scholarly journals Convergence of the spectral radius of a random matrix through its characteristic polynomial

2021 ◽  
Author(s):  
Charles Bordenave ◽  
Djalil Chafaï ◽  
David García-Zelada
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Random Matrix

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.

How random is the characteristic polynomial of a random matrix ?

1993 ◽  
Vol 114 (3) ◽  
pp. 507-515 ◽  
Author(s):  
Jennie C. Hansen ◽  
Eric Schmutz
Keyword(s):  
Finite Field ◽  
Characteristic Polynomial ◽  
Random Matrix ◽  
A Priori ◽  
Equal Frequency ◽  
Precise Version

AbstractEvery monic, degree n polynomial in Fq[x;] is the characteristic polynomial of at least one n × n matrix (with entries in the finite field Fq), but they do not appear with equal frequency. There is no a priori reason that the characteristic polynomial of a typical matrix should resemble a typical monic degree n polynomial. Nevertheless, we prove a precise version of the following heuristic statement: ‘Excepting its small factors, the characteristic polynomial of a random matrix is random.’

On the Laplacian characteristic polynomials of mixed graphs

2015 ◽  
Vol 30 ◽  
Author(s):  
Dariush Kiani ◽  
Maryam Mirzakhah
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Characteristic Polynomials ◽  
Laplacian Spectral Radius ◽  
Mixed Graph ◽  
First Derivative ◽  
Signless Laplacian ◽  
Mixed Graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial of a mixed graph is reconstructible from the collection of the Laplacian characteristic polynomials of its edge deleted subgraphs. Then, it is investigated how graph modifications affect the mixed Laplacian characteristic polynomial. Also, a connection between the Laplacian characteristic polynomial of a non-singular connected mixed graph and the signless Laplacian characteristic polynomial is provided, and it is used to establish a lower bound for the spectral radius of L(G). Finally, using Coates digraphs, the perturbation of the mixed Laplacian spectral radius under some graph transformations is discussed.

A Lagrange series approach to the spectrum of the Kite

2015 ◽  
Vol 30 ◽  
pp. 934-943
Author(s):  
Piet Van Mieghem
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Characteristic Polynomial ◽  
Clique Number ◽  
Electronic Journal ◽  
Largest Eigenvalue ◽  
Eigenvalues Of Graphs ◽  
Lagrange Series

A Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110–117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds.

scholarly journals The Limiting Characteristic Polynomial of Classical Random Matrix Ensembles

2019 ◽  
Vol 20 (4) ◽  
pp. 1093-1119
Author(s):  
Reda Chhaibi ◽  
Emma Hovhannisyan ◽  
Joseph Najnudel ◽  
Ashkan Nikeghbali ◽  
Brad Rodgers
Keyword(s):  
Characteristic Polynomial ◽  
Random Matrix ◽  
Random Matrix Ensembles ◽  
Matrix Ensembles

scholarly journals On the spectral radius of a random matrix: An upper bound without fourth moment

2018 ◽  
Vol 46 (4) ◽  
pp. 2268-2286 ◽  
Author(s):  
Charles Bordenave ◽  
Pietro Caputo ◽  
Djalil Chafaï ◽  
Konstantin Tikhomirov
Keyword(s):  
Spectral Radius ◽  
Random Matrix ◽  
Upper Bound ◽  
Fourth Moment

scholarly journals On the moments of the characteristic polynomial of a Ginibre random matrix

2018 ◽  
Vol 118 (5) ◽  
pp. 1017-1056 ◽  
Author(s):  
Christian Webb ◽  
Mo Dick Wong
Keyword(s):  
Characteristic Polynomial ◽  
Random Matrix
Sign in / Sign up

Export Citation Format

Share Document