Zero Eigenvalue Research Articles

ALGEBRAIC MULTIPLICITY OF EIGENVALUE ZERO OF CERTAIN BI-REGULAR GRAPHS

A bi-regular graph is a graph whose degree sequence is exactly two. In this article, we give the algebraic multiplicity of the eigenvalue zero of certain bi-regular graphs.

The controllability and structural controllability of Laplacian dynamics

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix L ∗ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs; however, it is always simple for the Laplacian L with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with L. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with L ∗ . In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between parent nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from parent nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant. Finally, we present a sufficient and necessary condition to identify the strong structural controllability. Moreover, a method to construct a strongly structurally controllable graph is proposed.

A low-rank ODE for spectral clustering stability

Spectral clustering is a well-known technique which identifies k clusters in an undirected graph, with n vertices and weight matrix W∈Rn×n, by exploiting its graph Laplacian L(W). In particular, the k clusters can be identified by the knowledge of the eigenvectors associated with the k−1 smallest non zero eigenvalues of L(W), say λ2,…,λk (recall that λ1=0). Identifying k is an essential task of a clustering algorithm, since if λk+1 is close to λk the reliability of the method is reduced. The k-th spectral gap λk+1−λk is often considered as a stability indicator. This difference can be seen as an unstructured distance between L(W) and an arbitrary symmetric matrix L⋆ with vanishing k-th spectral gap. A more appropriate structured distance to ambiguity such that L⋆ represents the Laplacian of a graph has been proposed in Andreotti et al. (2021) [2]. This is defined as the minimal distance between L(W) and Laplacians of graphs with the same vertices and edges, but with weights that are perturbed such that the k-th spectral gap vanishes. In this article we consider a slightly different approach, still based on the reformulation of the problem into the minimization of a suitable functional in the eigenvalues. After determining the gradient system associated with this functional, we introduce a low-rank projected system, suggested by the underlying low-rank structure of the extremizers of the problem. The integration of this low-rank system, requires both a moderate computational effort and a memory requirement, as it is shown in some illustrative numerical examples.

Classification of Cubic Tricirculant Nut Graphs

A nut graph is a simple graph whose adjacency matrix has the eigenvalue zero with multiplicity one such that its corresponding eigenvector has no zero entries. It is known that there exist no cubic circulant nut graphs. A bicirculant (resp. tricirculant) graph is defined as a graph that admits a cyclic group of automorphisms having two (resp. three) orbits of vertices of equal size. We show that there exist no cubic bicirculant nut graphs and we provide a full classification of cubic tricirculant nut graphs.

Reconstruction of averaging indicators for highly heterogeneous media

We propose a new imaging algorithm capable of producing quantitative indicator functions for unknown and possibly highly oscillating media from multi-static far field measurements of scattered fields at a fixed frequency. The algorithm exploits the notion of modified transmission eigenvalues and their determination from measurements. We propose in particular the use of a new modified background obtained as the limit of a metamaterial background. It has the advantage of having a unique non zero eigenvalue, which is particularly suited for the proposed imaging procedure. We show the efficiency of this new algorithm on some two dimensional experiments and emphasise its superiority with respect to some classical approaches such as the linear sampling method.

Asymptotics of Solutions to a Singularly Perturbed Problem

The work examines the asymptotics of solutions to singularly perturbed differential equations. The zeros of the matrix eigenvalues lie on the real axis. Moving on to the complex plane, we define the negative region in which the research is carried out. Level lines completely cover this area. One of the level lines divides the area into four parts. In each of these parts of the region we choose integration paths. The integration paths should be as decreasing from the starting point to the last point. Carrying out calculations along the chosen integration path, we obtain asymptotic estimates for solutions of singularly perturbed differential equations.

Structural obstruction to the simplicity of the eigenvalue zero in chemical reaction networks

Multistationarity is the property of a system to exhibit two distinct equilibria (steady‐states) under otherwise identical conditions, and it is a phenomenon of recognized importance for biochemical systems. Multistationarity may appear in the parameter space as a consequence of saddle‐node bifurcations, which necessarily require an algebraically simple eigenvalue zero of the Jacobian, at the bifurcating equilibrium. Matrices with a simple eigenvalue zero are generic in the set of singular matrices. Thus, one would expect that in applications singular Jacobians are always with a simple eigenvalue zero. However, chemical reaction networks typically consider a fixed network structure, while the freedom rests with the various choices of kinetics. Here, we present an example of a chemical reaction network, whose Jacobian is either nonsingular or has an algebraically multiple eigenvalue zero. This in particular constitutes an obstruction to standard saddle‐node bifurcations. The presented structural obstruction is based on the network structure alone, and it is independent of the value of the positive concentrations and the choice of kinetics.

On the local Gevrey integrability

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm.

Some Results on Semi-Symmetric Spaces

We give some properties of semi-symmetric pseudo-Riemannian manifolds as an indecomposable irreducible Ricci pseudo-Riemannian manifold (i.e. the minimal polynomial of its Ricci operator is irreducible) is semi symmetric if and only if it is locally symmetric. We also show that any semi-symmetric pseudo-Riemannian manifold will be foliated. Moreover, if the metric is Lorentzian, the Ricci operator has only real eigenvalues and more precisely, on each leaf, it is diagonalizable with at most a single non zero eigenvalue or isotropic.

Some Comments about Zero and Non-Zero Eigenvalues from Connected Undirected Planar Graph Adjacency Matrices

Two linear algebra problems implore a solution to them, creating the themes pursued in this paper. The first problem interfaces with graph theory via binary 0-1 adjacency matrices and their Laplacian counterparts. More contemporary spatial statistics/econometrics applications motivate the second problem, which embodies approximating the eigenvalues of massively large versions of these two aforementioned matrices. The proposed solutions outlined in this paper essentially are a reformulated multiple linear regression analysis for the first problem and a matrix inertia refinement adapted to existing work for the second problem.

On open neighborhood energy of graphs

The open neighborhood of a vertex u denoted by N(u) in a simple connected graph G is the collection of all vertices other than u and adjacent to u. In this paper, we introduce a square matrix of order n, called the open neighborhood matrix, ONM(G) of a graph G whose (i, j)th entry is |N(vi) ∩ N(vj)|/(di+dj) whenever vi~vj, i≠ j; and zero otherwise, where di and dj, are the degrees of vi and vj respectively. We then establish the relationship between the connectedness of the graph G and the multiplicity of the eigenvalue zero of the matrix ONM(G), if it exists. Furthermore, we found the bounds for the largest open neighbourhood eigenvalue and open neighbourhood energy of graphs.

Graph theory approach to exceptional points in wave scattering

In this paper, we use graph theory to solve wave scattering problems in the discrete dipole approximation. As a key result of this work, in the presence of active scatterers, we present a systematic method to find arbitrary large-order zero eigenvalue exceptional points (EPs). This is achieved by solving a set of non-linear equations that we interpret, in a graph theory picture, as vanishing sums of scattering events. We then show how the total field of the system responds to parameter perturbations at the EP. Finally, we investigate the sensitivity of the power output to imaginary perturbation in the design frequency. This perturbation can be employed to trade sensitivity for a different dissipation balance of the system. The purpose of the results of this paper is manifold. On the one hand, we aim to shed light on the link between graph theory and wave scattering. On the other hand, the results of this paper find application in all those settings where zero eigenvalue EPs play a unique role like in coherent perfect absorption structures.

Stability and local bifurcations of single-mode equilibrium states of the Ginzburg-Landau variational equation

One of the versions of the generalized variational Ginzburg-Landau equation is considered, supplemented by periodic boundary conditions. For such a boundary value problem, the question of existence, stability, and local bifurcations of single-mode equilibrium states is studied. It is shown that in the case of a nearly critical threefold zero eigenvalue, in the problem of stability of single-mode spatially inhomogeneous equilibrium states, subcritical bifurcations of two-dimensional invariant tori filled with spatially inhomogeneous equilibrium states are realized. The analysis of the stated problem is based on such methods of the theory of infinite-dimensional dynamical systems as the theory of invariant manifolds and the apparatus of normal forms. Asymptotic formulas are obtained for the solutions that form invariant tori.

Generalized Decay Synchronization of Directly Coupled Stochastic Dynamic Networks With Multiple Time Delays

This article is dedicated to addressing the generalized decay synchronization problem of directly coupled stochastic dynamic networks with multiple time delays. First, by means of the presentations of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi $ </tex-math></inline-formula> -type function and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\psi $ </tex-math></inline-formula> -type stability, the definitions of decay synchronization <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\psi $ </tex-math></inline-formula> -type synchronization) are introduced. After that, by constructing appropriate feedback controllers and the Lyapunov functional method, we study the decay synchronization for multiple coupled dynamic networks. Moreover, for different coupling matrices, their zero eigenvalues are correspondingly denoted by different normalized left eigenvectors (NLEVec), so it is difficult to establish the functional based on these NLEVec. With the help of the weighted combination of NLEVec for multiple coupling matrices, under the assumption that the Chebyshev gap among NLEVec is less than an allowable deviation interval, some generalized decay (anti-) synchronization conditions are established. Furthermore, we discuss exponential synchronization and polynomial synchronization as special cases, and more relevant criteria are acquired. Examples are provided to verify the effectiveness of those obtained conditions.

Characterizing Graphs with Nullity n-4

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in the spectrum of G. A unified approach is presented for the characterization of graphs of order n with η(G) = n−4. All known results on trees, unicyclic graphs, bicyclic graphs, graphs with minimum degree 1, and r-partite graphs, for which η(G) = n−4 are shown to be corollaries of a theorem of Chang, Huang and Yeh that characterizes all graphs with nullity n − 4.

Recent advances in coherent perfect absorber-lasers and their future applications

In recent years, peculiar physical phenomena enabled by non-Hermitian systems, especially the parity-time (PT)-symmetric systems, have drawn tremendous research interests. Particularly, special spectral degeneracies known as exceptional points (EPs) and coherent perfect absorber-laser (CPAL) points where zero and infinite large eigenvalues coexist are the most popular topics to be studied. To date, the discussions of EPs that serve as transition boundaries between broken PT-symmetry phase and exact PT-symmetry phase have been intensively presented. However, the theoretical analysis and experimental validations of CPAL points are inadequate. Different from EPs, CPAL points, as a special solution of broken PT-symmetry phase, may exhibit even further counterintuitive physical features, which may have significant implications to study non-Hermitian physics. Here, we review some recent advances of CPAL phenomena in different sub-disciplines of physics, including optics, electronics and electromagnetics, and acoustics. Additionally, we also provide an envision of future directions and applications of CPAL systems.

Bifurcations of a predator–prey system with cooperative hunting and Holling III functional response

In this paper, we consider the dynamics of a predator–prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known work numerically shows that the system exhibits saddle-node and Hopf bifurcations except homoclinic bifurcation for some special parameter values. Our results show that there are a weak focus of multiplicity three and a degenerate equilibrium with double zero eigenvalues (i.e., a cusp of codimension two) for general parameter conditions and the system can exhibit various bifurcations as perturbing the bifurcation parameters appropriately, such as the transcritical and the pitchfork bifurcations at the degenerate boundary equilibrium, the saddle-node and the Bogdanov–Takens bifurcations at the degenerate positive equilibrium and the Hopf bifurcation around the weak focus. The comparative study demonstrates that the dynamics are far richer and more complex than that of the system without cooperative hunting among predators. The analysis results reveal that appropriate intensity of cooperative hunting among predators is beneficial for the persistence of predators and the diversity of ecosystem.

Deep learning, stochastic gradient descent and diffusion maps

Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major challenge. It has been observed empirically that most eigenvalues of the Hessian of the loss functions on the loss landscape of over-parametrized deep neural networks are close to zero, while only a small number of eigenvalues are large. Zero eigenvalues indicate zero diffusion along the corresponding directions. This indicates that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian. Although the parameter space is very high-dimensional, these findings seems to indicate that the SGD dynamics may mainly live on a low-dimensional manifold. In this paper, we pursue a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter surface, and in particular, of the landscape traced out by SGD by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration, we use diffusion maps introduced by R. Coifman and coauthors.

Active Synchronization for Double-Integrator Network Systems Without Velocity Information

Of particular interest in this paper is to study the active synchronization (act-synchronization) problem for double-integrator network systems without velocity information. To this end, scaling parameters are suggested to quantify the degree of act-synchronization for different agents. And weighted gains are employed to assure that there is a simple zero eigenvalue with the other ones sharing positive real parts. An auxiliary variable is introduced to mitigate the unavailability on velocity information. Thus only the directed spanning tree requirement in the content of algebraic graph theory is required. This exhibits a clear distinction from signed-graph or scaled consensus as non-unitary signs for scaling parameter and weighted gain are allowed. It is shown that the condition for act-synchronization is related to weighted gain, scaling parameter, the real and imaginary parts of the involved eigenvalues. With the constraint on identical weighted gain and scaling parameter for position and auxiliary variables, it is proven that act-synchronization can be exponentially guaranteed using the Lyapunov-based technique. Finally, the multiple unmanned ground vehicles system and the 10-generator in IEEE 39-bus system are given to facilitate the proposed configuration and the theoretical findings.

Relation between the nullity of a graph and its matching number

Let G be a connected graph of order n(G) and size e(G). The nullity of G, denoted by η(G), is the multiplicity of eigenvalue zero of the adjacency matrix of G. In 2014, Wang and Wong (2014) bounded η(G) as n(G)−2m(G)−c(G)≤η(G)≤n(G)−2m(G)+2c(G), where m(G) is the matching number of G and c(G), defined by c(G)=e(G)−n(G)+1, is the dimension of cycle space of G. We find that the upper and the lower bounds for η(G) both fail to accurate if the edges in G are dense, namely if c(G) is large enough. In this paper, we aim to improve the above bounds. It is proved that n(G)−2m(G)−σ(G)≤η(G)≤n(G)−2m(G)+2ω(G), where σ(G) is the largest number of disjoint odd cycles in G and ω(G) is the number of even cycles in G. The cycle-disjoint connected graphs with nullity n(G)−2m(G)−σ(G) are characterized.