scholarly journals The circular law for random regular digraphs with random edge weights

2017 ◽  
Vol 06 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Nicholas Cook
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Spectral Distribution ◽  
Hadamard Product ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Unit Variance ◽  
The Matrix

We consider random [Formula: see text] matrices of the form [Formula: see text], where [Formula: see text] is the adjacency matrix of a uniform random [Formula: see text]-regular directed graph on [Formula: see text] vertices, with [Formula: see text] for some fixed [Formula: see text], and [Formula: see text] is an [Formula: see text] matrix of i.i.d. centered random variables with unit variance and finite [Formula: see text]th moment (here ∘ denotes the matrix Hadamard product). We show that as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges weakly in probability to the normalized Lebesgue measure on the unit disk.

scholarly journals Circular law for random matrices with unconditional log-concave distribution

2015 ◽  
Vol 17 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Radosław Adamczak ◽  
Djalil Chafaï
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Real Matrix ◽  
Random Real ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Ginibre Ensemble ◽  
Mean And Variance ◽  
Uniform Law ◽  
Special Case

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

scholarly journals On optimal bounds in the local semicircle law under four moment condition

2019 ◽  
Vol 484 (3) ◽  
pp. 265-268
Author(s):  
F. Götze ◽  
A. A. Naumov ◽  
A. N. Tikhomirov
Keyword(s):  
Spectral Distribution ◽  
Moment Condition ◽  
Finite Moment ◽  
Semicircle Law ◽  
Empirical Spectral Distribution ◽  
Local Semicircle Law ◽  
Unit Variance ◽  
Optimal Bounds ◽  
Order Four ◽  
Stieltjes Transforms

We consider symmetric random matrices with independent mean zero and unit variance entries in the upper triangular part. Assuming that the distributions of matrix entries have finite moment of order four, we prove optimal bounds for the distance between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law. Application concerning the convergence rate in probability of the empirical spectral distribution to the semicircle law is discussed as well.

scholarly journals The limit empirical spectral distribution of complex matrix polynomials

2021 ◽  
pp. 2250023
Author(s):  
Giovanni Barbarino ◽  
Vanni Noferini
Keyword(s):  
Spectral Distribution ◽  
Matrix Polynomial ◽  
Matrix Coefficient ◽  
Finite Variance ◽  
Complex Matrix ◽  
Hyperbolic Polynomials ◽  
Empirical Spectral Distribution ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Uniformly Bounded

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

Patterned sparse random matrices: A moment approach

2017 ◽  
Vol 06 (03) ◽  
pp. 1750011
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Random Matrices ◽  
Limit Distribution ◽  
Spectral Distribution ◽  
Explicit Description ◽  
Circulant Matrices ◽  
Hankel Matrices ◽  
Largest Eigenvalue ◽  
Empirical Spectral Distribution ◽  
Moment Approach ◽  
The Largest Eigenvalue

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

scholarly journals MONOTONE INDEPENDENCE, COMB GRAPHS AND BOSE–EINSTEIN CONDENSATION

2004 ◽  
Vol 07 (03) ◽  
pp. 419-435 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANIS BEN GHORBAL ◽  
NOBUAKI OBATA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Adjacency Matrix ◽  
Spectral Distribution ◽  
Vacuum State ◽  
Independent Random Variables ◽  
Explicit Calculation ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Bose Einstein

The adjacency matrix of a comb graph is decomposed into a sum of monotone independent random variables with respect to the vacuum state. The vacuum spectral distribution is shown to be asymptotically the arcsine law as a consequence of the monotone central limit theorem. As an example the comb lattice is studied with explicit calculation.

scholarly journals The Pre-Schwarzian Norm Estimate for Analytic Concave Functions

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Young Jae Sim ◽  
Oh Sang Kwon
Keyword(s):  
Unit Disk ◽  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Univalent Functions ◽  
Open Unit ◽  
Opening Angle ◽  
Sufficient Condition ◽  
Open Unit Disk ◽  
Concave Functions ◽  
Norm Estimate

LetDdenote the open unit disk and letSdenote the class of normalized univalent functions which are analytic inD. LetCo(α)be the class of concave functionsf∈S, which have the condition that the opening angle off(D)at infinity is less than or equal toπα,α∈(1,2]. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the classCo(α). And we define a classCo(α,A,B),(-1≤B<A≤1), which is a subclass ofCo(α)and we find the set of variabilities for the functional(1-|z|2)(f″(z)/f′(z))forf∈Co(α,A,B). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions inCo(α,A,B). We also give a characterization for functions inCo(α,A,B)in terms of Hadamard product.

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