scholarly journals On lower bounds for the largest eigenvalue of a symmetric matrix

2008 ◽  
Vol 429 (2-3) ◽  
pp. 519-526 ◽  
Author(s):  
S.G. Walker ◽  
P. Van Mieghem
Keyword(s):  
Lower Bounds ◽  
Symmetric Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

scholarly journals On the Spectrum of Laplacian Matrix

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Rana Khoeilar
Keyword(s):  
Lower Bounds ◽  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Matrix ◽  
Vertex Degrees ◽  
The Largest Eigenvalue

Let G be a simple graph of order n . The matrix ℒ G = D G − A G is called the Laplacian matrix of G , where D G and A G denote the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let l 1 G , l n − 1 G be the largest eigenvalue, the second smallest eigenvalue of ℒ G respectively, and λ 1 G be the largest eigenvalue of A G . In this paper, we will present sharp upper and lower bounds for l 1 G and l n − 1 G . Moreover, we investigate the relation between l 1 G and λ 1 G .

scholarly journals Bounds for sine and cosine via eigenvalue estimation

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Pentti Haukkanen ◽  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Alexander Kovacec
Keyword(s):  
Lower Bounds ◽  
Upper Bounds ◽  
Main Diagonal ◽  
Largest Eigenvalue ◽  
Tridiagonal Matrices ◽  
Eigenvalue Estimation ◽  
The Largest Eigenvalue

AbstractDefine n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue,Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain intervals. Also upper bounds can be obtained in this way.

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