Kirchhoff Index Research Articles

Extremal polyphenyl chains with respect to the Kirchhoff index

Let G be a simple connected graph. The Kirchhoff index of a graph is the sum of the resistance distance between all vertex pairs in the graph G. The resistance distance in a graph is equivalent to the effective resistance between any node pairs in the electrical network obtained by replacing each edge in the graph G with a unit resistance. The polyphenyl chain PL(S) is a graph with n hexagons where each hexagon can be reduced to a vertex and connect to a path. In this paper, we characterize polyphenyl chains with maximal and minimal Kirchhoff index.

Kirchhoff index of some networks based on the cluster of graph

Abstract The Kirchhoff index represents the sum of resistance distances between all pairs of nodes in a network. It reflects the integrity and connectivity of the network. In this paper, we propose three classes of network models, all generated based on the cluster of graphs. We derive exact expressions for their Kirchhoff indices through an iterative methodology. Leveraging the derived formulas, we compared the Kirchhoff indices of the network families corresponding to three classes of networks generated under the same base graph. This enhances our evaluation of network characteristics, potentially serving as a critical tool in the practical design of networks.

Resistance distance and Kirchhoff index of two kinds of double join operations on graphs

Resistance distance and Kirchhoff index of two kinds of double join operations on graphs

Extremal values on the Kirchhoff index of the line graph of trees

The computation of resistance distance and the Kirchhoff index is a classical problem that has been extensively investigated by numerous mathematicians, physicists, and scientists. Consider a simple connected graph G with vertex set V(G) and edge set E(G). If we replace each edge of G with a resistor of 1 ohm resistance, we create an electrical network N; in that case, the distance between any two nodes between network N is called resistance distance. The Kirchhoff index is a mathematical term that quantifies the complexity of a graph; it is defined as the total resistance distance among each pair of vertices in G. The line graph LG is constructed from G by swapping out the edges for vertices; in LG, if two vertices share endpoints in G, then they are connected in LG. In this study, extremal values for the Kirchhoff index of the line graph of trees are calculated. Further, we will also calculate the Kirchhoff index for the line graph of some special trees and establish the relationship between them.

Resistance Distance and Kirchhoff Index of Cayley Graphs on Generalized Quaternion Groups

Resistance Distance and Kirchhoff Index of Cayley Graphs on Generalized Quaternion Groups

Upper bounds for the global cyclicity index

Abstract We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.

On Resistance Distance and Kirchhoff Index of Cacti Networks

On Resistance Distance and Kirchhoff Index of Cacti Networks

Robustness of consensus in double‐layer degree positive and negative correlation ring networks

The robustness of consensus in multilayer networks is affected by the network structure, especially the interlayer coupling approach. The article investigates the impact of two interlayer coupling approaches on the robustness of consensus in double‐layer ring networks, where the interlayer relationships are denoted as degree positive correlation (PC) and degree negative correlation (NC), respectively. The robustness of network consensus can be derived from network coherence. Two types of network coherence are first described using the Kirchhoff index and the number of spanning trees, providing a new calculation method for network coherence. The number of network nodes and the coupling strength have been shown to impact the coherence of the two types of networks. Combined with numerical simulation experiments, the NC model shows better consensus than the PC model under the same parameter set. In particular, the parameter values when the network consensus is optimal are obtained.

Spectral analysis for weighted extended Vicsek polygons

Because of the application of fractal networks and their spectral properties in various fields of science and engineering, they have become a hot topic in network science. Moreover, deterministic weighted graphs are widely used to model complex real-world systems. This paper studys weighted extended Vicsek polygons W(G m,t ), which are based on the Vicsek fractal model and the extended fractal cactus model. The structure of these polygons is controlled by the positive integer coefficient m and the number of iterations t. From the construction of the graph, we derive recursive relations of all eigenvalues and their multiplicities of normalized Laplacian matrices from the two successive generations of the weighted extended Vicsek polygons. Then, we use the spectra of the normalized Laplacian matrices to study Kemeny’s constant, the multiplicative Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for the weighted extended Vicsek polygons. The above results help to analyze the topology and dynamic properties of the network model, so it has potential application prospects.

Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Let [Formula: see text], [Formula: see text], be a simple connected graph of order [Formula: see text] and size [Formula: see text]. Denote with [Formula: see text] eigenvalues of the Laplacian matrix [Formula: see text] of [Formula: see text]. The Kirchhoff index and the number of spanning trees of [Formula: see text] expressed in terms of Laplacian eigenvalues are given by [Formula: see text] and [Formula: see text], respectively. The characteristic polynomial of [Formula: see text] is given by [Formula: see text]. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of [Formula: see text]. Then, we use it to improve the previously obtained results on [Formula: see text] and [Formula: see text]. In addition, we present new Nordhaus–Gaddum-type inequalities for [Formula: see text] and [Formula: see text].

Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs

Let G be a simple connected graph with vertex set V(G) and edge set E(G). The resistance distance r(u,v) between two vertices u,v∈V(G) is the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. This function is known to be a metric on the vertex-set of any graph. The sum of resistance distances between pairs of vertices in a G is called Kirchhoff index and is denoted by Kf(G). In this study, we will compute the resistance distance between pairs of vertices of string of diamonds and ring of diamonds by using some methods from electrical network theory such as series and parallel principles, the principle of elimination, the star-triangle transformation, and the delta-wye transformation. Then we determine the exact formulas for the Kirchhoff index of the string of diamonds and ring of diamonds.

Computation of resistance distance with Kirchhoff index of body centered cubic structure

Computation of resistance distance with Kirchhoff index of body centered cubic structure

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>

Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let Pn(Pn′) represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of Pn contain the eigenvalues of matrices LR and LS. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of Pn are determined. The relationship between the Wiener and Kirchhoff indices of Pn is also established.

A New Probabilistic Molecular Index

We propose a new molecular index based on a distance on graphs defined through hitting times of random walks on connected graphs. We show its connections to previous probabilistic/electric indices such as the RW index and the Kirchhoff index, compute its value for some families of graphs, and present some open questions.

The minimal degree Kirchhoff index of bicyclic graphs

<abstract><p>The degree Kirchhoff index of graph $ G $ is defined as $ Kf^{*}(G) = \sum\limits_{{u, v}\subseteq V(G)}d(u)d(v)r_{G}(u, v) $, where $ d(u) $ is the degree of vertex $ u $ and $ r_{G}(u, v) $ is the resistance distance between the vertices $ u $ and $ v $. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchhoff index of order $ n\geq5 $. Moreover, we obtain the minimum degree Kirchhoff index on bicyclic graphs of order $ n\geq4 $ with exactly three cycles, and all bicyclic graphs of order $ n\geq4 $ where the minimum degree Kirchhoff index has been obtained.</p></abstract>

Construction of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [?]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs ℛ(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and ℛ(G2) and S(H1) and ℛ(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.

On the Laplacian spectra and the Kirchhoff indices of two types of networks

Let G be a connected simple graph. Denoted by the Kirchhoff index of G. In this note, we study the Laplacian spectra of two types of networks and . As a special case, explicit formulas for the Kirchhoff index and number of spanning trees of prisms are obtained. Some other interesting results, such as the expression for the coefficient of the Laplacian characteristic polynomials of cycles, are also derived.

Minimizing the effective graph resistance by adding links is NP-hard

The effective graph resistance, also known as the Kirchhoff index, is metric that is used to quantify the robustness of a network. We show that the optimisation problem of minimizing the effective graph resistance of a graph by adding a fixed number of links, is NP-hard.

Greedy optimization of resistance-based graph robustness with global and local edge insertions

The total effective resistance, also called the Kirchhoff index, provides a robustness measure for a graph G. We consider two optimization problems of adding k new edges to G such that the resulting graph has minimal total effective resistance (i.e., is most robust)—one where the new edges can be anywhere in the graph and one where the new edges need to be incident to a specified focus node. The total effective resistance and effective resistances between nodes can be computed using the pseudoinverse of the graph Laplacian. The pseudoinverse may be computed explicitly via pseudoinversion, yet this takes cubic time in practice and quadratic space. We instead exploit combinatorial and algebraic connections to speed up gain computations in an established generic greedy heuristic. Moreover, we leverage existing randomized techniques to boost the performance of our approaches by introducing a sub-sampling step. Our different graph- and matrix-based approaches are indeed significantly faster than the state-of-the-art greedy algorithm, while their quality remains reasonably high and is often quite close. Our experiments show that we can now process larger graphs for which the application of the state-of-the-art greedy approach was impractical before.