scholarly journals The Largest Eigenvalue of a Convex Function, Duality, and a Theorem of Slodkowski

2015 ◽  
Vol 26 (4) ◽  
pp. 3027-3055
Author(s):  
Matthew Dellatorre
Keyword(s):  
Convex Function ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals A note on a walk-based inequality for the index of a signed graph

2021 ◽  
Vol 9 (1) ◽  
pp. 19-21
Author(s):  
Zoran Stanić
Keyword(s):  
Adjacency Matrix ◽  
Signed Graph ◽  
Arbitrary Length ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

Abstract We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

scholarly journals The Laplacian Spread of Tricyclic Graphs

10.37236/169 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yanqing Chen ◽  
Ligong Wang
Keyword(s):  
Laplacian Matrix ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
Fixed Order ◽  
Tricyclic Graphs ◽  
The Difference ◽  
The Largest Eigenvalue

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

scholarly journals A note on distance spectral radius of trees

2017 ◽  
Vol 5 (1) ◽  
pp. 296-300
Author(s):  
Yanna Wang ◽  
Rundan Xing ◽  
Bo Zhou ◽  
Fengming Dong
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Distance Matrix ◽  
Largest Eigenvalue ◽  
Maximal Distance ◽  
The Largest Eigenvalue

Abstract The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

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.

scholarly journals PROBABILITY DENSITIES AND DISTRIBUTIONS FOR SPIKED AND GENERAL VARIANCE WISHART β-ENSEMBLES

2013 ◽  
Vol 02 (04) ◽  
pp. 1350011 ◽  
Author(s):  
PETER J. FORRESTER
Keyword(s):  
Random Matrices ◽  
Exact Form ◽  
Recursive Structure ◽  
Simple Derivation ◽  
Largest Eigenvalue ◽  
Wishart Matrices ◽  
Real Quaternion ◽  
Parameter Dependent ◽  
Dependent State ◽  
The Largest Eigenvalue

A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue b different from unity. As b increases through b = 2, a gap forms from the largest eigenvalue to the rest of the spectrum, and with b - 2 of order N-1/3 the scaled largest eigenvalues form a well-defined parameter dependent state. Recent works by Bloemendal and Virág [Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825], and Mo [Rank I real Wishart spiked model, Comm. Pure Appl. Math.65 (2012) 1528–1638], have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart β-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart β-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices (β = 4) the latter is recognized as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825]. We also use the construction of the spiked Wishart β-ensemble from [A. Bloemendal and B. Virág, Limits of spiked random matrices I, Probab. Theory Related Fields156 (2013) 795–825] to give a simple derivation of the explicit form of the eigenvalue PDF.

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