Tree Theorem Research Articles

The Stability of a Predator–Prey Model with Cross-Dispersal in a Multi-Patch Environment

This paper investigates the stability of predator–prey models within multi-patch environments, with a particular focus on the influence of cross-dispersion across patches. We apply Kirchhoff’s matrix tree theorem and Liapunov’s method to derive criteria related to the cross-dispersion topology, thus solving the challenge of determining global asymptotic stability conditions. The method incorporates realistic ecological interactions and spatial heterogeneity, offering a framework for stability analysis. Our findings demonstrate that an appropriate level of cross-dispersion can effectively mitigate oscillations and foster convergence toward equilibrium. Two numerical examples validate these theoretical results and demonstrate the feasibility and effectiveness of the model across multiple patches.

Top-down complementation of automata on finite trees

We present a new complementation construction for nondeterministic automata on finite trees. The traditional complementation involves determinization of the corresponding bottom-up automaton (recall that top-down deterministic automata are less powerful than nondeterministic automata, whereas bottom-up deterministic automata are equally powerful).The construction works directly in a top-down fashion, therefore without determinization. The main advantages of this construction are: (i) in the special case of finite words it boils down to the standard subset construction (which is not the case of the traditional bottom-up complementation construction), and (ii) it illustrates the core argument of the complementation lemma for infinite trees, central in the proof of Rabin's tree theorem, in a simpler setting where issues related to acceptance conditions over infinite words and determinacy of infinite games are not present.

Generating functions and counting formulas for spanning trees and forests in hypergraphs

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15].

Milliken’s Tree Theorem and Its Applications: A Computability-Theoretic Perspective

Milliken’s tree theorem is a deep result in combinatorics that generalizes a vast number of other results in the subject, most notably Ramsey’s theorem and its many variants and consequences. In this sense, Milliken’s tree theorem is paradigmatic of structural Ramsey theory, which seeks to identify the common combinatorial and logical features of partition results in general. Its investigation in this area has consequently been extensive. Motivated by a question of Dobrinen, we initiate the study of Milliken’s tree theorem from the point of view of computability theory. The goal is to understand how close it is to being algorithmically solvable, and how computationally complex are the constructions needed to prove it. This kind of examination enjoys a long and rich history, and continues to be a highly active endeavor. Applied to combinatorial principles, particularly Ramsey’s theorem, it constitutes one of the most fruitful research programs in computability theory as a whole. The challenge to studying Milliken’s tree theorem using this framework is its unusually intricate proof, and more specifically, the proof of the Halpern-Laüchli theorem, which is a key ingredient. Our advance here stems from a careful analysis of the Halpern-Laüchli theorem which shows that it can be carried out effectively (i.e., that it is computably true). We use this as the basis of a new inductive proof of Milliken’s tree theorem that permits us to gauge its effectivity in turn. The key combinatorial tool we develop for the inductive step is a fast-growing computable function that can be used to obtain a finitary, or localized, version of Milliken’s tree theorem. This enables us to build solutions to the full Milliken’s tree theorem using effective forcing. The principal result of this is a full classification of the computable content of Milliken’s tree theorem in terms of the jump hierarchy, stratified by the size of instance. As usual, this also translates into the parlance of reverse mathematics, yielding a complete understanding of the fragment of second-order arithmetic required to prove Milliken’s tree theorem. We apply our analysis also to several well-known applications of Milliken’s tree theorem, namely Devlin’s theorem, a partition theorem for Rado graphs, and a generalized version of the so-called tree theorem of Chubb, Hirst, and McNicholl. These are all certain kinds of extensions of Ramsey’s theorem for different structures, namely the rational numbers, the Rado graph, and perfect binary trees, respectively. We obtain a number of new results about how these principles relate to Milliken’s tree theorem and to each other, in terms of both their computability-theoretic and combinatorial aspects. In particular, we establish new structural Ramsey-theoretic properties of the Rado graph theorem and the generalized Chubb-Hirst-McNicholl tree theorem using Zucker’s notion of big Ramsey structure.

A generalization of the Graham-Pollak tree theorem to even-order Steiner distance

A generalization of the Graham-Pollak tree theorem to even-order Steiner distance

Exponential stability of stochastic complex networks with multi-weights driven by second-order process based on graph theory

<abstract><p>Stochastic complex networks with multi-weights which were driven by Brownian motion were widely investigated by many researchers. However, Brownian motion is not suitable for the modeling of engineering issues by reason of its variance, which is infinite at any time. So, in this paper, a novel kind of stochastic complex network with multi-weights driven by second-order process is developed. To disclose how the weights and second-order process affect the dynamical properties of stochastic complex networks with multi-weights driven by the second-order process, we discuss exponential stability of the system. Two types of sufficient criteria are provided to ascertain exponential stability of the system on the basis of Kirchhoff's matrix tree theorem and the Lyapunov method. Finally, some numerical examples are given to verify the correctness and validity of our results.</p></abstract>

A generalization of the Graham-Pollak tree theorem to Steiner distance

Graham and Pollak showed that the determinant of the distance matrix of a tree T depends only on the number of vertices of T. Graphical distance, a function of pairs of vertices, can be generalized to “Steiner distance” of sets S of vertices of arbitrary cardinality, by defining it to be the minimum number of edges in a connected subgraph containing all the vertices of S. Here, we show that the same is true for trees' Steiner distance hypermatrix of all odd orders, whereas the theorem of Graham-Pollak concerns order 2. We conjecture that the statement holds for all even orders as well.

The cosmological tree theorem

A number of diagrammatic “cutting rules” have recently been developed for the wavefunction of the Universe which determines cosmological correlation functions. These leverage perturbative unitarity to relate particular “discontinuities” in Feynman-Witten diagrams (with cosmological boundary conditions) to simpler diagrams, in much the same way that the Cutkosky rules relate different scattering amplitudes. In this work, we make use of a further causality condition to derive new cutting rules for Feynman-Witten diagrams on any time-dependent spacetime background. These lead to the cosmological analogue of Feynman’s tree theorem for amplitudes, which can be used to systematically expand any loop diagram in terms of (momentum integrals of) tree-level diagrams. As an application of these new rules, we show that certain singularities in the wavefunction cannot appear in equal-time correlators due to a cancellation between “real” and “virtual” contributions that closely parallels the KLN theorem. Finally, when combined with the Bunch-Davies condition that certain unphysical singularities are absent, these cutting rules completely determine any tree-level exchange diagram in terms of simpler contact diagrams. Altogether, these results remove the need to ever perform nested time integrals when computing cosmological correlators.

Finite-time synchronization and topology identification of stochastic multi-layer networks with Markovian switching

The finite-time synchronization and topology identification problems of stochastic multi-layer networks with Markovian switching are investigated in this paper. Compared with previous studies, Markovian switching is considered in stochastic multi-layer networks. First, by using Kirchhoff’s matrix tree theorem in graph theory, the Lyapunov function of the coupling system is constructed indirectly from the Lyapunov function of the vertex system. In view of finite-time stability theory and stochastic analysis technique, some finite-time synchronization criteria of stochastic multi-layer networks are proposed. Moreover, the intra-layer and inter-layer topological structures are successfully identified in finite time. Then, using the pinning control technique, the unknown partial topological structures are also identified in finite time. In the end, two examples for whole and partial topology identification of two-layer small-world networks are given.

Finite-time stabilisation issue for a class of highly nonlinear stochastic coupled systems

In this work, we deal with the stabilisation issue for a class of highly nonlinear stochastic coupled systems on networks (SCSNs, in short). Specifically, we obtain the criteria for the finite-time stabilisation of the highly nonlinear SCSNs based on two methods such as the Lyapunov-type condition related with Kirchhoff's Matrix Tree Theorem and coefficients-type criterion. Further, highly nonlinear SCSNs is finite-time stabilised through designing a state feedback controller. To demonstrate the effectiveness of the developed result, as an application, we show the finite-time stabilisation for a class of stochastic coupled oscillators whose coefficients satisfy highly nonlinear coefficients.

Integral input-to-state stability for delayed networks control systems and its applications

Integral input-to-state stability (iISS) for delayed networks control systems (DNCS) is studied. Based on Kirchhoff’s matrix tree theorem in graph theory, a novel Lyapunov-Krasovaskii functional with integral terms is constructed via the Lyapunov function of each node control system in DNCS and its corresponding topological structure. According to Lyapunov method, some analysis and inequality skills, we raise two sufficient criterions assuring iISS for DNCS, which are the Lyapunov-type criterion and the coefficient-type criterion, respectively. Then, our approach is used in some classical DNCS. Finally, some numerical examples and simulation process are given to verify the effectiveness and applicability of the theoretical results.

Generalized fusible numbers and their ordinals

Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function x+y+12. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε0. They also investigated a recursively defined function M:R→R. They showed that the set of points of discontinuity of M is a subset of F of order type ε0. They also showed that, although M is a total function on R, the fact that the restriction of M to Q is total is not provable in first-order Peano arithmetic PA.In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets F of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function g:Rn→R.The most straightforward generalization of x+y+12 to an n-ary function is the function x1+⋯+xn+1n. We show that this function generates a set Fn whose order type is just φn−1(0). For this, we develop recursively defined functions Mn:R→R naturally generalizing the function M.Furthermore, we prove that for any linear function g:Rn→R, the order type of the resulting F is at most φn−1(0).Finally, we show that there do exist continuous functions g:Rn→R for which the order types of the resulting sets F approach the small Veblen ordinal.

Discrete-time state observations control to synchronization of hybrid-impulses complex-valued multi-links coupled systems

ABSTRACT The synchronization of stochastic hybrid-impulses complex-valued multi-links coupled systems (SHCMCS) based on discrete-time state observations control is studied. Therein, hybrid impulses, multiple links and complex-valued factors are considered simultaneously, which make the model more common. By using the Lyapunov method and Kirchhoff's Matrix Tree Theorem, average impulsive interval and average impulsive gain, the synchronization is achieved. Moreover, synchronization criteria are obtained, whose sufficient conditions reflect that the realization of synchronization depends on coupled strength and stochastic disturbance strength. In addition, stochastic complex-valued multi-links coupled oscillators with hybrid impulses are studied. Finally, a numerical test is presented to illustrate the validity of results.

Constructing an Evolutionary Tree and Path–Cycle Graph Evolution along It

The paper solves the problem of constructing an evolutionary tree and the evolution of structures along it. This problem has long been posed and extensively researched; it is formulated and discussed below. As a result, we construct an exact cubic-time algorithm which outputs a tree with the minimum cost of embedding into it and of embedding it into a given network (Theorem 1). We construct an algorithm that outputs a minimum embedding of a tree into a network, taking into account incomplete linear sorting; the algorithm depends linearly on the number of nodes in the network and is exact if the sorting cost is not less than the sum of the duplication cost and the loss cost (Theorem 3). We construct an exact approximately quadratic-time algorithm which, for arbitrary costs of SCJ operations, solves the problem of reconstruction of given structures on any two-star tree (Theorem 4). We construct an exact algorithm which reduced the problem of DCJ reconstruction of given structures on any star to a logarithmic-length sequence of SAT problems, each of them being of approximately quadratic size (Theorem 5). The theorems have rigorous and complete proofs of correctness and complexity of the algorithms, and are accompanied by numerical examples and numerous explanatory illustrations, including flowcharts.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Dual Ramsey Theorem for Trees

Dual Ramsey Theorem for Trees

Matrix tree theorem for the net Laplacian matrix of a signed graph

For a simple signed graph G with the adjacency matrix A and net degree matrix , the net Laplacian matrix is . We introduce a new oriented incidence matrix which can keep track of the sign as well as the orientation of each edge of G. Also . Using this decomposition, we find the number of both positive and negative spanning trees of G in terms of the principal minors of generalizing the Matrix Tree Theorem for an unsigned graph. We present similar results for the signless net Laplacian matrix along with a combinatorial formula for its determinant.

Watermelons on the half-plane

We study the watermelon probabilities in the uniform spanning forests on the two-dimensional semi-infinite square lattice near either the open or closed boundary to which the forests can or cannot be rooted, respectively. We derive universal power laws describing the asymptotic decay of these probabilities with the distance between the reference points growing to infinity, as well as their non-universal constant prefactors. The obtained exponents match with the previous predictions made for the related dense polymer models using the Coulomb gas technique and conformal field theory, as well as with the lattice calculations made by other authors in different settings. We also discuss the logarithmic corrections some authors argued to appear in the watermelon correlation functions on the infinite lattice. We show that the full account for diverging terms of the lattice Green function, which ensures the correct probability normalization, provides the pure power law decay in the case of semi-infinite lattice with the closed boundary studied here, as well as in the case of infinite lattice discussed elsewhere. The solution is based on the all-minors generalization of the Kirchhoff matrix tree theorem, the image method and the developed asymptotic expansion of the Kirchhoff determinants.

A Meyer–Vietoris Formula for the Determinant of the Dirichlet-to-Neumann Operator on Riemann Surfaces

This paper presents a Meyer–Vietoris type gluing formula for a conformal invariant of a Riemannian surface with boundary that is defined by the determinant of the Dirichlet-to-Neumann operator. The formula is used to bound the asymptotics of the invariant under degeneration. It is shown that the associated height function on the moduli space of hyperbolic surfaces with geodesic boundary is proper only in genus zero. Properness implies a compactness theorem for Steklov isospectral metrics in the case of genus zero. The formula also provides asymptotics for the determinant of the Laplacian with Dirichlet or Neumann boundary conditions. For the proof, we derive an extension of Kirchhoff’s weighted matrix tree theorem for graph Laplacians with an external potential.

Generalized Markov chain tree theorem and Kemeny’s constant for a class of non-Markovian matrices

Given an ergodic Markov chain with transition matrix P and stationary distribution π, the classical Markov chain tree theorem expresses π in terms of graph-theoretic parameters associated with the graph of P. For a class of non-stochastic matrices M2 associated with P, recently introduced by the first author in Choi (2020) and Choi and Huang (2020), we prove a generalized version of Markov chain tree theorem in terms of graph-theoretic quantities of M2. This motivates us to define generalized version of mean hitting time, fundamental matrix and Kemeny’s constant associated with M2, and we show that they enjoy similar properties as their counterparts of P even though M2 is non-stochastic. We hope to shed lights on how concepts and results originated from the Markov chain literature, such as the Markov chain tree theorem, Kemeny’s constant or the notion of hitting time, can possibly be extended and generalized to a broader class of non-stochastic matrices via introducing appropriate graph-theoretic parameters. In particular, when P is reversible, the results of this paper reduce to the results of P.