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.

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.

Spectral analysis for weighted iterated quadrilateral graphs

2018 ◽  
Vol 29 (11) ◽  
pp. 1850113 ◽  
Author(s):  
Meifeng Dai ◽  
Yufei Chen ◽  
Xiaoqian Wang ◽  
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 matrix of weighted iterated quadrilateral 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 the multiplicative degree Kirchhoff index and the Kemeny’s constant, as well as the number of weighted spanning trees.

Spectral analysis for weighted iterated pentagonal graphs and its applications

2020 ◽  
Vol 34 (28) ◽  
pp. 2050308
Author(s):  
Qun Liu
Keyword(s):  
Spectral Analysis ◽  
Complex Systems ◽  
Closed Form ◽  
Real World ◽  
Spanning Trees ◽  
Laplacian Matrix ◽  
Kirchhoff Index ◽  
Weighted Networks ◽  
Successive Generations ◽  
Recursive Relations

Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated pentagonal networks. From the construction of the network, we derive recursive relations of all eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated pentagonal networks. As applications of spectra of the normalized Laplacian matrix, we study the Kemeny’s constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for the weighted iterated pentagonal networks.

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 NONLINEAR OPTICAL WAVEGUIDE STRUCTURE FOR SENSOR APPLICATION: TM CASE

2007 ◽  
Vol 21 (30) ◽  
pp. 5075-5089 ◽  
Author(s):  
HALA M. KHALIL ◽  
MOHAMMED M. SHABAT ◽  
SOFYAN A. TAYA ◽  
MAZEN M. ABADLA
Keyword(s):  
Refractive Index ◽  
Theoretical Analysis ◽  
Closed Form ◽  
Optical Waveguide ◽  
Effective Refractive Index ◽  
Nonlinear Optical ◽  
Waveguide Structure ◽  
Analytical Expressions ◽  
Evanescent Waveguide ◽  
Waveguide Sensor

In this work, we present an extensive theoretical analysis of nonlinear optical waveguide sensor. The waveguide under consideration consists of a thin dielectrica film surrounded by a self-focused nonlinear cladding and a linear substrate. The nonlinearity of the cladding is considered to be of Kerr-type. Both cases, when the effective refractive index is greater and when it is smaller than the index of the guiding layer, are discussed. The sensitivity of the effective refractive index to any change in the cladding index in evanescent optical waveguide sensor is derived for TM modes. Closed form analytical expressions and normalized charts are given to provide the conditions required for the sensor to exhibit its maximum sensitivity. The results are compared with those of the well-known linear evanescent waveguide sensors.

scholarly journals Conditional and Unconditional Deterministic Bounds on the MSE of the Non-Uniform Linear Co-centered Orthogonal Loop and Dipole Array

2021 ◽  
Vol 1 (1) ◽  
pp. 13-20
Author(s):  
Tao Bao ◽  
Mohammed Nabil EL KORSO
Keyword(s):  
Theoretical Analysis ◽  
Closed Form ◽  
Mean Square Error ◽  
Source Localization ◽  
Threshold Region ◽  
Observation Model ◽  
Mean Square ◽  
Numerical Examples ◽  
Cramer Rao Bound ◽  
Deterministic Bounds

The co-centered orthogonal loop and dipole (COLD) array exhibits some interesting properties, which makes it ubiquitous in the context of polarized source localization. In the literature, one can find a plethora of estimation schemes adapted to the COLD array. Nevertheless, their ultimate performance in terms the so-called threshold region of mean square error (MSE), have not been fully investigated. In order to fill this lack, we focus, in this paper, on conditional and unconditional bounds that are tighter than the well known Cramér-Rao Bound (CRB). More precisely, we give some closed form expressions of the McAulay-Hofstetter, the Hammersley-Chapman-Robbins, the McAulaySeidman bounds and the recent Todros-Tabrikian bound, for both the conditional and unconditional observation model. Finally, numerical examples are provided to corroborate the theoretical analysis and to reveal a number of insightful properties.

scholarly journals Resistance Distance and Kirchhoff Index of Graphs with Pockets

2018 ◽  
Author(s):  
Qun Liu ◽  
Jiabao Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

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.

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