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 The Kirchhoff Index of Toroidal Meshes and Variant Networks

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jia-Bao Liu ◽  
Xiang-Feng Pan ◽  
Jinde Cao ◽  
Xia Huang
Keyword(s):  
Asymptotic Behavior ◽  
Graph Theory ◽  
Distance Function ◽  
Network Theory ◽  
Spectral Graph Theory ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Spectral Graph ◽  
Explicit Formulae

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the toroidal meshes networkTm×nand its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofL(Tm×n),S(Tm×n),T(Tm×n), andC(Tm×n)were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

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.

Optimal networks for exact controllability

2020 ◽  
Vol 31 (10) ◽  
pp. 2050144
Author(s):  
Yunhua Liao ◽  
Mohamed Maama ◽  
M. A. Aziz-Alaoui
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complex Networks ◽  
Open Problem ◽  
Topological Structure ◽  
Exact Controllability ◽  
Spectral Graph Theory ◽  
Algebraic Multiplicity ◽  
Spectral Graph ◽  
Optimal Networks

The exact controllability can be mapped to the problem of maximum algebraic multiplicity of all eigenvalues. In this paper, we focus on the exact controllability of deterministic complex networks. First, we explore the eigenvalues of two famous networks, i.e. the comb-of-comb network and the Farey graph. Due to their special structure, we find that the eigenvalues of each network are mutually distinct, showing that these two networks are optimal networks with respect to exact controllability. Second, we study how to optimize the exact controllability of a deterministic network. Based on the spectral graph theory, we find that reducing the order of duplicate sets or co-duplicate sets which are two special vertex subsets can decrease greatly the exact controllability. This result provides an answer to an open problem of Li et al. [X. F. Li, Z. M. Lu and H. Li, Int. J. Mod. Phys. C 26, 1550028 (2015)]. Finally, we discuss the relation between the topological structure and the multiplicity of two special eigenvalues and the computational complexity of our method.

scholarly journals On the spectral invariants of symmetric matrices with applications in the spectral graph theory

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2925-2932
Author(s):  
Abdullah Alazemi ◽  
Milica Andjelic ◽  
Slobodan Simic
Keyword(s):  
Graph Theory ◽  
Spectral Radius ◽  
Weighted Graph ◽  
Spectral Graph Theory ◽  
Combinatorial Structure ◽  
Symmetric Matrices ◽  
Spectral Invariants ◽  
Spectral Graph ◽  
Principal Submatrices ◽  
The Relationship

We first prove a formula which relates the characteristic polynomial of a matrix (or of a weighted graph), and some invariants obtained from its principal submatrices (resp. vertex deleted subgraphs). Consequently, we express the spectral radius of the observed objects in the form of power series. In particular, as is relevant for the spectral graph theory, we reveal the relationship between spectral radius of a simple graph and its combinatorial structure by counting certain walks in any of its vertex deleted subgraphs. Some computational results are also included in the paper.

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 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.

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