportation system. Due to the complex temporal and spatial dependence of traffic flow sequence, it is very challenging to construct accurate traffic flow prediction in its ring neural network, graph network and Transformer model. Much of the existing work is based on very good models. Considering the advantages of convolutional networks, such as high computational efficiency and strong feature extraction ability, a traffic flow prediction model based on multi-view spatiotemporal convolution is proposed.

Self-Organized Polygon Formation Control Based on Distributed Estimation

This paper studies the problem of controlling a multi-robot system to achieve a polygon formation in a self-organized manner. Different from the typical formation control strategies where robots are steered to satisfy the predefined control variables, such as pair-wise distances, relative positions and bearings, the foremost idea of this paper is to achieve polygon formations by injecting control inputs randomly to a few robots (say, vertex robots) of the group, and the rest follow the simple principles of moving towards the midpoint of their two nearest neighbors in the ring graph without any external inputs. In our problem, a fleet of robots is initially distributed in the plane. The so-called vertex robots take the responsibility of determining the geometric shape of the entire formation and its overall size, while the others move so as to minimize the differences with two direct neighbors. In the first step, each vertex robot estimates the number of robots in its associated chain. Two types of control inputs that serve for the estimation are designed using the measurements from the latest and the last two time instants respectively. In the second step, the self-organized formation control law is proposed where only vertex robots receive external information. Comparisons between the two estimation strategies are carried out in terms of the convergence speed and robustness. The effectiveness of the whole control framework is further validated in both simulation and physical experiments.

Lyapunov exponents and extensivity of strongly coupled chaotic maps in regular graphs

In Thermodynamics and Statistical Physics, a system’s property is extensive when it grows with the system size. When it happens, the system can be decomposed into separate components, which has been done in many systems with weakly interacting components, such as for various gas models. Similarly, Ruelle conjectured 40 years ago that the Lyapunov exponents (LEs) of some sufficiently large chaotic systems are extensive, which led to study the extensivity properties of chaotic systems with strong interactions. Because of the complexities in these systems, most results achieved so far are restricted to numerical simulations. Here, we derive closed-form expressions for the LEs and entropy rate of coupled maps in finite- and infinite-sized regular graphs, according to the coupling strength, map’s chaoticity, and graph’s spectral properties. We show that this type of system has either 4 or 5 cases for the LEs, depending on the graph’s extreme Laplacian eigenvalues. These cases represent qualitatively different collective behaviours emerging in parameter space, including chaotic synchronisation (N−1 negative LEs) and incoherent chaos (N positive LEs). From the entropy rate, we show that the ring and complete graphs (nearest-neighbour and all-to-all couplings, respectively) are extensive in all parameter regions outside the chaotic synchronisation region. Although our derivations are restricted to one-dimensional maps with constant positive derivative (i.e., chaotic), our approach can be used to find LE and entropy rates for other regular graphs (such as for cyclic graphs) or be the basis for tackling small world graphs via perturbative methods.

Passivity-Based Trajectory Tracking and Formation Control of Nonholonomic Wheeled Robots Without Velocity Measurements

This note proposes a passivity-based control method for trajectory tracking and formation control of nonholonomic wheeled robots without velocity measurements. Coordinate transformations are used to incorporate the nonholonomic constraints, which are then avoided by controlling the front end of the robot rather than the center of the wheel axle into the differential equations. Starting from the passivity-based coordination design, the control goals are achieved via an internal controller for velocity tracking and heading control and an external controller for formation in the port-Hamiltonian framework. This approach endows the resulting controller with a physical interpretation. To avoid unavailable velocity measurements or unreliable velocity estimations, we derive the distributed control law with only position measurements by introducing a dynamic extension. In addition, we prove that our approach is suitable not only for acyclic graphs but also for a class of non-acyclic graphs, namely, ring graphs. Simulations are provided to illustrate the effectiveness of the approach.

Energy landscapes of some matching-problem ensembles

The maximum-weight matching problem and the behavior of its energy landscape is numerically investigated. We apply a perturbation method adapted from the analysis of spin glasses. This method provides insight into the complexity of the energy landscape of different ensembles. Erdős–Rényi graphs and ring graphs with randomly added edges are considered, and two types of distributions for the random edge weights are used. Fast and scalable algorithms exist for maximum weight matching, allowing us to study large graphs with more than 105 nodes. Our results show that the structure of the energy landscape for standard ensembles of matching is simple, comparable to the energy landscape of a ferromagnet. Nonetheless, for some of the ensembles presented here, our results allow for the presence of complex energy landscapes in the spirit of a replica-symmetry breaking scenario.

A Real‐Time and Fast LiDAR–IMU–GNSS SLAM System with Point Cloud Semantic Graph Descriptor Loop‐Closure Detection

Herein, a real‐time, fast, tightly coupled simultaneous localization and mapping (SLAM) system is proposed. The deep neural network is used to segment the point cloud semantically to construct the point cloud semantic map descriptor, and the global navigation satellite system real‐time kinematic position is used to detect the loop closure. Finally, a factor‐graph model is used for global optimization. The working principle of the SLAM system is introduced in detail, including the construction of the factor graph, the generation of the point cloud semantic graph descriptor, the generation of the ring graph, the loop‐closure detection process, and the global optimization. Indoor and outdoor experiments are conducted to validate the performance of overall trajectory estimation and loop‐closure detection. Compared to traditional methods, the proposed method offers improved accuracy and real‐time performance in trajectory estimation. It effectively addresses issues such as limited feature constraints, large computational consumption, and subpar real‐time performance, allowing for quick and accurate loop‐closure detection. Moreover, the method demonstrates a time consumption reduction of two‐thirds compared to traditional approaches while exhibiting superior overall loop‐closure detection performance.

Detour Pebbling Number on Some Commutative Ring Graphs

Detour Pebbling Number on Some Commutative Ring Graphs

Scale fragilities in localized consensus dynamics

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher (n≥2). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents (n≥3), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus (n=2) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh–Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue.

Line graphs associated to Von Neumann regular graphs of rings

Let [Formula: see text] be a commutative ring with unity. Taloukolaei and Sahebi [Von Neumann regular graphs associated with rings, Discrete Math. Algorithms Appl. 10(3) (2018) 1850029, doi:10.1142/S1793830918500295] introduced the Von Neumann regular graph [Formula: see text] of a ring [Formula: see text] whose vertices are the elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a Von Neumann regular element of [Formula: see text]. In this paper, we investigate and determine some graph theoretic properties of the line graph [Formula: see text] associated to [Formula: see text]. We give some characterization results regarding the completeness, bipartiteness, traversability, diameter and girth. We also prove Beck’s conjecture for [Formula: see text]. Finally we characterize rings having the planarity, the outerplanarity and also being the ring graph of the line graphs associated to Von Neumann regular graphs [Formula: see text] of rings.

Rigid-Body Attitude Control, Synchronization, and Bipartite Consensus Using Feedback Reshaping

Improved attitude control laws based on a feedback reshaping strategy using rotation matrices are proposed for attitude control of a single rigid body on and for multibody attitude synchronization on . For attitude control of a single body the proposed control law improves the closed-loop response for initial principal rotation angles near 180° while producing a globally continuous control torque. For the problem of multibody attitude synchronization, an alternative feedback reshaping strategy is proposed, which destabilizes nonconsensus equilibria and produces attitude synchronization with almost global stability and globally continuous control. Finally, a previously proposed bipartite consensus algorithm based on the concept of a structurally balanced graph with antagonistic interactions represented by negative edge weights is applied for the first time to rigid-body attitude consensus control with feedback reshaping in which the consensus attitudes and angular velocities are divided into two distinct groups. Simulations serve as illustrative examples of each of these problems and confirm the advantages of the proposed strategies. For multibody synchronization the focus is on ring graph topologies for which locally stable nonconsensus equilibria arise when using an existing attitude consensus control protocol in the literature.

Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is a graph with vertex set $Z(R)\setminus \{0\}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.

Crystal structure, molecular structural analysis, optical absorption and thermal analysis of 2-amino-6-methyl pyridine-1-ium 2-carboxy-6-nitro benzoate – A third harmonic nonlinear optical organic single crystal

Single crystals of the newly concocted 2-amino-6-methyl pyridine-1-ium 2-carboxy-6-nitro benzoate (MPNP) were grown by slow evaporation approach using an equimolar ethanol-water mixed solvent. The crystal structure of MPNP with the single-crystal X-ray diffraction technique revealed a centrosymmetric triclinic molecular arrangement. A robust hydrogen bond network with a cloud of supramolecular assemblies, S (4), S (8), S (10) and S (11), ring graph set motifs, R22 (8), R44 (18) and R (10) and π – π linkages establish molecular packing. Hirshfeld surface analysis substantiated non-bonded interactions in the structure. Experimental FT-IR and Raman vibration modes were analyzed in order to establish the presence of functional groups and the amine-imine tautomeric form. Molecular structure confirmation was corroborated with experimental NMR spectra. UV–Visible spectral profiling with Tauc, Urbach, and dispersion plots for refractive index variations were performed to understand the origin of optical band in the material. TCSPC lifetime studies envisaged the immense scintillating efficacy of MPNP. The experimental Z-scan data of MPNP in the closed and open aperture modes were deployed to estimate the magnitude of the nonlinear optical properties of MPNP, thereby establishing a good scope for optical applications. Thermogravimetry coupled with Differential Scanning Calorimetry (DSC) declares high thermal stability of the material till 177.6 °C.

On Another Class of Strongly Perfect Graphs

For a commutative ring R with unity, the associate ring graph, denoted by AG(R), is a simple graph with vertices as nonzero elements of R and two distinct vertices are adjacent if they are associates. The graph AG(R) contains components equal in number to the number of distinct orbits, except for the orbit of an element 0. Moreover, each component is a complete graph. An important finding is that this is a class of strongly perfect graphs. In this article we describe the structure of the associate ring graph of the ring of integers modulo n, denoted by AG(Zn). We carried out computer experiments and provide a program for the same. We further characterize cases in which AG(Zn), its complement AG(Zn)¯, and their line graphs are planar, ring graphs, and outerplanar. We also discuss the properties of the associate ring graph of a commutative ring R with unity.

Stability of heteroclinic cycles in ring graphs.

Networks of interacting nodes connected by edges arise in almost every branch of scientific inquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring graphs with nearest-neighbor or nearest- m-neighbor coupling and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle that can be asymptotically stable, and, thus, the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behavior persists in less structured networks and as such, non-ergodic behavior is somehow typical.

Sensing Enhancement on Social Networks: The Role of Network Topology.

Sensing and processing information from dynamically changing environments is essential for the survival of animal collectives and the functioning of human society. In this context, previous work has shown that communication between networked agents with some preference towards adopting the majority opinion can enhance the quality of error-prone individual sensing from dynamic environments. In this paper, we compare the potential of different types of complex networks for such sensing enhancement. Numerical simulations on complex networks are complemented by a mean-field approach for limited connectivity that captures essential trends in dependencies. Our results show that, whilst bestowing advantages on a small group of agents, degree heterogeneity tends to impede overall sensing enhancement. In contrast, clustering and spatial structure play a more nuanced role depending on overall connectivity. We find that ring graphs exhibit superior enhancement for large connectivity and that random graphs outperform for small connectivity. Further exploring the role of clustering and path lengths in small-world models, we find that sensing enhancement tends to be boosted in the small-world regime.

Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph.

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. Our analysis involves use of the reflection time difference, as well as a comprehensive use of complex time delay, to analyze experimental scattering data.