scholarly journals Perron and Frobenius Meet Carathéodory

10.37236/9996 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Márton Naszódi ◽  
Polyanskii Alexandr
Keyword(s):  
Classical Result ◽  
Matrix Theory ◽  
Frobenius Theorem ◽  
New Approach ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

We present a new approach of proving certain Carathéodory-type theorems using the Perron–Frobenius Theorem, a classical result in matrix theory describing the largest eigenvalue of a matrix with positive entries. At the end, we list some results and conjectures that we hope can be approached with this method.

scholarly journals Analysis of the Global Banking Network by Random Matrix Theory

2021 ◽  
Vol 8 ◽  
Author(s):  
Ali Namaki ◽  
Reza Raei ◽  
Jamshid Ardalankia ◽  
Leila Hedayatifar ◽  
Ali Hosseiny ◽  
...  
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Final Analysis ◽  
The United States ◽  
Mean Value ◽  
Largest Eigenvalue ◽  
Global Banking ◽  
The Largest Eigenvalue

Since the financial crisis of 2008, the network analysis of financial systems has attracted a lot of attention. In this paper, we analyze the global banking network via the method of Random Matrix Theory. By applying that method on a cross border lending network, it is shown that while the connectivity between different parts of the network has risen and the profile of transactions has diversified, the role of hubs remains important in the weighted perspective. The largest eigenvalue of the transaction matrix as the leading mode of the system shows sharp growth since 2002. As well, it is observed that its growth has diminished since 2008. This indicates that the crisis of 2008 has left a long-lasting footprint on the financial system. Analyzing the mean value of the participation ratio reveals the fact that the role of countries in forming small modes, has increased since 2002. In our final analysis, we provide snapshots of the hubs in the network over time. We observe that the share of countries in total transactions is not equal to their share in shaping the eigenvector of the largest eigenvalue. In 2018 for example, while the United Kingdom leads the share of transactions, it is the United States that has the largest value in the leading eigenvector. The proposed technique in the paper can be useful for analyzing different types of interaction networks between countries.

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.

On the Free Vibration of Mono-Coupled Periodic Distributed Parameter Systems

1993 ◽  
Author(s):  
Gerhard G. G. Lueschen ◽  
Lawrence A. Bergman
Keyword(s):  
Exact Solution ◽  
Free Vibration ◽  
Periodic Structure ◽  
Classical Result ◽  
Discrete Systems ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
New Approach

Abstract A new approach to the exact solution is given for the free vibration of a periodic structure comprised of a multiplicity of identical linear distributed parameter substructures, closely coupled through identical linear springs. The method used is an extension of a classical result for periodic discrete systems.

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.

scholarly journals Invertible harmonic and harmonic quasiconformal mappings

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1953-1967
Author(s):  
Miodrag Mateljevic
Keyword(s):  
Classical Result ◽  
Quasiconformal Mappings ◽  
Lipschitz Property ◽  
Planar Case ◽  
New Approach

Recently G. Alessandrini - V. Nesi and Kalaj generalized a classical result of H. Kneser (RKCTheorem). Using a new approach we get some new results related to RKC-Theorem and harmonic quasiconformal (HQC) mappings. We also review some results concerning bi-Lipschitz property for HQC-mappings between Lyapunov domains and related results in planar case using some novelty.

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