scholarly journals More on Spectral Analysis of Signed Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Guihai Yu ◽  
Hui Qu
Keyword(s):  
Spectral Analysis ◽  
Graph Theory ◽  
Necessary Conditions ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Signed Networks ◽  
Spectral Graph ◽  
Sufficient And Necessary Conditions ◽  
Signed Network ◽  
Determinant Of Laplacian

Spectral graph theory plays a key role in analyzing the structure of social (signed) networks. In this paper we continue to study some properties of (normalized) Laplacian matrix of signed networks. Sufficient and necessary conditions for the singularity of Laplacian matrix are given. We determine the correspondence between the balance of signed network and the singularity of its Laplacian matrix. An expression of the determinant of Laplacian matrix is present. The symmetry about 1 of eigenvalues of normalized Laplacian matrix is discussed. We determine that the integer 2 is an eigenvalue of normalized Laplacian matrix if and only if the signed network is balanced and bipartite. Finally an expression of the coefficient of normalized Laplacian characteristic polynomial is present.

scholarly journals The Kirchhoff Index of Hypercubes and Related Complex Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jiabao Liu ◽  
Jinde Cao ◽  
Xiang-Feng Pan ◽  
Ahmed Elaiw
Keyword(s):  
Graph Theory ◽  
Complex Networks ◽  
Laplacian Matrix ◽  
Exact Formula ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Relationship Of ◽  
The Relationship

The resistance distance between any two vertices ofGis defined as the network effective resistance between them if each edge ofGis replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices inG. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networksQnby utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networksQnand its three variant networksl(Qn),s(Qn),t(Qn)by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes ofl(Qn),s(Qn), andt(Qn)were proposed, respectively.

scholarly journals Laplacian Matrix Based Spectral Graph Clustering

2019 ◽  
Vol 9 (2S) ◽  
pp. 666-670
Keyword(s):  
Graph Theory ◽  
Spectral Clustering ◽  
Clustering Algorithms ◽  
Laplacian Matrix ◽  
Graph Clustering ◽  
Research Field ◽  
Spectral Graph Theory ◽  
Clustering Methods ◽  
Spectral Graph ◽  
Core Technology

Recent attention in the research field of clustering is focused on grouping of clusters based on structure of a graph. At present, there are plentiful literature work has been proposed towards the clustering techniques but it is still an open challenge to find the best technique for clustering. This paper present a comprehensive review of our insights towards emerging clustering methods on graph based spectral clustering. Graph Laplacians have become a core technology for the spectral clustering which works based on the properties of the Laplacian matrix. In our study, we discuss correlation between similarity and Laplacian matrices within a graph and spectral graph theory concepts. Current studies on graph-based clustering methods requires a well defined good quality graph to achieve high clustering accuracy. This paper describes how spectral graph theory has been used in the literature of clustering concepts and how it helps to predict relationships that have not yet been identified in the existing literature. Some application areas on the graph clustering algorithms are discussed. This survey outlines the problems addressed by the existing research works on spectral clustering with its problems, methodologies, data sets and advantages. This paper identifies fundamental issues of graph clustering which provides a better direction for more applications in social network analysis, image segmentation, computer vision and other domains.

scholarly journals The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jiabao Liu ◽  
Xiang-Feng Pan ◽  
Yi Wang ◽  
Jinde Cao
Keyword(s):  
Graph Theory ◽  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Kirchhoff Index ◽  
Spectral Graph ◽  
Explicit Formulae ◽  
Folded Hypercube

Then-dimensional folded hypercubeFQnis an important and attractive variant of then-dimensional hypercubeQn, which is obtained fromQnby adding an edge between any pair of vertices complementary edges.FQnis superior toQnin many measurements, such as the diameter ofFQnwhich is⌈n/2⌉, about a half of the diameter in terms ofQn. The Kirchhoff indexKf(G)is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the folded hypercubes networksFQnand its three variant networksl(FQn),s(FQn), andt(FQn)on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofFQn,l(FQn),s(FQn), andt(FQn)were proposed, respectively.

Laplacian Energy of Digraphs and a Minimum Laplacian Energy Algorithm

2015 ◽  
Vol 26 (03) ◽  
pp. 367-380 ◽  
Author(s):  
Xingqin Qi ◽  
Edgar Fuller ◽  
Rong Luo ◽  
Guodong Guo ◽  
Cunquan Zhang
Keyword(s):  
Oriented Graph ◽  
Laplacian Matrix ◽  
Upper Bounds ◽  
Spectral Graph Theory ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Spectral Graph ◽  
Laplacian Energy ◽  
Energy Formula ◽  
Matrix Formula

In spectral graph theory, the Laplacian energy of undirected graphs has been studied extensively. However, there has been little work yet for digraphs. Recently, Perera and Mizoguchi (2010) introduced the directed Laplacian matrix [Formula: see text] and directed Laplacian energy [Formula: see text] using the second spectral moment of [Formula: see text] for a digraph [Formula: see text] with [Formula: see text] vertices, where [Formula: see text] is the diagonal out-degree matrix, and [Formula: see text] with [Formula: see text] whenever there is an arc [Formula: see text] from the vertex [Formula: see text] to the vertex [Formula: see text] and 0 otherwise. They studied the directed Laplacian energies of two special families of digraphs (simple digraphs and symmetric digraphs). In this paper, we extend the study of Laplacian energy for digraphs which allow both simple and symmetric arcs. We present lower and upper bounds for the Laplacian energy for such digraphs and also characterize the extremal graphs that attain the lower and upper bounds. We also present a polynomial algorithm to find an optimal orientation of a simple undirected graph such that the resulting oriented graph has the minimum Laplacian energy among all orientations. This solves an open problem proposed by Perera and Mizoguchi at 2010.

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