Largest Eigenvalue In Absolute Value Research Articles

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.

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.

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.

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.

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.

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).

A lower bound on the spectral radius of the universal cover of a graph

For a finite connected graph G let ρ( G ̃ ) be the spectral radius of its universal cover. We prove that ρ( G ̃ )⩾2 d−1 for any graph G of average degree d⩾2 and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph G after deleting any radius r⩾2 ball is at least d⩾2, then its second largest eigenvalue in absolute value λ( G) is at least 2 d−1 (1−c log r r ) for some absolute constant c. This result is tight in the sense that we can construct graphs with high average degree and diameter but small λ( G). For bipartite graphs with minimal degree at least two, we prove that ρ( G ̃ )⩾ d L −1 + d R −1 , where d L, d R are the average degrees on the left and right hand sides.

ON THE CONVERGENCE OF METROPOLIS-TYPE RELAXATION AND ANNEALING WITH CONSTRAINTS

We discuss the asymptotic behavior of time-inhomogeneous Metropolis chains for solving constrained sampling and optimization problems. In addition to the usual inverse temperature schedule (βn)n∈[hollow N]*, the type of Markov processes under consideration is controlled by a divergent sequence (θn)n∈[hollow N]* of parameters acting as Lagrange multipliers. The associated transition probability matrices (Pβn,θn)n∈[hollow N]* are defined by Pβ,θ = q(x, y)exp(−β(Wθ(y) − Wθ(x))+) for all pairs (x, y) of distinct elements of a finite set Ω, where q is an irreducible and reversible Markov kernel and the energy function Wθ is of the form Wθ = U + θV for some functions U,V : Ω → [hollow R]. Our approach, which is based on a comparison of the distribution of the chain at time n with the invariant measure of Pβn,θn, requires the computation of an upper bound for the second largest eigenvalue in absolute value of Pβn,θn. We extend the geometric bounds derived by Ingrassia and we give new sufficient conditions on the control sequences for the algorithm to simulate a Gibbs distribution with energy U on the constrained set [Ω with tilde above] = {x ∈ Ω : V(x) = minz∈ΩV(z)} and to minimize U over [Ω with tilde above].

Geometric inequalities for the eigenvalues of concentrated Markov chains

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

Geometric inequalities for the eigenvalues of concentrated Markov chains

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of same size of a regular graph in terms of the ith largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the ith largest eigenvalue for any integer i. Our bounds are shown to be asymptotically tight for explicit families of graphs having an asymptotically optimal ith largest eigenvalue. A result by Quenell [Adv. Math., 106 (1994), pp. 122--148] relating the diameter, the second eigenvalue, and the girth of a regular graph is obtained as a by-product.

On the second eigenvalue and random walks in randomd-regular graphs

The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, λ2, of (the adjacency matrix of) a randomd-regular graph,G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at thek-th step, for various values ofk. Our main theorem about eigenvalues is that $$E|\lambda _2 (G)|^m \leqslant \left( {2\sqrt {2d - 1} \left( {1 + \frac{{\log d}}{{\sqrt {2d} }} + 0\left( {\frac{1}{{\sqrt d }}} \right)} \right) + 0\left( {\frac{{d^{3/2} \log \log n}}{{\log n}}} \right)} \right)^m $$ for any\(m \leqslant 2\left\lfloor {log n\left\lfloor {\sqrt {2d - } 1/2} \right\rfloor /\log d} \right\rfloor \), where E denotes the expected value over a certain probability space of 2d-regular graphs. It follows, for example, that for fixedd the second eigenvalue's magnitude is no more than\(2\sqrt {2d - 1} + 2\log d + C'\) with probability 1−n −C for constantsC andC′ for sufficiently largen.

Diameter, Covering Index, Covering Radius and Eigenvalues

Fan Chung has recently derived an upper bound on the diameter of a regular graph as a function of the second largest eigenvalue in absolute value. We generalize this bound to the case of bipartite biregular graphs, and regular directed graphs.We also observe the connection with the primitivity exponent of the adjacency matrix. This applies directly to the covering number of Finite Non Abelian Simple Groups (FINASIG). We generalize this latter problem to primitive association schemes, such as the conjugacy scheme of Paige's simple loop.By noticing that the covering radius of a linear code is the diameter of a Cayley graph on the cosets, we derive an upper bound on the covering radius of a code as a function of the scattering of the weights of the dual code. When the code has even weights, we obtain a bound on the covering radius as a function of the dual distance dl which is tighter, for d⊥ large enough, than the recent bounds of Tietäväinen.