Smallest Spectral Radius Research Articles

A CMOS-compatible oscillation-based VO2 Ising machine solver

Phase-encoded oscillating neural networks offer compelling advantages over metal-oxide-semiconductor-based technology for tackling complex optimization problems, with promising potential for ultralow power consumption and exceptionally rapid computational performance. In this work, we investigate the ability of these networks to solve optimization problems belonging to the nondeterministic polynomial time complexity class using nanoscale vanadium-dioxide-based oscillators integrated onto a Silicon platform. Specifically, we demonstrate how the dynamic behavior of coupled vanadium dioxide devices can effectively solve combinatorial optimization problems, including Graph Coloring, Max-cut, and Max-3SAT problems. The electrical mappings of these problems are derived from the equivalent Ising Hamiltonian formulation to design circuits with up to nine crossbar vanadium dioxide oscillators. Using sub-harmonic injection locking techniques, we binarize the solution space provided by the oscillators and demonstrate that graphs with high connection density (η > 0.4) converge more easily towards the optimal solution due to the small spectral radius of the problem’s equivalent adjacency matrix. Our findings indicate that these systems achieve stability within 25 oscillation cycles and exhibit power efficiency and potential for scaling that surpasses available commercial options and other technologies under study. These results pave the way for accelerated parallel computing enabled by large-scale networks of interconnected oscillators.

Uniform supertrees with extremal spectral radii

A supertree is a connected and acyclic hypergraph. We investigate the supertrees with the extremal spectral radii among several kinds of r-uniform supertrees. First, by using the matching polynomials of supertrees, a new and useful grafting operation is proposed for comparing the spectral radii of supertrees, and its applications are shown to obtain the supertrees with the extremal spectral radii among some kinds of r-uniform supertrees. Second, the supertree with the third smallest spectral radius among the r-uniform supertrees is deduced. Third, among the r-uniform supertrees with a given maximum degree, the supertree with the smallest spectral radius is derived. At last, among the r-uniform starlike supertrees, the supertrees with the smallest and the largest spectral radii are characterized.

Gradient Iterative Method with Optimal Convergent Factor for Solving a Generalized Sylvester Matrix Equation with Applications to Diffusion Equations

We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.

The effect on the spectral radius of r-graphs by grafting or contracting edges

Let Hn(r) be the set of all connected r-graphs with given size n. In this paper, we investigate the effect on the spectral radius of r-uniform hypergraphs by grafting or contracting an edge and then give the ordering of the r-graphs with small spectral radius over Hn(r), when n≥20.

Uniform hypergraphs with the first two smallest spectral radii

The spectral radius of a uniform hypergraph G is the maximum modulus of the eigenvalues of the adjacency tensor of G. For k≥2, among connected k-uniform hypergraphs with m≥1 edges, we show that the k-uniform loose path with m edges is the unique one with smallest spectral radius, and we also determine the unique ones with second smallest spectral radius when m≥2.

On minimally 2-(edge)-connected graphs with extremal spectral radius

A graph is minimally 2-(edge)-connected if it is 2-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not 2-(edge)-connected. In this paper, we completely characterize the minimally 2-(edge)-connected graphs having the largest and the smallest spectral radius, respectively.

Connected hypergraphs with small spectral radius

In 1970 Smith classified all connected graphs with spectral radius at most 2. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to r-uniform hypergraphs. In this paper, we generalize Smith's theorem to r-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected r-uniform hypergraphs is ρr=4r. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected r-uniform hypergraphs with spectral radius at most ρr.

Ordering trees by their spectral radii

Let T be a tree with n vertices and let A(T) be the adjacency matrix of T. Spectral radius of T is the largest eigenvalue of A(T). Wu et al. [Wu, B.F., Yuan, X.Y., and Xiao, E.L. On the spectral radii of trees, Journal of East China Normal University (Natural Science), 3: 22–28 (2004)] determined the first seven trees of order n with the smallest spectral radius. In this paper, we extend this ordering by determining the trees with the eighth to the tenth smallest spectral radius among all trees with n vertices.

Spectral characterizations of graphs with small spectral radius

A graph is said to have a small spectral radius if it does not exceed the corresponding Hoffmann limit value. In the case of (signless) Laplacian matrix, the Hoffmann limit value is equal to ϵ+2=4.38+, with ϵ being the real root of x3-4x-4. Here the spectral characterization of connected graphs with small (signless) Laplacian spectral radius is considered. It is shown that all connected graphs with small Laplacian spectral radius are determined by their Laplacian spectra, and all but one of connected graphs with small signless Laplacian spectral radius are determined by their signless Laplacian spectra.

SMALL ESSENTIAL SPECTRAL RADIUS PERTURBATIONS OF OPERATORS WITH TOPOLOGICAL UNIFORM DESCENT

AbstractIn this paper we consider small essential spectral radius perturbations of operators with topological uniform descent—small essential spectral radius perturbations which cover compact, quasinilpotent and Riesz perturbations.

Integer Symmetric Matrices of Small Spectral Radius and Small Mahler Measure

In a previous paper we completely described cyclotomic matrices--integer symmetric matrices of spectral radius at most 2. In this paper we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular we solve the strong version of problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least Lehmer's number 1.17628... .

A lower bound for the spectral radius of graphs with fixed diameter

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.

An extended [formula omitted] method for dense linear systems

Davey and Rosindale [K. Davey, I. Rosindale, An iterative solution scheme for systems of boundary element equations, Internat. J. Numer. Methods Engrg. 37 (1994) 1399–1411] derived the GSOR method, which uses an upper triangular matrix Ω in order to solve dense linear systems. By applying functional analysis, the authors presented an expression for the optimum Ω . Moreover, Davey and Bounds [K. Davey, S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Comput. 19 (1998) 953–967] also introduced further interesting results. In this note, we employ a matrix analysis approach to investigate these schemes, and derive theorems that compare these schemes with existing preconditioners for dense linear systems. We show that the convergence rate of the Gauss–Seidel method with preconditioner P G is superior to that of the GSOR method. Moreover, we define some splittings associated with the iterative schemes. Some numerical examples are reported to confirm the theoretical analysis. We show that the EGS method with preconditioner P G ( γ opt ) produces an extremely small spectral radius in comparison with the other schemes considered.

A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.

Fourier Convergence Analysis of Two-Dimensional/One-Dimensional Coupling Methods for the Three-Dimensional Neutron Diffusion Eigenvalue Problems

The purpose of this paper is to present the Fourier convergence analysis of four methods for performing two-dimensional/one-dimensional (2-D/1-D) coupling to solve neutron diffusion eigenvalue problems (EVPs). The four methods differ principally in the manner of using the interface currents or node average fluxes to perform the 2-D/1-D coupling. Method A uses net currents, method B employs partial currents, method C uses a current correction factor, and method D uses an analytic expression for the axial net currents. In a previous paper, we analyzed the convergence behavior of these methods for the 2-D/1-D coupling of the fixed source problem (FSP). In this paper, the convergence performance of these methods is analyzed for the EVP using a one-group neutron diffusion EVP in a homogeneous infinite slab geometry. Among the four methods, method A diverges for small mesh sizes as it did in the FSP, whereas the other methods are stable regardless of the mesh size. The spectral radii of methods C and D are identical while the latter had a smaller spectral radius than the former in an FSP. The spectral radii of methods C and D are smaller than that of method B in the range of practical mesh size. The spectral radii approach one for all the methods as the mesh size increases, while in the FSP the spectral radii of method B approached a finite positive value and those of the other methods approached zero. For practical applications, method C has several advantages over the other methods and is the preferred 2-D/1-D coupling method for EVPs.

Iterative Two- and One-Dimensional Methods for Three-Dimensional Neutron Diffusion Calculations

Two methods are proposed for solving the three-dimensional neutron diffusion equation by iterating between solutions of the two-dimensional (2-D) radial and one-dimensional (1-D) axial solutions. In the first method, the 2-D/1-D equations are coupled using a current correction factor (CCF) with the average fluxes of the lower and upper planes and the axial net currents at the plane interfaces. In the second method, an analytic expression for the axial net currents at the interface of the planes is used for planar coupling. A comparison of the new methods is made with two previously proposed methods, which use interface net currents and partial currents for planar coupling. A Fourier convergence analysis of the four methods was performed, and results indicate that the two new methods have at least three advantages over the previous methods. First, the new methods are unconditionally stable, whereas the net current method diverges for small axial mesh size. Second, the new methods provide better convergence performance than the other methods in the range of practical mesh sizes. Third, the spectral radii of the new methods asymptotically approach zero as the mesh size increases, while the spectral radius of the partial current method approaches a nonzero value as the mesh size increases. Of the two new methods proposed here, the analytic method provides a smaller spectral radius than the CCF method, but the CCF method has several advantages over the analytic method in practical applications.

ERROR PROPAGATION OF HHT-α METHOD FOR PSEUDODYNAMIC TESTS

Error propagation for the direct and improved implementations of the subfamily of HHT-α method with favourable numerical dissipation is studied herein for a pseudo-dynamic test. Analytical results show that the direct implementation of the subfamily of HHT-α method is inappropriate for the pseudodynamic testing although it possesses desired numerical dissipation. However, it is found that the error propagation of high frequency responses can be effectively suppressed by numerical dissipation for the improved implementation of this subfamily of integration methods. Furthermore, the error amplification factor for the improved implementation highly depends on the number of remaining time steps to be performed and the spectral radius of the amplification matrix. In fact, a small spectral radius and/or a large number of remaining time steps will lead to a small error amplification factor. This means an error introduced in an early time step will contribute less to the cumulative error at the end of the test as compared to an error introduced in a later time step.

Iterative Solution of Cyclically Reduced Systems Arising from Discretization of the Three-Dimensional Convection-Diffusion Equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection-diffusion model problem. This system is typically nonsymmetric. We show that performing one step of cyclic reduction, followed by reordering of the unknowns, yields a system of equations for which the block Jacobi method generally converges faster than for the original system, using lexicographic ordering. The matrix representing the system of equations can be symmetrized for a large range of the coefficients of the underlying partial differential equation, and the associated iteration matrix has a smaller spectral radius than the one associated with the original system. In this sense, the three-dimensional problem is similar to the one-dimensional and two-dimensional problems, which have been studied by Elman and Golub. The process of reduction, the suggested orderings, and bounds on the spectral radii of the associated iteration matrices are presented, followed by a comparison of the reduced system with the full system and by details of the numerical experiments.

The Effectiveness ofP2and SimplifiedP2Synthetic Accelerations in the Solutions of Discrete Ordinates Transport Equations

Using the operator form of a synthetic acceleration, the P{sub 1} acceleration [diffusion synthetic acceleration (DSA)] and P{sub 2} acceleration schemes for one-dimensional slab and the P{sub 1} and simplified P{sub 2} acceleration schemes for two-dimensional x-y geometry are derived. The convergence rate of each scheme for a simple model problem is compared, and the result is generalized by performing a Fourier analysis. In the one-dimensional case, the new second-moment P{sub 2} acceleration outperforms an earlier third-moment P{sub 2} acceleration developed by Miller and Larsen. However, it is still less efficient than P{sub 1} acceleration. Similar results show that the P{sub 1} acceleration converges faster than the simplified P{sub 2} acceleration in two-dimensional x-y geometry. These results confirm that one cannot simply assume that replacement of the DSA method with a higher order operator will lead to a smaller spectral radius.