Distinct Eigenvalues Research Articles

A Note on Eigenvalues and Asymmetric Graphs

This note is intended as a contribution to the study of quantitative measures of graph complexity that use entropy measures based on symmetry. Determining orbit sizes of graph automorphism groups is a key part of such studies. Here we focus on an extreme case where every orbit contains just a single vertex. A permutation of the vertices of a graph G is an automorphism if, and only if, the corresponding permutation matrix commutes with the adjacency matrix of G. This fact establishes a direct connection between the adjacency matrix and the automorphism group. In particular, it is known that if the eigenvalues of the adjacency matrix of G are all distinct, every non-trivial automorphism has order 2. In this note, we add a condition to the case of distinct eigenvalues that makes the graph asymmetric, i.e., reduces the automorphism group to the identity alone. In addition, we prove analogous results for the Google and Laplacian matrices. The condition is used to build an O(n3) algorithm for detecting identity graphs, and examples are given to demonstrate that it is sufficient, but not necessary.

Geometry of SU(3)-character varieties of torus knots

We describe the geometry of the character variety of representations of the knot group Γm,n=〈x,y|xn=ym〉 into the group SU(3), by stratifying the character variety into strata corresponding to totally reducible representations, representations decomposing into a 2-dimensional and a 1-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity 2. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for SU(3) into the character variety for SL(3,C) is a homotopy equivalence.

On bounds of Aα-eigenvalue multiplicity and the rank of a complex unit gain graph

Let Φ=(G,φ) be a connected complex unit gain graph (T-gain graph) of n vertices with largest vertex degree Δ, adjacency matrix A(Φ), and degree matrix D(Φ). Let mα(Φ,λ) be the multiplicity of λ as an eigenvalue of Aα(Φ):=αD(Φ)+(1−α)A(Φ), for α∈[0,1). In this article, we establish that mα(Φ,λ)≤(Δ−2)n+2Δ−1 and characterize the sharpness. Then, we obtain some lower bounds for the rank r(Φ) in terms of n and Δ including r(Φ)≥n−2Δ−1 and characterize their sharpness. Besides, we introduce zero-2-walk gain graphs and study their properties. It is shown that a zero-2-walk gain graph is always regular. Furthermore, we prove that Φ has exactly two distinct eigenvalues with equal magnitude if and only if it is a zero-2-walk gain graph. Using this, we establish a lower bound of r(Φ) in terms of the number of edges and characterize the sharpness. Result about mα(Φ,λ) extends the corresponding known result for undirected graphs and simplifies the existing proof, and other bounds of r(Φ) obtained in this article work better than the bounds given elsewhere.

Invariant spinors on homogeneous spheres

Using the characterization of the spin representation in terms of exterior forms, we give a complete classification of invariant spinors on the nine homogeneous realizations of the sphere Sn. In each of the cases we determine the dimension of the space of such spinors, give their explicit description, and study the underlying related geometric structures depending on the metric. We recover some known results in the Sasaki and 3-Sasaki cases and find several new examples: in particular we give the first known examples of generalized Killing spinors with four distinct eigenvalues.

Signed graphs with integral net Laplacian spectrum

Given a signed graph , let and be its standard adjacency matrix and the diagonal matrix of net-degrees, respectively. The net Laplacian matrix of is defined as . In this paper we investigate signed graphs whose net Laplacian spectrum consists entirely of integers. The focus is mainly on the two extreme cases, the one in which all eigenvalues of are simple and the other in which has 2 or 3 (distinct) eigenvalues. Both cases include structure theorems, degree constraints and particular constructions of new examples. Several applications in the framework of control theory are reported.

Synthesizing efficient circuits for Hamiltonian simulation

We provide an approach for compiling quantum simulation circuits that appear in Trotter, qDRIFT and multi-product formulas to Clifford and non-Clifford operations that can reduce the number of non-Clifford operations. The total number of gates, especially CNOT, reduce in many cases. We show that it is possible to implement an exponentiated sum of commuting Paulis with at most m (controlled)-rotation gates, where m is the number of distinct non-zero eigenvalues (ignoring sign). Thus we can collect mutually commuting Hamiltonian terms into groups satisfying one of several symmetries identified in this work. This allows an inexpensive simulation of the entire group of terms. We further show that the cost can in some cases be reduced by partially allocating Hamiltonian terms to several groups and provide a polynomial time classical algorithm that can greedily allocate the terms to appropriate groupings.

√2-estimation for smooth eigenvectors of matrix-valued functions

Summary Modern statistical methods for multivariate time series rely on the eigendecomposition of matrix-valued functions such as time-varying covariance and spectral density matrices. The curse of indeterminacy or misidentification of smooth eigenvector functions has not received much attention. We resolve this important problem and recover smooth trajectories by examining the distance between the eigenvectors of the same matrix-valued function evaluated at two consecutive points. We change the sign of the next eigenvector if its distance with the current one is larger than the square root of 2. In the case of distinct eigenvalues, this simple method delivers smooth eigenvectors. For coalescing eigenvalues, we match the corresponding eigenvectors and apply an additional signing around the coalescing points. We establish consistency and rates of convergence for the proposed smooth eigenvector estimators. Simulation results and applications to real data confirm that our approach is needed to obtain smooth eigenvectors.

Graphs with three distinct distance eigenvalues

In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue −1, and determine all triangle-free connected graphs with three distinct distance eigenvalues of which the smallest one is equal to −3. Furthermore, we characterize all trees with three distinct distance eigenvalues.

On graphs with a few distinct reciprocal distance Laplacian eigenvalues

<abstract><p>For a $ \nu $-vertex connected graph $ \Gamma $, we consider the reciprocal distance Laplacian matrix defined as $ RD^L(\Gamma) = RT(\Gamma)-RD(\Gamma) $, i.e., $ RD^L(\Gamma) $ is the difference between the diagonal matrix of the reciprocal distance degrees $ RT(\Gamma) $ and the Harary matrix $ RD(\Gamma) $. In this article, we determine the graphs with exactly two distinct reciprocal distance Laplacian eigenvalues.We completely characterize the graph classes with a $ RD^L $ eigenvalue of multiplicity $ \nu-2 $. Moreover, we characterize families of graphs with reciprocal distance Laplacian eigenvalue whose multiplicity is $ \nu-3 $.</p></abstract>

Distributed Estimation of the Laplacian Spectrum via Wave Equation and Distributed Optimization

This paper presents a distributed algorithm to estimate all distinct eigenvalues of the Laplacian matrix encoding the unknown topology of a multi-agent system. The agents interact according to the discrete-time wave equation so that their state trajectory persistently oscillates with modes that depend on the eigenvalues of the Laplacian matrix. In this way, the problem of distributed estimation of the eigenvalues of the Laplacian is recast into that of estimating the modes of evolution of the state-trajectory of a linear dynamical system. Unlike previous literature, this paper formulates a distributed optimization problem where, by considering its own state trajectory, each agent estimates all distinct eigenvalues of the Laplacian matrix. The main advantages of the proposed algorithm are the ability of each agent to estimate also eigenvalues corresponding to modes unobservable from its own state trajectory, a much greater numerical stability, and therefore improved scalability to large networks wit h respect to competing approaches, as evidenced by the numerical comparisons.

Rotating anisotropic stringy spheroid in a modified Hartle formalism

Here we look at an application of the Hartle metric to describe a rotating version of the spherical string cloud/ global monopole solution. While rotating versions of this solution have previously been constructed via the Newman-Janis algorithm, that process does not preserve the equation of state. The Hartle method allows for preservation of equation of state, at least in the sense of a slowly rotating perturbative solution. In addition to the direct utility of generating equations which could be used to model a region of a rotating string cloud or similar system, this work shows that it is possible to adapt the Hartle metric to slowly rotating anisotropic systems with Segre type [(11)(1,1)] following an equation of state between the distinct eigenvalues.

On strongly walk regular graphs, triple sum sets and their codes

Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues k> theta _1> theta _2 > theta _3, and that the triple (theta _1, theta _2, theta _3) satisfies a certain homogeneous polynomial equation of degree s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for s = 5 and s = 7, there are only the obvious solutions, and we conjecture this to remain true for all (odd) s ge 9.

Synthesis of unequally-spaced arrays using the fractional Fourier series

This paper introduces an algebraic method to synthesize the radiation pattern of unequally-spaced arrays. In this method, first, the characteristic matrix, containing the sampled data of the desired pattern, is established. Then, the fractional Fourier series is calculated using the distinct eigenvalues and eigenvectors of the characteristic matrix. In the following, the location of the array elements is estimated using the eigenvalues by considering the mutual coupling effect. The magnitude and phase of the excitation currents are computed using the least square method. Also, it is proved that the number of elements can be reduced using the eigenvalue decomposition and differential spectrum method. Several important patterns are investigated to verify the performance of the proposed method, and then a comprehensive discussion is expressed for all cases. It is shown that using the proposed method, a greater average spacing is achieved which allows bettering mitigate the phenomenon of mutual coupling. Furthermore, the various results show that the proposed method offers a better approximation of the desired array factor especially for beam-shaped patterns. This method also has a noble ability to reduce the number of array elements compared to a generic reference array, while still retaining a good ability to approximate the desired pattern fairly faithfully.

General solutions to decay chain equations

We present an explicit compact form of the general solution for linear decay chains with repeated eigenvalues. Comparing with previous solutions of the Bateman equations with identical decay constants, our expression brings new contributions in two ways. First, it is a closed analytical expression valid for arbitrary initial conditions, whereas available solutions typically concern the simplest case of null initial population for all but the first member of the chain. Second, our expression is written in terms of a minimal number of factors and sums, avoiding large nested sums, large products, complicated implicit conditions and high order derivatives. The efficiency of our expression is illustrated with computational implementations of both analytical and numerical applications. To complement our result for linear chains, we present also the solution, for arbitrary initial conditions, of the basic cyclic chain, restricted to the case of negative and distinct eigenvalues.

Eigenvalues and critical groups of Adinkras

Adinkras are signed graphs used to study supersymmetry in physics. We provide an introduction to these objects, and study the properties of their signed adjacency and signed Laplacian matrices. These matrices each have exactly two distinct eigenvalues (of equal multiplicity), making Adinkras closely related to the notions of strongly regular graphs. We also study the critical groups of Adinkras, and in particular determine their odd components. A novel technique of independent interest is used which considers modules over polynomial rings in order to study critical groups as Z-modules.

Inverse eigenvalue and related problems for hollow matrices described by graphs

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow inverse eigenvalue problems for paths and complete bipartite graphs are presented. Results for related subproblems such as possible ordered multiplicity lists, maximum multiplicity of an eigenvalue, and minimum number of distinct eigenvalues are presented for additional families of graphs.

Quantum walks driven by quantum coins with two multiple eigenvalues

We consider a spectral analysis on the quantum walks on graph $$G=(V,E)$$ with the local coin operators $$\{C_u\}_{u\in V}$$ and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues $$\kappa ,\kappa '$$ and $$p=\dim (\ker (\kappa -C_u))$$ for any $$u\in V$$ with $$1\le p\le \delta (G)$$ , where $$\delta (G)$$ is the minimum degrees of G. We show that this quantum walk can be decomposed into a cellular automaton on $$\ell ^2(V;\mathbb {C}^p)$$ whose time evolution is described by a self adjoint operator T and its remainder. We obtain how the eigenvalues and its eigenspace of T are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on $$\mathbb {Z}^d$$ with the moving shift in the Fourier space.

Nonlinear Spectrum Modulation in the Anomalous Dispersion Regime Using Second- and Third-Order Solitons

We study the robustness of a nonlinear frequency-division multiplexing (NFDM) system, based on the Zakharov-Shabat spectral problem (ZSSP), that is comprised of two independent quadrature phase-shift keyed (QPSK) channels modulated in the discrete spectrum associated with two distinct eigenvalues. Among the many fiber impairments that may limit this system, we focus on determining the limits due to third-order dispersion, the Raman effect, amplified spontaneous emission (ASE) noise from erbium-doped fiber amplifiers (EDFAs), and fiber losses with lumped gain from EDFAs. We examine the impact of these impairments on a 1600-km system by analyzing the Q-factor calculated from the error vector magnitude (EVM) of the received symbols. We found that the maximum launch power due to these impairments is: 13 dBm due to third-order dispersion, 11 dBm due to the Raman effect, 3 dBm due to fiber losses with lumped gain, and 2 dBm due to these three impairments combined with ASE noise. The maximum launch power due to all these impairments combined is comparable to that of a conventional wavelength-division multiplexing (WDM) system, even though WDM systems can operate over a much larger bandwidth and, consequently, have a much higher data throughput when compared with NFDM systems. We find that fiber losses in practical fiber transmission systems with lumped gain from EDFAs is the most stringent limiting factor in the performance of this NFDM system.

Minimum number of distinct eigenvalues allowed by a sign pattern

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n×n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2×2 and the irreducible 3×3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern.

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only C6. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.