scholarly journals The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-25
Author(s):  
Jia-Bao Liu ◽  
Qian Zheng ◽  
Sakander Hayat
Keyword(s):  
Structural Properties ◽  
Decomposition Theorem ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Triangular Matrices ◽  
Irregular Graphs ◽  
The Relationship ◽  
Explicit Expressions

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let L n 8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of L n 8,4 , we get the corresponding Möbius graph M Q n 8,4 . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of M Q n 8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒ A and ℒ S of order 4 n . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of M Q n 8,4 .

scholarly journals On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1374
Author(s):  
Umar Ali ◽  
Hassan Raza ◽  
Yasir Ahmed
Keyword(s):  
Structural Properties ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Molecular Graph ◽  
Decomposition Theorem ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Polynomial Decomposition ◽  
Number Of Spanning Trees

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

scholarly journals On the Normalized Laplacian and the Number of Spanning Trees of Linear Heptagonal Networks

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 314 ◽  
Author(s):  
Jia-Bao Liu ◽  
Jing Zhao ◽  
Zhongxun Zhu ◽  
Jinde Cao
Keyword(s):  
Spanning Trees ◽  
Decomposition Theorem ◽  
Laplacian Spectrum ◽  
Explicit Formulas ◽  
Kirchhoff Index ◽  
Regular Networks ◽  
Number Of Spanning Trees ◽  
Structure Properties

The normalized Laplacian plays an important role on studying the structure properties of non-regular networks. In fact, it focuses on the interplay between the structure properties and the eigenvalues of networks. Let H n be the linear heptagonal networks. It is interesting to deduce the degree-Kirchhoff index and the number of spanning trees of H n due to its complicated structures. In this article, we aimed to first determine the normalized Laplacian spectrum of H n by decomposition theorem and elementary operations which were not stated in previous results. We then derived the explicit formulas for degree-Kirchhoff index and the number of spanning trees with respect to H n .

ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS

2015 ◽  
Vol 91 (3) ◽  
pp. 353-367 ◽  
Author(s):  
JING HUANG ◽  
SHUCHAO LI
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Subdivision Graph ◽  
Number Of Spanning Trees ◽  
Sharp Lower Bound

Given a connected regular graph $G$, let $l(G)$ be its line graph, $s(G)$ its subdivision graph, $r(G)$ the graph obtained from $G$ by adding a new vertex corresponding to each edge of $G$ and joining each new vertex to the end vertices of the corresponding edge and $q(G)$ the graph obtained from $G$ by inserting a new vertex into every edge of $G$ and new edges joining the pairs of new vertices which lie on adjacent edges of $G$. A formula for the normalised Laplacian characteristic polynomial of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$) in terms of the normalised Laplacian characteristic polynomial of $G$ and the number of vertices and edges of $G$ is developed and used to give a sharp lower bound for the degree-Kirchhoff index and a formula for the number of spanning trees of $l(G)$ (respectively $s(G),r(G)$ and $q(G)$).

scholarly journals Idealistic Soft Topological Hyperrings

2020 ◽  
Vol 4 (3) ◽  
Author(s):  
Gülay OĞUZ
Keyword(s):  
Structural Properties ◽  
The Family ◽  
The Relationship

The aim of this article is to introduce the concept of an idealistic soft topological hyperring over a hyperring. Some structural properties of this concept are also studied. Moreover, this study investigates the relationship between the idealistic soft topological hyperrings and the idealistic soft hyperrings. Finally, the restricted (extended) intersection and ∧-intersection of the family of the idealistic soft topological hyperrings are examined.

The relationship between thermodynamic and structural properties of low and high amylose maize starches

2001 ◽  
Vol 44 (2) ◽  
pp. 151-160 ◽  
Author(s):  
Y.I Matveev ◽  
J.J.G van Soest ◽  
C Nieman ◽  
L.A Wasserman ◽  
V.A Protserov ◽  
...  
Keyword(s):  
Structural Properties ◽  
High Amylose ◽  
The Relationship

The relationship between internal supply chain structure and operational performance: survey results from Japanese manufacturers

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mikihisa Nakano ◽  
Kazuki Matsuyama
Keyword(s):  
Supply Chain ◽  
Structural Properties ◽  
Operations Management ◽  
Chain Structure ◽  
Operational Performance ◽  
Equation Modeling ◽  
Content Type ◽  
Wide Range ◽  
Supply Chain Structure ◽  
The Relationship

Purpose The purpose of this paper is to discuss the roles of a supply chain management (SCM) department. To achieve that, this study empirically examines the relationship between internal supply chain structure and operational performance, using survey data collected from 108 Japanese manufacturers. Design/methodology/approach Based on a literature review of not only organizational theory but also other fields such as marketing, logistics management, operations management and SCM, this study focused on two structural properties, formalization and centralization and divided operational performance to firm-centric efficiency and customer-centric responsiveness. To examine the analytical model using these dimensions, this study conducted a structural equation modeling. Findings The correlation between centralization of operational tasks and centralization of strategic tasks, the impacts of centralization of both tasks on formalization and the effect of formalization on responsiveness performance were demonstrated. In addition, the reasons for formalization not positively influencing efficiency performance were explored through follow-up interviews. Practical implications Manufacturers need to formalize, as much as possible, a wide range of SCM tasks to realize operational excellence. To establish such formalized working methods, it is effective to centralize the authorities of both operational and strategic tasks in a particular department. In addition, inefficiency due to strict logistics service levels is a problem that all players involved in the supply chain of various industries should work together to solve. Originality/value The theoretical contribution of this study is that the authors established an empirical process that redefined the constructs of formalization and centralization, developed these measures and examined the impacts of these structural properties on operational performance.

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 Macro- and mesoscale pattern interdependencies in complex networks

2019 ◽  
Vol 16 (159) ◽  
pp. 20190553 ◽  
Author(s):  
M. J. Palazzi ◽  
J. Borge-Holthoefer ◽  
C. J. Tessone ◽  
A. Solé-Ribalta
Keyword(s):  
Complex Networks ◽  
Empirical Evidence ◽  
Structural Properties ◽  
Modular Architecture ◽  
Modular Networks ◽  
As Species ◽  
Species Abundances ◽  
Organizational Patterns ◽  
Natural Combination ◽  
The Relationship

Identifying and explaining the structure of complex networks at different scales has become an important problem across disciplines. At the mesoscale, modular architecture has attracted most of the attention. At the macroscale, other arrangements—e.g. nestedness or core–periphery—have been studied in parallel, but to a much lesser extent. However, empirical evidence increasingly suggests that characterizing a network with a unique pattern typology may be too simplistic, since a system can integrate properties from distinct organizations at different scales. Here, we explore the relationship between some of these different organizational patterns: two at the mesoscale (modularity and in-block nestedness); and one at the macroscale (nestedness). We show experimentally and analytically that nestedness imposes bounds to modularity, with exact analytical results in idealized scenarios. Specifically, we show that nestedness and modularity are interdependent. Furthermore, we analytically evidence that in-block nestedness provides a natural combination between nested and modular networks, taking structural properties of both. Far from a mere theoretical exercise, understanding the boundaries that discriminate each architecture is fundamental, to the extent that modularity and nestedness are known to place heavy dynamical effects on processes, such as species abundances and stability in ecology.

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