Absolute Values Of Eigenvalues Research Articles

Bounds on the minimum edge dominating energy of induced subgraphs of a graph

Let [Formula: see text] be a graph with [Formula: see text] vertices and [Formula: see text] edges. The minimum edge dominating energy is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph [Formula: see text]. In this paper, we establish some bounds for the minimum edge dominating energy of subgraphs of a graph.

Near-perfect clique-factors in sparse pseudorandom graphs

AbstractWe prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

AbstractLet (λ, v) be a known real eigenpair of an n×n real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.

The relation between the minimum edge dominating energy and the other energies

Let [Formula: see text] be a graph of the order [Formula: see text] and size [Formula: see text]. The minimum edge dominating energy is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of the graph [Formula: see text]. In this paper, we establish relations between the minimum edge dominating energy of a graph [Formula: see text] and the graph energy, the energy of the line graph, signless Laplacian energy of [Formula: see text].

Some properties of ergodicity coefficients with applications in spectral graph theory

The main result is Corollary 2.9 which provides upper bounds on, and even better, approximates the largest non-trivial eigenvalue in absolute value of real constant row-sum matrices by the use of vector norm-based ergodicity coefficients . If the constant row-sum matrix is nonsingular, then it is also shown how its smallest non-trivial eigenvalue in absolute value can be bounded by using . In the last section, these two results are applied to bound the spectral radius of the Laplacian matrix as well as the algebraic connectivity of its associated graph. Many other results are obtained. In particular, Theorem 2.15 is a convergence theorem for and Theorem 4.7 says that is less than or equal to for the Laplacian matrix of every simple graph. An application related to the stability of Markov chains is discussed. Other discussions, open questions and examples are provided.

A note on the minimum edge dominating energy of graphs

Let G be a graph with n vertices and m edges. The edges energy is defined as the sum of the absolute values of eigenvalues of the adjacency matrix of line graph of G. In this paper, the minimum edge dominating energy of the graph G is introduced and the minimum edge dominating energy of some graphs is computed. We also investigate the bounds of the minimum edge dominating energy of graphs.

Clique-factors in sparse pseudorandom graphs

We prove that for any t≥3 there exist constants c>0 and n0 such that any d-regular n-vertex graph G with t∣n≥n0 and second largest eigenvalue in absolute value λ satisfying λ≤cdt∕nt−1 contains a Kt-factor, that is, vertex-disjoint copies of Kt covering every vertex of G. The result generalizes to broader setting of jumbled graphs, which were introduced by Thomason in the eighties.

Parametric Stability Analysis of Active Magnetic Bearing Supported Rotor System With a Novel Control Law Subject to Periodic Base Motion

Active magnetic bearings (AMBs) are being continuously explored for industrial applications mainly because of their friction-free operation. However, if a rotor-shaft-AMB system is used in applications such as turbo engine of an aircraft or in the propeller shaft of a ship, it would be subject to parametric excitation because of the moving base of the system. Parametric excitation is known to cause instability in systems. This paper addresses the issue of such flexible rotor-AMB system and performs the parametric stability analysis. A novel control law based on the constitutive relationship of a viscoelastic material called the four-element control law is developed. Because of the presence of periodic base motion terms in the system state matrix, the system can be classified as a linear periodic system. Thus, the stability analysis is carried out using Floquet–Liapunov theory on stability of periodic system, for which an approximate state transition matrix is first found and stability of the system is then decided based on the absolute value of eigenvalue. Three different base motions (roll, pitch, and yaw) were considered for evaluating the performance of the proposed control law. Simulation results reveal that the proposed control law has better stability characteristics than the proportional-integral-derivative (PID) control law.

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam & Sotirov on the smallest eigenvalue of (distance-j) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-j) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical P- and Q-polynomial association schemes.

Near-perfect clique-factors in sparse pseudorandom graphs

We prove that, for any t≥3, there exists a constant c=c(t)>0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying λ≤cdt−1/nt−2 contains (1−o(1))n/t vertex-disjoint copies of Kt. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), pp. 403–426] that (n,d,λ)-graphs with n∈3N and λ≤cd2 for a suitably small absolute constant c>0 contain triangle-factors.

Size biased couplings and the spectral gap for random regular graphs

Let $\lambda$ be the second largest eigenvalue in absolute value of a uniform random $d$-regular graph on $n$ vertices. It was famously conjectured by Alon and proved by Friedman that if $d$ is fixed independent of $n$, then $\lambda=2\sqrt{d-1} +o(1)$ with high probability. In the present work we show that $\lambda=O(\sqrt{d})$ continues to hold with high probability as long as $d=O(n^{2/3})$, making progress towards a conjecture of Vu that the bound holds for all $1\le d\le n/2$. Prior to this work the best result was obtained by Broder, Frieze, Suen and Upfal (1999) using the configuration model, which hits a barrier at $d=o(n^{1/2})$. We are able to go beyond this barrier by proving concentration of measure results directly for the uniform distribution on $d$-regular graphs. These come as consequences of advances we make in the theory of concentration by size biased couplings. Specifically, we obtain Bennett-type tail estimates for random variables admitting certain unbounded size biased couplings.

Irregular Hodge Theory

We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\exp\varphi$ for any meromorphic function $\varphi$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also show that those rigid irreducible holonomic D-modules on the complex projective line whose local formal monodromies have eigenvalues of absolute value one, are equipped with such an irregular Hodge filtration in a canonical way, up to a shift of the filtration. In a chapter written jointly with Jeng-Daw~Yu, we make explicit the case of irregular mixed Hodge structures, for which we prove in particular a Thom-Sebastiani formula.

Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials

In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing $e^{-\tau A}v$ for a given $\tau>0$ and $v\in\mathbb{C}^n$, where $A$ is a large sparse non-Hermitian matrix. The a priori error bounds relate the convergence to $\lambda_{\min}(\frac{A+A^*}{2})$, $\lambda_{\max}(\frac{A+A^*}{2})$ (the smallest and the largest eigenvalue of the Hermitian part of $A$), and $|\lambda_{\max}(\frac{A-A^*}{2})|$ (the largest eigenvalue in absolute value of the skew-Hermitian part of $A$), which define a rectangular region enclosing the field of values of $A$. In particular, our bounds explain an observed convergence behavior where the error may first stagnate for a certain number of iterations before it starts to converge. The special case that $A$ is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.

Extremal values of matching energies of one class of graphs

In 1978, Gutman proposed the concept of graph energy, defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph, which is related to the energy of π-electrons in conjugated hydrocarbons. Recently, Gutman and Wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s. In this paper, we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter. Our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies.

On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

The second largest eigenvalue in absolute value determines the rate of convergence of the Markov chain Monte Carlo methods. In this paper we consider the Gibbs sampler for the 1-D Ising model. We apply the geometric bound by Diaconis and Stroock (1991) to calculate an upper bound of the second largest eigenvalue, which we show is also a bound of the second largest eigenvalue in absolute value. Based on this upper bound, we derive that the convergence time is O(n2), where n is the number of sites. Our result includes a constant of proportionality, which enables us to give a precise bound of the convergence time. The results presented in this paper provide the lowest bound compared to those with a constant of proportionality in the literature.

Maximal Energy of Subdivisions of Graphs with a Fixed Chromatic Number

The energy of a simple graph \(G\), denoted by \(E(G)\), is defined as the sum of the absolute values of eigenvalues of \(G\). In this paper, we show that, among all subdivisions of graphs with \(n\) vertices and chromatic number \(k\), the subdivision of the Turan graph \(T(n,k)\) has the maximal energy.

Lower bounds for boxicity

An axis-parallel b-dimensional box is a Cartesian product R1×–2×...×Rb where Ri is a closed interval of the form [ai; bi] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree δ is at least n/2(n−δ−1). 2. Consider the G(n;p) model of random graphs. Let p ≤ 1 − 40logn/n2. Then with high probability, box(G) = Ω(np(1 − p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Ω(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and \(m \leqslant c\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\) edges have boxicity Ω(m/n). 3. Let G be a connected k-regular graph on n vertices. Let λ be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is at least \(\left( {\frac{{k^2 /\lambda ^2 }} {{\log \left( {1 + k^2 /\lambda ^2 } \right)}}} \right)\left( {\frac{{n - k - 1}} {{2n}}} \right)\). 4. For any positive constant c < 1, almost all balanced bipartite graphs on 2n vertices and m≤cn2 edges have boxicity Ω(m/n).

Bicyclic digraphs with extremal skew energy

Let −→ G be a digraph and S( −→ G) be the skew-adjacency matrix of −→ G . The skew energy of −→ G is the sum of the absolute values of eigenvalues of S( −→ G). In this paper, the bicyclic digraphs with minimal and maximal skew energy are determined.

Logarithmic vector-valued modular forms

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

Performance Analysis Based on Channel Matrix Eigenvalue for MIMO Systems in LOS Environments

This paper makes an examination of BER (bit error rate) performance in 2 × 2 MIMO (Multi-Input Multi-Output) systems using spatial multiplexing in a LOS (line-of-sight) environment by analysis using the eigenvalue and eigenvector of the channel matrix and simulation. It is shown that BER performance in a LOS environment is dependent on the phase relation of direct paths and the minimum squared absolute value of the eigenvalue, which are determined by the propagated distance. A technique for improving BER performance in a LOS environment is then considered based on these findings. A receive antenna selection (RAS) technique is presented which uses the minimum squared absolute value of the channel matrix eigenvalue. Proof is then given that BER performance is improved using this RAS technique, thus making clear the validity of this approach for obtaining optimum BER performance in a LOS environment.