scholarly journals The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight

2019 ◽  
Vol 42 (9) ◽  
pp. 3272-3288 ◽  
Author(s):  
Mengkun Zhu ◽  
Niall Emmart ◽  
Yang Chen ◽  
Charles Weems
Keyword(s):  
Hankel Matrices ◽  
Smallest Eigenvalue

scholarly journals The Smallest Eigenvalue of Hankel Matrices

2010 ◽  
Vol 34 (1) ◽  
pp. 107-133 ◽  
Author(s):  
Christian Berg ◽  
Ryszard Szwarc
Keyword(s):  
Hankel Matrices ◽  
Smallest Eigenvalue

scholarly journals Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices

2015 ◽  
Vol 18 (1) ◽  
pp. 104-124 ◽  
Author(s):  
Niall Emmart ◽  
Yang Chen ◽  
Charles C. Weems
Keyword(s):  
Message Passing ◽  
High Efficiency ◽  
Matrix Theory ◽  
Parallel Implementation ◽  
Hankel Matrices ◽  
Smallest Eigenvalue ◽  
Matrix Operations ◽  
Core Cluster ◽  
Large Matrix ◽  
Class Of Matrices

AbstractThis paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned. Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic. Surprisingly, we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than previously possible, with the potential for extending this work to systems with greater levels of parallelism. The contributions of this work are in three areas: determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations; the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster, and the numerical results themselves.

scholarly journals The smallest eigenvalue of large Hankel matrices

2018 ◽  
Vol 334 ◽  
pp. 375-387 ◽  
Author(s):  
Mengkun Zhu ◽  
Yang Chen ◽  
Niall Emmart ◽  
Charles Weems
Keyword(s):  
Hankel Matrices ◽  
Smallest Eigenvalue

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

Problems on Graphs with Fixed Smallest Eigenvalue

2020 ◽  
Vol 27 (01) ◽  
pp. 51-54
Author(s):  
Jack Koolen ◽  
Qianqian Yang
Keyword(s):  
Smallest Eigenvalue

In this note we give several problems and conjectures on graphs with fixed smallest eigenvalue.

A note on the eigenvalues of $$g$$ g -circulants (and of $$g$$ g -Toeplitz, $$g$$ g -Hankel matrices)

CALCOLO ◽  
2013 ◽  
Vol 51 (4) ◽  
pp. 639-659 ◽  
Author(s):  
Stefano Serra-Capizzano ◽  
Debora Sesana
Keyword(s):  
Hankel Matrices
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