Multiplicative degree-Kirchhoff Index Research Articles

The multiplicative degree-Kirchhoff index and complexity of a class of linear networks

<abstract><p>In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a pentagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.</p></abstract>

Topological indices of linear crossed phenylenes with respect to their Laplacian and normalized Laplacian spectrum

<abstract><p>As a powerful tool for describing and studying the properties of networks, the graph spectrum analyses and calculations have attracted substantial attention from the scientific community. Let $ C_{n} $ represent linear crossed phenylenes. Based on the Laplacian (normalized Laplacian, resp.) polynomial of $ C_{n} $, we first investigated the Laplacian (normalized Laplacian, resp) spectrum of $ C_{n} $ in this paper. Furthermore, the Kirchhoff index, multiplicative degree-Kirchhoff, index and complexity of $ C_{n} $ were obtained through the relationship between the roots and the coefficients of the characteristic polynomials. Finally, it was found that the Kirchhoff index and multiplicative degree-Kirchhoff index of $ C_{n} $ were approximately one quarter of their Wiener index and Gutman index, respectively.</p></abstract>

Analyzing the normalized Laplacian spectrum and spanning tree of the cross of the derivative of linear networks

<abstract><p>In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a hexagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.</p></abstract>

Development of some novel resistance distance based topological indices for certain special types of graph networks

A graph network consisting of nodes and edges, forms an electrical network if its edges are replaced by a unit resistor. The resistance distance between any two nodes of a connected network is defined as the net effective resistance between them. This work derives mathematical closed form expressions for the resistance distance based topological indices of four families of graph networks such as and Firstly, we compute the multiplicative eccentricity resistance-distance sum index, multiplicative degree-Kirchhoff index and the reciprocal degree resistance-distance index for the four families of graphs. Secondly, we introduce some new versions of resistance distance based topological indices viz. the additive reciprocal degree-Kirchhoff index, the additive reciprocal eccentric resistance-distance index, the reciprocal eccentric resistance distance index, modified eccentric resistance distance index and the modified degree resistance distance index for the above four families of graph networks. These resistance distance based topological indices are introduced for the first time in the literature, which can surely be useful to investigate the electrical networks.

Statistical Analyses of a Class of Random Cyclooctatetraene Chain Networks with Respect to Several Topological Properties

In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists’ interests. By applying the concept of symmetry and probability theory, we obtain the explicit analytical expressions for the variances of Schultz index, multiplicative degree-Kirchhoff index Gutman index, and additive degree-Kirchhoff index of a random cyclooctatetraene chain with n octagons, which plays a crucial role in the research and application of topological indices.

Modeling spatial networks by contact graphs of disk packings

Spatial networks are ubiquitous in the real world, with examples including Internet, power grids, neural networks, and so on. In this paper, based on a proposed disk packing, we present a contact graph as an exactly solvable model for spatial networks. We then derive analytically some critical structural properties of the model, including degree distribution, clustering coefficient, and diameter, and show that it displays simultaneously the striking scale-free small-world phenomena observed in most real networks. We also determine all the eigenvalues and their multiplicities of the transition probability matrix for the contact graph, and exploit the eigenvalues to derive closed-form expressions for Kemeny constant of random walks and the number of spanning trees on the contact graph. Finally, we derive closed-form expressions for some interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance. As an application of resistance distances, we study the coherence of the first-order consensus dynamics, which tends to a small constant as the graph grows, implying that the graph is resistant to noise.

Kirchhoff Index and Additive Kirchhoff Index Based on Multiplicative Degree for a Random Polyomino Chain

Several topological indices are known to have widespread implications in a variety of research areas. Over the years, the Kirchhoff index has turned out to be an extremely significant and efficient index. In this paper, we propose the exact formulas for the expected values of the random polyomino chain to construct the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index. We also carefully examine the highest degree of the expected values for a random polyomino chain through the multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index.

Statistical Analyses of a Class of Random Pentagonal Chain Networks with respect to Several Topological Properties

There has been an upsurge of research on complex networks in recent years. The purpose of this paper is to study the mathematical properties of the random pentagonal chain networks PE C n with the help of graph theory. Based on the networks PE C n , we first obtain the expected value expressions of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index, and then, we get the explicit expression formulas of their variances. Finally, we find that their limiting distributions all have the probabilistic and statistical significance of normal distribution.

The normalized Laplacian spectrum of n-polygon graphs and applications

Given an arbitrary connected graph G, the n-polygon graph τ n ( G ) is obtained by adding a path with length n ( n ≥ 2 ) to each edge of graph G, and the iterated n-polygon graphs τ n g ( G ) ( g ≥ 0 ) are obtained from the iteration τ n g ( G ) = τ n ( τ n g − 1 ( G ) ) , with the initial condition τ n 0 ( G ) = G . In this paper, a method for calculating the eigenvalues of the normalized Laplacian matrix for graph τ n ( G ) is presented if the eigenvalues of a normalized Laplacian matrix for graph G is first given. The normalized Laplacian spectra for the graph τ n ( G ) and graphs τ n g ( G ) ( g ≥ 0 ) can also then be derived. Finally, as applications, we calculate the multiplicative degree-Kirchhoff index, Kemeny's constant, and the number of spanning trees for the graph τ n ( G ) and graphs τ n g ( G ) by exploring their connections with the normalized Laplacian spectrum, and obtain exact results for these quantities.

Enumeration of the Multiplicative Degree-Kirchhoff Index in the Random Polygonal Chains.

Multiplicative degree-Kirchhoff index is a very interesting topological index. In this article, we compute analytical expression for the expected value of the Multiplicative degree-Kirchhoff index in a random polygonal. Based on the result above, we also get the Multiplicative degree-Kirchhoff index of all polygonal chains with extremal values and average values.

Resistance Distances In Simplicial Networks

Abstract It is well known that in many real networks, such as brain networks and scientific collaboration networks, there exist higher order nonpairwise relations among nodes, i.e. interactions between more than two nodes at a time. This simplicial structure can be described by simplicial complexes and has an important effect on topological and dynamical properties of networks involving such group interactions. In this paper, we study analytically resistance distances in iteratively growing networks with higher order interactions characterized by the simplicial structure that is controlled by a parameter $q$. We derive exact formulas for interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance, which have found applications in various areas elsewhere. We show that the average resistance distance tends to a $q$-dependent constant, indicating the impact of simplicial organization on the structural robustness measured by average resistance distance.

Expected Value of Multiplicative Degree-Kirchhoff Index in Random Polygonal Chains.

The multiplicative degree-Kirchhoff index is a significant topological index. This paper is devoted to the exact formulas for the expected value of the multiplicative degree-Kirchhoff index in random polygonal chains. Moreover, on the basis of the result above, the multiplicative degree-Kirchhoff index of all polygonal chains with extremal values and average values are obtained.

On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Let denote a molecular graph of linear phenylene, containing n hexagons and squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of is derived.

Multiplicative Degree-Kirchhoff Index of Random Polyphenyl Chains

Multiplicative Degree-Kirchhoff Index of Random Polyphenyl Chains

The (degree-) Kirchhoff indices in random polygonal chains

In this paper, the explicit analytical expressions for the expected values of the Kirchhoff index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index in random polygonal chains are established by solving a difference equation. Meanwhile, the average values and variance of these three indices with respect to the set of all polygonal chains with n polygons are obtained. Moreover, we compare the expected values, average values, and the variance between the Wiener index and the Kirchhoff index in random polygonal chains, and analyze their correlation coefficient. The results obtained here generalize the corresponding conclusions in this field.

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

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.

The limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain

In this article, we establish the explicit analytical expressions for the variance of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain. We also prove these four indices of the random polyphenylene chain are asymptotic to normal distributions.

The (normalized) Laplacian spectrum and related indexes of generalized quadrilateral graphs

In this paper, we introduce the generalized quadrilateral graph Q (n) (G), which can be got by replacing each edge of the given graph G with a complete bipartite graph K n,n . We characterize all the spectrum of the graph Q (n) (G) in terms of the given graph. Then we derive the formula for the multiplicative degree-Kirchhoff index, the Kemeny’s constant and the number of spanning trees of Q (n) (G). Finally, we can obtain more about the iterative graph Q r (n) (G).

Some Indices in the Random Spiro Chains

The Gutman index, Schultz index, multiplicative degree-Kirchhoff index, additive degree-Kirchhoff index are four well-studied topological indices, which are useful tools in QSPR and QSAR investigations. Spiro compounds are an important class of cycloalkanes in organic chemistry. In this paper, we determine the expected values of these indices in the random spiro chains, and the extremal values among all spiro chains with n hexagons.

Spectral analysis for weighted iterated pentagonal graphs and its applications

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.