Tree Theorem Research Articles

The “0,1-property” of exceptional objects for nilpotent operators of degree 6 with one invariant subspace

Each exceptional object M in the category S(6˜) of nilpotent operators with nilpotency degree bounded by 6, acting on finite dimensional vector spaces with invariant subspaces in a graded sense, can be presented by matrices with 0-1 entries (Theorem 1.2). Moreover it, and even every member of all but one of the exceptional tubes, has the “interval tree”-property in the sense that the graph associated to the matrix of the structure map with respect to the interval decompositions of M restricted to the two copies of the cover of the algebra k[x]/(x6), is a tree (Theorem 1.7 and Corollary 1.8).

An analog of Matrix Tree Theorem for signless Laplacians

A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph G is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of G. We show a similar combinatorial interpretation for principal minors of signless Laplacian Q. We also prove that the number of odd cycles in G is less than or equal to det⁡(Q)4, where the equality holds if and only if G is a bipartite graph or an odd-unicyclic graph.

Finite‐Time Stabilization of Coupled Systems on Networks with Time‐Varying Delays via Periodically Intermittent Control

AbstractIn this paper, finite‐time stabilization of coupled systems on networks with time‐varying delays (CSNTDs) via periodically intermittent control is studied. Both delayed subsystems and delayed couplings are considered; the self‐delays of different subsystems in delayed couplings are not identical. A periodically intermittent controller is designed to stabilize CSNTDs within finite time, and the stabilization duration is closely related to the topological structures of networks. Furthermore, two sufficient criteria are developed to ensure CSNTDs under periodically intermittent control can be stabilized within finite time by using an approach that combines the Lyapunov method with Kirchhoff's Matrix Tree Theorem. Then finite‐time stabilization of coupled oscillators with time‐varying delays is given as a practical application and sufficient criteria is obtained. Finally, a numerical simulation is proposed to support our results and show the effectiveness of the controller.

Tree formulas, mean first passage times and Kemeny’s constant of a Markov chain

In this paper, we aim to provide probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson's algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let $m_{ij}$ be the mean first passage time from $i$ to $j$ for an irreducible chain with finite state space $S$ and transition matrix $(p_{ij}; i, j \in S)$. It is well-known that $m_{jj} = 1/\pi_j = \Sigma^{(1)}/\Sigma_j$, where $\pi$ is the stationary distribution for the chain, $\Sigma_j$ is the tree sum, over $n^{n-2}$ trees $\textbf{t}$ spanning $S$ with root $j$ and edges $i \rightarrow k$ directed to $j$, of the tree product $\prod_{i \rightarrow k \in \textbf{t} }p_{ik}$, and $\Sigma^{(1)}:= \sum_{j \in S} \Sigma_j$. Chebotarev and Agaev derived further results from {\em Kirchhoff's matrix tree theorem}. We deduce that for $i \ne j$, $m_{ij} = \Sigma_{ij}/\Sigma_j$, where $\Sigma_{ij}$ is the sum over the same set of $n^{n-2}$ spanning trees of the same tree product as for $\Sigma_j$, except that in each product the factor $p_{kj}$ is omitted where $k = k(i,j,\textbf{t})$ is the last state before $j$ in the path from $i$ to $j$ in $\textbf{t}$. It follows that Kemeny's constant $\sum_{j \in S} m_{ij}/m_{jj}$ equals to $ \Sigma^{(2)}/\Sigma^{(1)}$, where $\Sigma^{(r)}$ is the sum, over all forests $\textbf{f}$ labeled by $S$ with $r$ trees, of the product of $p_{ij}$ over edges $i \rightarrow j$ of $\textbf{t}$. We show that these results can be derived without appeal to the matrix tree theorem. A list of relevant literature is also reviewed.

Synchronization of stochastic complex networks with time delay via feedback control based on discrete-time state observations

Abstract In this paper, synchronization of stochastic complex networks (SCNs) with time delay is researched. A feedback control method is introduced to drive such SCNs to achieve inner synchronization in mean square. Different from previous researches, the feedback control we designed is based on discrete-time state observations. Moreover, by exploiting Lyapunov method and Kirchhoff’s Matrix Tree Theorem in graph theory, some sufficient criteria are obtained to guarantee synchronization in mean square and asymptotical synchronization in mean square of SCNs. What is more, we make use of the theoretical results to analyze asymptotical synchronization in mean square of second-order Kuramoto oscillators with time delay and stochastic disturbances and get a sufficient criterion. At the end of this paper, a numerical example is provided to validate the effectiveness and feasibility of our theoretical results.

Counting trees with random walks

We give a simple proof of Tutte’s matrix-tree theorem, a well-known result providing a closed-form expression for the number of rooted spanning trees in a directed graph. Our proof stems from placing a random walk on a directed graph and then applying the Markov chain tree theorem to count trees. The connection between the two theorems is not new, but it appears that only one direction of the formal equivalence between them is readily available in the literature. The proof we now provide establishes the other direction. More generally, our approach is another example showing that random walks can serve as a powerful glue between graph theory and Markov chain theory, allowing formal statements from one side to be carried over to the other.

Quantized feedback control scheme on coupled systems with time delay and distributed delay: A finite-time inner synchronization analysis

In this paper, finite-time inner synchronization of coupled systems on a network with time delay and distributed delay (CSNTD) is investigated. And here, time delay and distributed delay are both taken into consideration when modelling a realistic network. Different from common feedback control, the controller we design is quantized, which is more realistic and reasonable. By using Lyapunov method and Kirchhoff’s Matrix Tree Theorem, some sufficient criteria are derived to guarantee finite-time inner synchronization of CSNTD. It should be underlined that the method is first applied to studying the issue of finite-time inner synchronization of CSNTD and the synchronization time we obtain has a close relationship with the topological structure of the network. Moreover, to verify our theoretical results, we present an application to coupled oscillators with time delay and distributed delay, and a sufficient criterion is obtained. Ultimately, a numerical example is given to verify the validity and feasibility of theoretical results.

Synchronization analysis of stochastic coupled systems with time delay on networks by periodically intermittent control and graph-theoretic method

In this paper, the issue of inner exponential synchronization (IES) for a class of stochastic coupled systems with time-varying delay on networks (SCSTVDNs) is investigated. A periodically intermittent control strategy is introduced to drive such SCSTVDNs to achieve IES. By exploiting Lyapunov method and Kirchhoffs matrix tree theorem in graph theory, two types of sufficient IES criteria are respectively established in the form of Lyapunov-type theorem and coefficients-type theorem. Furthermore, to show the practicability of the theoretical results, the analysis of IES for the second-order Kuramoto oscillators with stochastic perturbations and time-varying delay is performed. A numerical example is finally provided to demonstrate the validity and feasibility of our analytical results.

Stabilization of stochastic complex-valued coupled delayed systems with Markovian switching via periodically intermittent control

This paper focuses on exponential stabilization of stochastic complex-valued coupled delayed systems with Markovian switching (SCCDM) via periodically intermittent control. The features of complex-valued systems, time-varying delays, stochastic disturbances and Markovian switching are taken into account. The mathematical model of this kind of complex-valued coupled systems is studied for the first time. To study the stabilization problem of SCCDM, two new differential inequalities on delayed Markovian jump systems are established. Then by utilizing Lyapunov method combined with Kirchhoff’s Matrix Tree Theorem, sufficient criteria promising the stability of SCCDM via periodically intermittent control are derived, which have a close relationship with control period, control rate, control gain and the topological structure of the considered coupled network. Then we employ the theoretical results to study the stabilization problem of stochastic complex-valued coupled oscillators with Markovian switching via periodically intermittent control. Finally, numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed results.

Synchronization of coupled stochastic complex-valued dynamical networks with time-varying delays via aperiodically intermittent adaptive control.

This paper deals with the synchronization problem of a class of coupled stochastic complex-valued drive-response networks with time-varying delays via aperiodically intermittent adaptive control. Different from the previous works, the intermittent control is aperiodic and adaptive, and the restrictions on the control width and time delay are removed, which lead to a larger application scope for this control strategy. Then, based on the Lyapunov method and Kirchhoff's Matrix Tree Theorem as well as differential inequality techniques, several novel synchronization conditions are derived for the considered model. Specially, impulsive control is also considered, which can be seen as a special case of the aperiodically intermittent control when the control width tends to zero. And the corresponding synchronization criteria are given as well. As an application of the theoretical results, a class of stochastic complex-valued coupled oscillators with time-varying delays is studied, and the numerical simulations are also given to demonstrate the effectiveness of the control strategies.

Synchronization of stochastic coupled systems with time-varying coupling structure on networks via discrete-time state feedback control

Abstract This paper is concerned with the exponential synchronization of stochastic coupled systems on networks. Time-varying coupling structure of coupled systems and discrete-time state feedback control are focuses and difficulties of our research. Based on the Lyapunov method and Kirchhoff’s Matrix Tree Theorem, two sufficient criteria are obtained, which guarantee the exponential synchronization of stochastic coupled systems with time-varying coupling structure on networks via discrete-time state feedback control. Finally, we apply theoretical results to stochastic coupled oscillators and propose two numerical examples to illustrate the effectiveness and feasibility of the obtained theoretical results.

A Ramsey theorem for trees with applications to weakly compact operators

A Ramsey theorem for trees with applications to weakly compact operators

THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL

AbstractSeveral variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. Forκweakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finited≥ 2, we prove the consistency of the Halpern–Läuchli Theorem ondmany normalκ-trees at a measurable cardinalκ, given the consistency of aκ+d-strong cardinal. This follows from a more general consistency result at measurableκ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method

We consider the outer synchronization between drive-response systems on networks with time-varying delays, where we focus on the case when the underlying networks are not strongly connected. A hierarchical method is proposed to characterize large-scale networks without strong connectedness. The hierarchical algorithm can be implemented by some programs to overcome the difficulty resulting from the scale of networks. This method allows us to obtain two different kinds of sufficient outer synchronization criteria without the assumption of being strongly connected, by combining the theory of asymptotically autonomous systems with Lyapunov method and Kirchhoff's Matrix Tree Theorem in graph theory. The theory improves some existing results obtained by graph theory. As illustrations, the theoretic results are applied to delayed coupled oscillators and a numerical example is also given.

Coloring the rationals in reverse mathematics

Ramsey’s theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey’s theorem for pairs due to Erdős and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type η, where η is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey’s theorem for pairs. This Erdős–Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.

The Stability of Stochastic Coupled Systems With Time-Varying Coupling and General Topology Structure.

We introduce a class of novel stochastic coupled systems in which the coupling structure is time-varying and the topology structure is not strongly connected, and first establish the system on a digraph with a time-varying weight matrix. Motivated by Du and Li (2014), we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach. Also, an example is given to illustrate our hierarchical algorithm and its feasibility. In the sequel, based on the theory of asymptotically autonomous systems, Kirchhoff's matrix tree theorem, and Lyapunov method, several moment exponential stability criteria are presented, including a Lyapunov-type theorem and a coefficient-type criterion. Furthermore, theoretical results are applied to stochastic coupled oscillators with time-varying coupling structure (SCTCS), and the stability criterion of SCTCS is obtained. Finally, the effectiveness of theoretical results is illustrated by two numerical examples.

Graph-theoretic approach to exponential synchronization of discrete-time stochastic coupled systems with time-varying delay

Abstract In this paper, we investigate the exponential synchronization problem for discrete-time stochastic drive-response coupled systems with time-varying delay. By employing the Lyapunov method combined with Kirchhoff’s matrix tree theorem in graph theory as well as stochastic analysis technique, some novel sufficient criteria are established to guarantee the exponential synchronization of two identical delayed coupled systems with stochastic disturbances. These sufficient criteria have a close relationship with the topological property of the coupled network. Moreover, the theoretical results are applied to a coupled oscillators system to demonstrate the applicability of the proposed synchronization approaches. Finally, a numerical example is provided to illustrate the effectiveness of our theoretical results.

Coloring trees in reverse mathematics

The tree theorem for pairs (TT22), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree 2<ω, there is a set of nodes isomorphic to 2<ω which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs (RT22), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of TT22, by showing that this principle does not imply the arithmetic comprehension axiom (ACA0) over the base system, recursive comprehension axiom (RCA0), of second-order arithmetic. Combined with a recent result of Patey's that TT22 is strictly stronger than RT22, this establishes TT22 as the first known example of a natural combinatorial principle to occupy the interval strictly between ACA0 and RT22. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on ω.

Periodic solutions of discrete time periodic time-varying coupled systems on networks

In this paper, we consider the existence of periodic solutions for discrete time periodic time-varying coupled systems on networks (DPTCSN). Some novel sufficient conditions are obtained to guarantee the existence of periodic solutions for DPTCSN, which have a close relation to the topology property of the corresponding network. Our approach is based on the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some new analysis techniques. The approach is applied to the existence of periodic solutions for discrete time Cohen–Grossberg Neural Networks (CGNN). Finally, an example and numerical simulations are provided to illustrate the effectiveness of our theoretical results.

Razumikhin method conjoined with graph theory to input-to-state stability of coupled retarded systems on networks

This paper is concerned with input-to-state stability (ISS) of coupled retarded systems on networks (CRSNs). By incorporating Razumikhin method with generalized Kirchhoff’s Matrix Tree Theorem, sufficient conditions are established to guarantee ISS of CRSNs. For the convenience of application, sufficient conditions in the form of coefficients are also obtained. What is more, a sufficient criterion on ISS of the coupled retarded oscillators system on networks is derived. Finally, as an illustration, a numerical example is given to demonstrate the effectiveness of the theoretical results.