Applications of Laplacian spectrum for the vertex–vertex graph

2019 ◽  
Vol 33 (17) ◽  
pp. 1950184 ◽  
Author(s):  
Tingting Ju ◽  
Meifeng Dai ◽  
Changxi Dai ◽  
Yu Sun ◽  
Xiangmei Song ◽  
...  
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
First Order ◽  
Laplacian Energy ◽  
Wide Range ◽  
Vertex Graph ◽  
Network Properties ◽  
Successive Generations ◽  
Relationship Of

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

Coherence analysis of a family of weighted star-composed networks

2019 ◽  
Vol 33 (23) ◽  
pp. 1950264
Author(s):  
Meifeng Dai ◽  
Tingting Ju ◽  
Yongbo Hou ◽  
Jianwei Chang ◽  
Yu Sun ◽  
...  
Keyword(s):  
Kronecker Product ◽  
Future Research ◽  
Weight Factor ◽  
Node Number ◽  
Laplacian Eigenvalues ◽  
First Order ◽  
Successive Generations ◽  
Network Consistency ◽  
Relationship Of ◽  
The Relationship

Recently, the study of many kinds of weighted networks has received the attention of researchers in the scientific community. In this paper, first, a class of weighted star-composed networks with a weight factor is introduced. We focus on the network consistency in linear dynamical system for a class of weighted star-composed networks. The network consistency can be characterized as network coherence by using the sum of reciprocals of all nonzero Laplacian eigenvalues, which can be obtained by using the relationship of Laplacian eigenvalues at two successive generations. Remarkably, the Laplacian matrix of the class of weighted star-composed networks can be represented by the Kronecker product, then the properties of the Kronecker product can be used to obtain conveniently the corresponding characteristic roots. In the process of finding the sum of reciprocals of all nonzero Laplacian eigenvalues, the key step is to obtain the relationship of Laplacian eigenvalues at two successive generations. Finally, we obtain the main results of the first- and second-order network coherences. The obtained results show that if the weight factor is 1 then the obtained results in this paper coincide with the previous results on binary networks, otherwise the scalings of the first-order network coherence are related to the node number of attaching copy graph, the weight factor and generation number. Surprisingly, the scalings of the first-order network coherence are independent of the node number of initial graph. Consequently, it will open up new perspectives for future research.

On skew Laplacian spectrum and energy of digraphs

2020 ◽  
pp. 2150051 ◽  
Author(s):  
Hilal A. Ganie ◽  
S. Pirzada ◽  
Bilal A. Chat ◽  
X. Li
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Arbitrary Direction ◽  
Laplacian Eigenvalues ◽  
Underlying Graph ◽  
Laplacian Energy ◽  
Energy Formula

We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

SPECTRAL ANALYSIS FOR WEIGHTED ITERATED TRIANGULATIONS OF GRAPHS

Fractals ◽  
2018 ◽  
Vol 26 (01) ◽  
pp. 1850017 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Spectral Analysis ◽  
Closed Form ◽  
Structural Properties ◽  
Spanning Trees ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Successive Generations

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

scholarly journals On the Adjacency, Laplacian, and Signless Laplacian Spectrum of Coalescence of Complete Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
S. R. Jog ◽  
Raju Kotambari
Keyword(s):  
Line Graph ◽  
Laplacian Matrix ◽  
Spectral Graph Theory ◽  
Complete Graphs ◽  
Laplacian Spectrum ◽  
Spectral Graph ◽  
Signless Laplacian Spectrum ◽  
Subdivision Graph ◽  
Laplacian Energy ◽  
Signless Laplacian

Coalescence as one of the operations on a pair of graphs is significant due to its simple form of chromatic polynomial. The adjacency matrix, Laplacian matrix, and signless Laplacian matrix are common matrices usually considered for discussion under spectral graph theory. In this paper, we compute adjacency, Laplacian, and signless Laplacian energy (Qenergy) of coalescence of pair of complete graphs. Also, as an application, we obtain the adjacency energy of subdivision graph and line graph of coalescence from itsQenergy.

Coherence analysis of a class of weighted tree-like polymer networks

2018 ◽  
Vol 32 (05) ◽  
pp. 1850064 ◽  
Author(s):  
Jiaojiao He ◽  
Meifeng Dai ◽  
Yue Zong ◽  
Jiahui Zou ◽  
Yu Sun ◽  
...  
Keyword(s):  
Polymer Networks ◽  
Laplacian Matrix ◽  
Network Size ◽  
Second Order ◽  
Weight Factor ◽  
Stochastic Disturbances ◽  
Weighted Tree ◽  
Consensus Dynamics ◽  
Successive Generations ◽  
Relationship Of

Complex networks have elicited considerable attention from scientific communities. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. Firstly, we introduce a class of weighted tree-like polymer networks with the weight factor. Then, we deduce the recursive relationship of the eigenvalues of Laplacian matrix at two successive generations. Finally, we calculate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor and the scalings of second-order coherence with network size obey five laws along with the range of the weight factor.

Spectral analysis for weighted iterated q-triangulations of graphs

2020 ◽  
Vol 31 (03) ◽  
pp. 2050042
Author(s):  
Yufei Chen ◽  
Wenxia Li
Keyword(s):  
Spectral Analysis ◽  
Theoretical Analysis ◽  
Closed Form ◽  
Structural Properties ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Successive Generations

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighed iterated [Formula: see text]-triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As examples of application of these results, we then derive closed-form expressions for their Kemeny’s constant and multiplicative Kirchhoff index. Simulation example is also provided to demonstrate the effectiveness of the theoretical analysis.

CHARACTERISTIC POLYNOMIAL OF ADJACENCY OR LAPLACIAN MATRIX FOR WEIGHTED TREELIKE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950074 ◽  
Author(s):  
MEIFENG DAI ◽  
YONGBO HOU ◽  
CHANGXI DAI ◽  
TINGTING JU ◽  
YU SUN ◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Future Study ◽  
Laplacian Matrix ◽  
Block Matrix ◽  
Weight Factor ◽  
Laplacian Spectrum ◽  
Weighted Networks ◽  
Adjacency Spectrum ◽  
Analytic Expression ◽  
Successive Generations

In recent years, weighted networks have been extensively studied in various fields. This paper studies characteristic polynomial of adjacency or Laplacian matrix for weighted treelike networks. First, a class of weighted treelike networks with a weight factor is introduced. Then, the relationships of adjacency or the Laplacian matrix at two successive generations are obtained. Finally, according to the operation of the block matrix, we obtain the analytic expression of the characteristic polynomial of the adjacency or the Laplacian matrix. The obtained results lay the foundation for the future study of adjacency spectrum or Laplacian spectrum.

Applications of Laplacian spectrum for the weighted scale-free network with a weight factor

2018 ◽  
Vol 32 (32) ◽  
pp. 1850353 ◽  
Author(s):  
Meifeng Dai ◽  
Tingting Ju ◽  
Jingyi Liu ◽  
Yu Sun ◽  
Xiangmei Song ◽  
...  
Keyword(s):  
Laplacian Matrix ◽  
Weight Matrix ◽  
Weight Factor ◽  
Scale Free Network ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Scale Free ◽  
Free Network ◽  
Eigenvalue And Eigenvector ◽  
Binary Scale

Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network.

EIGENTIME IDENTITY FOR WEIGHT-DEPENDENT WALK ON A CLASS OF WEIGHTED FRACTAL SCALE-FREE HIERARCHICAL-LATTICE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950138
Author(s):  
BO WU ◽  
ZHIZHUO ZHANG ◽  
YINGYING CHEN ◽  
TINGTING JU ◽  
MEIFENG DAI ◽  
...  
Keyword(s):  
Iteration Process ◽  
Laplacian Matrix ◽  
Network Size ◽  
Hierarchical Lattice ◽  
Analytical Expression ◽  
Scale Free ◽  
Scale Free Networks ◽  
Reciprocal Sum ◽  
Successive Generations ◽  
Relationship Of

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates [Formula: see text] connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when [Formula: see text] is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks.

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