Algebraic Graph Theory Research Articles

The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $$e_1,e_2$$ , let $$X_i$$ denote the set of $$e_i$$ -dimensional subspaces of a fixed finite vector space $$V=({\mathbb F}_q)^{e_1+e_2}$$ . Let $$Y_i$$ be a non-empty subset of $$X_i$$ and let $$\alpha _i = |Y_i|/|X_i|$$ . We give a positive lower bound, depending only on $$\alpha _1,\alpha _2,e_1,e_2,q$$ , for the proportion of pairs $$(S_1,S_2)\in Y_1\times Y_2$$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

Decentralized adaptive finite‐time fault‐tolerant control for a class of high‐power interconnected nonlinear systems via graph theory

SummaryThis article investigates the decentralized adaptive finite‐time fault‐tolerant control (FTC) for unknown strong interconnected nonlinear systems with high‐powers and actuator faults. The powers are the ratio of positive odd rational integers and may be different, which makes the presented scheme different from existing work on interconnected nonlinear systems. The considered interconnections are governed by some boundary functions that grow nonlinearly in the states of all subsystems. First, the difficulty brought by high powers to controller design is overcome by introducing the adding one power integrator method. Then, based on backstepping method and FTC theory, a decentralized adaptive algorithm is proposed without approximate structure (neural network, fuzzy logic, etc.), which adaptively compensates the impact of actuator faults. Besides, with the aid of algebraic graph theory, the system is stable under practical finite‐time concept. Finally, the effectiveness of the theoretical findings are validated by conducting two simulation examples.

The Quantum States of a Graph

Quantum codes are crucial building blocks of quantum computers. With a self-dual quantum code is attached, canonically, a unique stabilised quantum state. Improving on a previous publication, we show how to determine the coefficients on the basis of kets in these states. Two important ingredients of the proof are algebraic graph theory and quadratic forms. The Arf invariant, in particular, plays a significant role.

Fractional-Order Nonlinear Multi-Agent Systems: A Resilience-Based Approach to Consensus Analysis with Distributed and Input Delays

In this article, a resilient consensus analysis of fractional-order nonlinear leader and follower systems with input and distributed delays is assumed. To make controller design more practical, it is considered that the controller is not implemented as it is, and a disturbance term is incorporated into the controller part. A multi-agent system’s topology ahead to a weighted graph which may be directed or undirected is used. The article examines a scenario of leader–follower consensus through the application of algebraic graph theory and the fractional-order Razumikhin method. Numerical simulations are also provided to show the effectiveness of the proposed design for the leader–follower consensus.

Distributed finite‐time optimization algorithms for multi‐agent systems under directed graphs

AbstractExisting distributed finite‐time optimization algorithms for continuous‐time multi‐agent systems require either undirected graphs or weight balanced digraphs, and distributed finite‐time optimization problems for weight unbalanced digraphs are still great challenging. Thus, the distributed finite‐time optimization problems for continuous‐time multi‐agent systems with strongly convex local cost functions are investigated under directed graphs in this article. First, a distributed finite‐time gradient estimator is constructed by using non‐smooth analysis and algebraic graph theory, then distributed finite‐time optimization algorithms and piecewise distributed finite‐time optimization algorithms are proposed based on the designed gradient estimator. The new proposed distributed finite‐time optimization algorithms which only require strongly connected graphs relax the balanced requirement. Furthermore, the communication bandwidth of systems could be saved by deploying the proposed piecewise distributed finite‐time optimization algorithms since the information exchange in the optimization process is reduced. Finally, simulation examples are given to verify the effectiveness of proposed distributed optimization algorithms.

A unified perspective on some autocorrelation measures in different fields: A note

Abstract Using notions from linear algebraic graph theory, this article provides a unified perspective on some autocorrelation measures in different fields. They are as follows: (a) Orcutt’s first serial correlation coefficient, (b) Anderson’s first circular serial correlation coefficient, (c) Moran’s r 11 {r}_{11} , and (d) Moran’s I I . The first two are autocorrelation measures for one-dimensional data equally spaced, such as time series data, and the last two are for spatial data. We prove that (a)–(c) are a kind of (d). For example, we show that (d) such that its spatial weight matrix equals the adjacency matrix of a path graph is the same as (a). The perspective is beneficial because studying the properties of (d) leads to studying the properties of (a)–(c) at the same time. For example, the bounds of (a)–(c) can be found from the bounds of (d).

Totally 2-closed finite groups with trivial Fitting subgroup

A finite permutation group [Formula: see text] is called [Formula: see text]-closed if [Formula: see text] is the largest subgroup of [Formula: see text] which leaves invariant each of the [Formula: see text]-orbits for the induced action on [Formula: see text]. Introduced by Wielandt in 1969, the concept of [Formula: see text]-closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total [Formula: see text]-closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group [Formula: see text] is said to be totally [Formula: see text]-closed if [Formula: see text] is [Formula: see text]-closed in each of its faithful permutation representations. There are infinitely many finite soluble totally [Formula: see text]-closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly [Formula: see text] totally [Formula: see text]-closed finite nonabelian simple groups: the Janko groups [Formula: see text], [Formula: see text] and [Formula: see text], together with [Formula: see text], [Formula: see text] and the Monster [Formula: see text]. Moreover, if a finite totally [Formula: see text]-closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely [Formula: see text] examples. In the course of obtaining this classification, we develop a general framework for studying [Formula: see text]-closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Fuzzy Adaptive Group Obstacle Avoidance Control for Second-Order Multi-Agent Systems Under Fixed and Switching Topologies

This paper investigates the multi-group formation obstacle avoidance control problem of second-order nonlinear leader-follower multi-agent systems (MASs) with unknown nonlinear function and uncertain input disturbance. To make multi-group MASs achieve global consensus, a distributed global fuzzy adaptive control method is presented. Then, navigation functions are employed to make agents bypass obstacles and reach the target positions. The adaptive method is applied to adjust parameters online which can substantially reduce computation burdens and make trajectories of agents smooth. Without loss of generality, both fixed and switching interaction topologies are all taken into account. After that, through Lyapunov stability theory and algebraic graph theory, the feasibility of the method proposed in this paper has been proved. Finally, simulation examples are given to validate the effectiveness of the method proposed in this paper.

Fuzzy Observer-Based Consensus Tracking Control for Fractional-Order Multi-Agent Systems Under Cyber-Attacks and Its Application to Electronic Circuits

Consensus control of multi-agent systems (MASs) has applications in various domains. As MASs work in networked environments, their security control becomes critically desirable in response to cyber-attacks. In this paper, the observer-based consensus tracking control problem is investigated for a class of Takagi-Sugeno fuzzy fractional-order multi-agent systems (FOMASs) under cyber-attacks. The malicious cyber attacks can impact the security of topologies of the communication networks of both controllers and observers. To estimate unmeasurable system states, a fuzzy observer is built. It is found that the topology of contact for observer states may be different from that of the feedback signals. A novel mathematical model for T-S fuzzy FOMASs with cyber-attacks is proposed. By using algebraic graph theory, Lyapunov functional, and fractional calculus theory, a distributed feed-back controller is developed for each agent, which guarantee the secure performance of tracking consensus error and observer error. Finally, two numerical examples demonstrate the effectiveness of the suggested control scheme, and the controller design for electronic network circuits shows the applicability of the proposed theoretical results. Simulations results for different differential-orders and coupling strength scenarios are given.

High-speed train cooperative control based on fractional-order sliding mode adaptive algorithm

PurposeThis study aims to propose an adaptive fractional-order sliding mode controller to solve the problem of train speed tracking control and position interval control under disturbance environment in moving block system, so as to improve the tracking efficiency and collision avoidance performance.Design/methodology/approachThe mathematical model of information interaction between trains is established based on algebraic graph theory, so that the train can obtain the state information of adjacent trains, and then realize the distributed cooperative control of each train. In the controller design, the sliding mode control and fractional calculus are combined to avoid the discontinuous switching phenomenon, so as to suppress the chattering of sliding mode control, and a parameter adaptive law is constructed to approximate the time-varying operating resistance coefficient.FindingsThe simulation results show that compared with proportional integral derivative (PID) control and ordinary sliding mode control, the control accuracy of the proposed algorithm in terms of speed is, respectively, improved by 25% and 75%. The error frequency and fluctuation range of the proposed algorithm are reduced in the position error control, the error value tends to 0, and the operation trend tends to be consistent. Therefore, the control method can improve the control accuracy of the system and prove that it has strong immunity.Originality/valueThe algorithm can reduce the influence of external interference in the actual operating environment, realize efficient and stable tracking of trains, and ensure the safety of train control.

Topological Indices of Graphs from Vector Spaces

Topological indices are numbers that are applied to a graph and can be used to describe specific graph properties through algebraic structures. Algebraic graph theory is a helpful tool in a range of chemistry domains. Because it helps explain how the different symmetries of molecules and crystals affect their structure and dynamics, it is a powerful theoretical approach for forecasting both the common and uncommon characteristics of molecules. A topological index converts the chemical structure into a number and contributes a lot in chemical graph theory. In this article, we compute the Wiener index, Zagreb indexes, Wiener polynomial, Hyper-Wiener index, ABC index and eccentricity-based topological index of a nonzero component union graph from vector space.

Predefined-Time Consensus of Nonlinear Multi-Agent Input Delay/Dynamic Event-Triggered Under Switching Topology

The consensus control problem of multi-agent systems (MASs) is studied. Firstly, a predefined-time consensus control algorithm is proposed for nonlinear uncertain systems with the input delay and switching topology. Then, the problem of resource consumption is considered, a dynamic event-triggered predefined-time consensus control algorithm is presented by introducing internal dynamic variables, which can make the MASs achieve consensus in the preset time. The correctness of the algorithm is proved by algebraic graph theory and Lyapunov theory, and there is no Zeno behavior. Simulation comparison experiments verify the effectiveness and superiority of the proposed algorithm. Compared with the finite-time control algorithm, the convergence time of this algorithm is independent of the initial state. The upper bound of the system convergence time can be set by selecting a time parameter. Compared with the fixed-time control algorithm, the convergence time of this algorithm is independent of the controller parameters, only related to a single parameter, the setting is simple, and the estimated convergence time is less conservative. Compared with the static triggering mechanism, the dynamic triggering mechanism can avoid a large number of triggering.

Fully distributed consensus control for second-order multi-agent systems based on adaptive dynamic clock communication

This paper proposes a novel event-triggered control strategy based on an adaptive dynamic clock to address the consensus problem of second-order multi-agent systems with undirected communication topology. The agent determines the triggering time according to its dynamic clock, adaptively resets the clock, and broadcasts its state information to neighbors at the triggering time. Each agent only obtains the status information of its neighbors at the triggering time and updates the clock and control signals based on its own state and the status of its neighbors at the triggering time. It does not require real-time status information from neighbors and does not rely on any global communication topology information. The designed control strategy is completely distributed and effectively avoids continuous communication. Using the Lyapunov stability analysis method and algebraic graph theory, it is proved that the proposed control strategy can ensure that the system is asymptotically stable and does not exhibit Zeno behavior. Finally, the effectiveness of the proposed dynamic event-triggered control strategy is demonstrated by a simulation example.

Time Cost for Consensus of Stochastic Multiagent Systems With Pinning Control

In this article, the estimation of time cost for stochastic consensus of second-order nonlinear multiagent systems (MASs) is investigated. Compared with previous studies, the effect of noise and nonlinear inherent dynamics of agents are considered. In order to achieve the finite/fixed-time stochastic consensus while minimizing energy consumption, the pinning protocols are designed, which only need to control a small fraction of agents. Sufficient conditions for the stochastic consensus are established by employing the stability theory of stochastic differential equations and the algebra graph theory. Finally, the proposed protocols are verified through numerical simulation on the consensus of unmanned aerial vehicles (UAVs) system.

A Multi-Subobject Approach to Dynamic Formation Target Tracking Using Random Matrices

Bird flocks are typical group targets with various linear formations and high dynamics due to swarm intelligence. This leads to several problems in traditional multi-subobject group target tracking such as shape model mismatch and false correlations. This paper proposes a multi-subobject approach to dynamic formation target tracking. The algebraic graph theory is introduced to analyze the structure of formation targets, then measurements are clustered and combined into subobjects. Based on the existing random matrix approach, two additional filtering branches, including collective filtering and internal structure filtering, are introduced to achieve the robust tracking performance of formation targets. The KL divergence between the prediction and updated densities of the collective filtering is used to determine the change of formation shape. The correct association between the measurements and the subobjects is realized by the guidance of the internal structure filter. Finally, the effectiveness of the proposed method is verified by the simulation and experimental results.

Distributed adaptive control for nonlinear multi-agent systems with nonlinear parametric uncertainties.

This paper considers the distributed tracking control problem for a class of nonlinear multi-agent systems with nonlinearly parameterized control coefficients and inherent nonlinearities. The essential of multi-agent systems makes it difficult to directly generalize the existing works for single nonlinearly parameterized systems with uncontrollable unstable linearization to the case in this paper. To dominate the inherent nonlinearities and nonlinear parametric uncertainties, a powerful distributed adaptive tracking control is presented by combing the algebra graph theory with the distributed backstepping method, which guarantees that all the closed-loop system signals are global bounded while the range of the tracking error between the follower's output and the leader's output can be tuned arbitrarily small. Finally, a numerical example is provided to verify the validity of the developed methods.

Finite-time scaled consensus tracking of a class of high-order nonlinear multi-agent systems with unstable linearization

In this paper, we consider the finite-time scaled consensus tracking of a class of high-order nonlinear multiagent systems(MASs)who owns unstable modes in its Jacobian linearized system. The presence of unstable linearization makes the high-order MASs in question essentially different from those in the existing works. Under a directed interaction topology, to overcome the difficulties caused by the asymmetry property of Laplacian matrix, the finite-time scaled consensus control scheme is developed by the modified addition of a power integrator method. Based on finite-time Lyapunov stability theorem and algebra graph theory, for high-order MASs with unstable linearization even in the presence of non-lipschitz nonlinear dynamic, all system states are bounded and the output tracking errors are finite-time uniformly ultimately bounded(FUUB). Finally, a numerical example is given to demonstrate the effectiveness of the theoretical results.

Finite 3-connected-set-homogeneous locally 2Kn graphs and s-arc-transitive graphs

In this paper, all graphs are assumed to be finite. For s≥1 and a graph Γ, if for every pair of isomorphic connected induced subgraphs on at most s vertices there exists an automorphism of Γ mapping the first to the second, then we say that Γ is s-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most s vertices can be extended to an automorphism of Γ, then we say that Γ is s-connected-homogeneous. For n≥1, a graph Γ is said to be locally 2Kn if the subgraph [Γ(u)] induced on the set of vertices of Γ adjacent to a given vertex u is isomorphic to 2Kn.Note that 2-connected-set-homogeneous but not 2-connected-homogeneous graphs are just the half-arc-transitive graphs which are a quite active topic in algebraic graph theory. Motivated by this, we posed the problem of characterizing or classifying 3-connected-set-homogeneous graphs of girth 3 which are not 3-connected-homogeneous in Zhou (2021) [54]. Until now, there have been only two known families of 3-connected-set-homogeneous graphs of girth 3 which are not 3-connected-homogeneous, and these graphs are locally 2Kn with n=2 or 4. In this paper, we complete the classification of finite 3-connected-set-homogeneous graphs which are locally 2Kn with n≥2, and all such graphs are line graphs of some specific 2-arc-transitive graphs. Furthermore, we give a good description of finite 3-connected-set-homogeneous but not 3-connected-homogeneous graphs which are locally 2Kn and have solvable automorphism groups. This is then used to construct some new 3-connected-set-homogeneous but not 3-connected-homogeneous graphs as well as some new 2-arc-transitive graphs.

Formation control of multiple unmanned vehicles based on graph theory: A Comprehensive Review

In recent years, formation control for multiple unmanned vehicles becomes an active research topic that has received a lot of attention from scientists due to its superior advantages compared with other conventional systems. Algebraic graph and graph rigidity theories are the two main mathematical backgrounds of the formation control theory. The graph theory is used to describe the interconnections among vehicles in formation while rigid graph theory - an important subset of graph theory - ensured that the inter-vehicle distance constraints of the desired formation are enforced via the graph rigidity. This paper provides a comprehensive review of graph theory supporting formation control for groups of unmanned aerial vehicles (UAV) or swarm UAVs. The background of the theory and the recent developments of graph-theory-based formation control are reviewed. We provide a cohesive overview of the formation control and coordination of multiple vehicles. Finally, some challenges and future potential directions in formation control are discussed.

Event-triggered [formula omitted] leader-follower consensus for nonlinear multi-agent systems with fixed and switching network topologies

This paper investigates H∞ leader-follower consensus problems of nonlinear multi-agent systems via event-triggered control. To significantly reduce the communication burdens, a distributed estimator-based event-triggered communication protocol is proposed. By using Lyapunov stability theory, matrix theory and algebraic graph theory, we derive some sufficient conditions to guarantee the leader-follower consensus with desired H∞ performance. Distributed triggering functions are designed for three cases: fixed topologies, switching topologies with connected sub-topologies and frequently connected switching topologies. Furthermore, we prove that the Zeno-behavior can be excluded for three cases, respectively. Finally, simulations are presented to illustrate the effectiveness of the theoretical results.