Metrics of graph Laplacian eigenvectors

Mathematics of networks

10.1093/oso/9780198805090.003.0006 ◽

2018 ◽

Author(s):

Mark Newman

Keyword(s):

Random Walks ◽

Adjacency Matrix ◽

Graph Partitioning ◽

Dynamic Networks ◽

Graph Laplacian ◽

Network Visualization ◽

Final Part ◽

Bipartite Networks ◽

Component Structure ◽

Basic Properties

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

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Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

Communications in Mathematics and Statistics ◽

10.1007/s40304-020-00222-7 ◽

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 .

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Rational Krylov methods for fractional diffusion problems on graphs

BIT Numerical Mathematics ◽

10.1007/s10543-021-00881-0 ◽

2021 ◽

Author(s):

Michele Benzi ◽

Igor Simunec

Keyword(s):

Analytic Function ◽

Diffusion Equation ◽

Graph Laplacian ◽

Fractional Diffusion ◽

Singular Matrix ◽

Krylov Methods ◽

Fractional Powers ◽

Directed Networks ◽

Rank One ◽

Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

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A Spectral Graph Theoretical Approach to Oriented Energy Features

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001417550011 ◽

2017 ◽

Vol 31 (01) ◽

pp. 1755001 ◽

Cited By ~ 1

Author(s):

Güleser Kalaycı Demir

Keyword(s):

Theoretical Approach ◽

Graph Laplacian ◽

Texture Segmentation ◽

Contour Detection ◽

Feature Extraction Method ◽

Spectral Graph ◽

Graph Theoretical Approach ◽

Novel Method ◽

Oriented Energy ◽

Complex Valued

In this work, we propose a novel method for determining oriented energy features of an image. Oriented energy features, useful for many machine vision applications like contour detection, texture segmentation and motion analysis, are determined from the filters whose outputs are enhanced at the edges of the image at a given orientation. We use the eigenvectors and eigenvalues of graph Laplacian for determining the oriented energy features of an image. Our method is based on spectral graph theoretical approach in which a graph is assigned complex-valued edge weights whose phases encode orientation information. These edge weights give rise to a complex-valued Hermitian Laplacian whose spectrum enables us to extract oriented energy features of the image. We perform a set of numerical experiments to determine the efficiency and characteristics of the proposed method. In addition, we apply our feature extraction method to texture segmentation problem. We do this in comparison with other known methods, and show that our method performs better for various test textures.

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