Kirchhoff Index Research Articles

Biharmonic Distance-Based Performance Metric for Second-Order Noisy Consensus Networks

We study second-order consensus dynamics with random additive disturbances. To quantify the robustness of these networks, we investigate three different performance measures: the steady-state variance of pairwise differences between vertex states, the steady-state variance of the deviation of each vertex state from the average, and the total steady-state variance of the system. We show that these performance measures are closely related to the concept of biharmonic distance; the square of the biharmonic distance plays a similar role in the system performance as resistance distance plays in the performance of first-order noisy consensus dynamics. We then define the new concepts of biharmonic Kirchhoff index and vertex centrality based on the biharmonic distance. We further derive analytical results for the performance measures and concepts for complete graphs, star graphs, cycles, and paths, and we use this analysis to compare the asymptotic behavior of the steady-state variance in first-and second-order systems. Finally, we propose a theoretically guaranteed approximation algorithm to estimate the total steady-state variance, which has a complexity of nearly linear time with respect to the number of edges. Extensive experiments results validate both efficiency and accuracy of our algorithm.

An Improved Lower Bound for the Degree Kirchhoff Index of Bipartite Graphs

An Improved Lower Bound for the Degree Kirchhoff Index of Bipartite Graphs

On the Kirchhoff index of a graph and the matchings of the subdivision

Suppose that G is a connected simple graph with n vertices. Let Kf(G) and m(G,i) denote the Kirchhoff index of G and the number of i-matchings of G, respectively. First, in 2006, Yan and Yeh showed that if G is a tree, then Kf(G)=m(S(G),n−2),where S(G) is the subdivision of G. Then, in 2021, Chen and Yan showed that if G is a unicyclic graph with the cycle Ck, then Kf(G)=[m(S(G),n−2)−2m(S(G)−S(Ck),n−2−k)]/k.In this paper, we generalize the above results to any connected simple graph. In fact, we obtain the expressions for all Laplacian coefficients of G in terms of m(S(G),i), which provides much more information about relations between various parameters of G and the matchings of the subdivision S(G).

The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n ! / r ! n − r ! , respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .

The (Multiplicative Degree‐) Kirchhoff Index of Graphs Derived from the Cartesian Product of Sn and K2

It is well known that many topological indices have widespread use in lots of fields about scientific research, and the Kirchhoff index plays a major role in many different sectors over the years. Recently, Li et al. (Appl. Math. Comput. 382 (2020) 125335) proposed the problem of determining the Kirchhoff index and multiplicative degree‐Kirchhoff index of graphs derived from Sn × K2, the Cartesian product of the star Sn, and the complete graph K2. In the present study, we completely solve this problem, that is, the explicit closed‐form formulae of the Kirchhoff index, multiplicative degree‐Kirchhoff index, and number of spanning trees are obtained for some graphs derived from Sn × K2.

Two Kinds of Laplacian Spectra and Degree Kirchhoff Index of the Weighted Corona Networks

Recently, the study related to network has aroused wide attention of the scientific community. Many problems can be usefully represented by corona graphs or networks. Meanwhile, the weight is a vital factor in characterizing some properties of real networks. In this paper, we give complete information about the signless Laplacian spectra of the weighted corona of a graph G1 and a regular graph G2 and the complete information about the normalized Laplacian spectra of the weighted corona of two regular graphs. The corresponding linearly independent eigenvectors of all these eigenvalues are also obtained. The spanning trees’ total number and the degree Kirchhoff index of the weighted corona graph are computed.

Mean Hitting Time for Random Walks on a Class of Sparse Networks.

For random walks on a complex network, the configuration of a network that provides optimal or suboptimal navigation efficiency is meaningful research. It has been proven that a complete graph has the exact minimal mean hitting time, which grows linearly with the network order. In this paper, we present a class of sparse networks in view of a graphic operation, which have a similar dynamic process with the complete graph; however, their topological properties are different. We capture that has a remarkable scale-free nature that exists in most real networks and give the recursive relations of several related matrices for the studied network. According to the connections between random walks and electrical networks, three types of graph invariants are calculated, including regular Kirchhoff index, M-Kirchhoff index and A-Kirchhoff index. We derive the closed-form solutions for the mean hitting time of , and our results show that the dominant scaling of which exhibits the same behavior as that of a complete graph. The result could be considered when designing networks with high navigation efficiency.

NETWORK COHERENCE ANALYSIS ON A FAMILY OF NESTED WEIGHTED n-POLYGON NETWORKS

In this paper, we propose a family of nested weighted [Formula: see text]-polygon networks, which is a kind of promotion of infinite fractal dimension networks. We study the coherence of the networks with recursive features that contain the initial states dominated by a weighted parameter. The network coherence is a consensus problem with additive noises, and it is known that coherence is defined by the eigenvalues of the Laplacian matrix. According to the structure of recursive nested model, we get the recursive expressions of Laplacian eigenvalues and further derive the exact results of first- and second-order coherence. Finally, we investigate the influential impacts on the coherence for different large parameters and discuss the relationship between Laplacian energy and network coherence. Furthermore, we obtain the expressions of Kirchhoff index, mean first-passage time and average path length of the networks.

Second-stage spectrum of corona of two graphs

The derived graph of a simple graph [Formula: see text], denoted by [Formula: see text], is the graph having the same vertex set as [Formula: see text], in which two vertices are adjacent if and only if their distance in [Formula: see text] is two. The spectrum of derived graph of [Formula: see text] is called the second-stage spectrum of [Formula: see text]. In this paper, we will determine the second-stage spectrum (Laplace and signless Laplace spectrum) of the corona of two regular graphs with diameter less than or equal to two. The energy of the derived graph is called the second-stage energy of [Formula: see text]. Here, we proved that the class of graphs [Formula: see text] is integral if and only if [Formula: see text] is a perfect square and the second-stage energy depends only on number of vertices. Moreover, we discuss some applications like the number of spanning trees, the Kirchhoff index and Laplacian-energy-like invariant of the newly constructed graph.

On the Laplacians for Strong Product Graphs Based on Polyacene Graphs

Let Hn be the molecular graph of a linear polyacene chain. In this paper, we introduce a kind of linear polyacene cylinder graph which is the strong product of two automorphism polyacene graphs Hn and Next, we obtain that the Laplacian spectrum of consists of the eigenvalues of symmetric matrices LA and LS of order according to the decomposition theorem of Laplacian characteristic polynomial. Furthermore, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we study the Kirchhoff index and the number of spanning trees, and derive their explicit expressions in terms of n for the graph

Resistance distance and Kirchhoff index in dihedral Cayley graphs

Let D2n=〈a,b∣an=b2=1,bab=a−1〉={1,a,…,an−1,b,ba,…,ban−1} be the dihedral group of order 2n and let S be a symmetric subset of D2n. The dihedral Cayley graph Cay(D2n,S) is the graph with vertex set VCay(D2n,S)=D2n and edge set ECay(D2n,S)={g,h}∣gh−1∈S,g,h∈D2n. In this paper, the complete information for the eigenvalues and their corresponding orthonormal eigenvectors of Cay(D2n,S) is provided. The closed-form formulae of the Kirchhoff index and the resistance distances between any two vertices of Cay(D2n,S) are then derived.

The Kirchhoff index and spanning trees of Möbius/cylinder octagonal chain

The Kirchhoff index and degree-Kirchhoff index have attracted extensive attentions due to their wide applications in physics and chemistry. These indices have been computed for many interesting graphs, such as linear polyomino chain, linear/Möbius/cylinder hexagonal chain, and linear octagonal chain. In the present paper, we consider Möbius octagonal chain (Mn) and cylinder octagonal chain (Mn′). Explicit closed-form formulae of the Kirchhoff index and the number of spanning trees are obtained for Mn and Mn′.

Comparison of the Wiener and Kirchhoff Indices of Random Pentachains

Let G be a connected (molecule) graph. The Wiener index W G and Kirchhoff index K f G of G are defined as the sum of distances and the resistance distances between all unordered pairs of vertices in G , respectively. In this paper, explicit formulae for the expected values of the Wiener and Kirchhoff indices of random pentachains are derived by the difference equation and recursive method. Based on these formulae, we then make comparisons between the expected values of the Wiener index and the Kirchhoff index in random pentachains and present the average values of the Wiener and Kirchhoff indices with respect to the set of all random pentachains with n pentagons.

A Note On Bipartite Graphs with Domination Number 2 and 3

When each edge of a connected G graph is replaced by a unit resistor, the resistance distance is computed as the effective resistance between any two vertices in G. The Kirchhoff index of G is given by the sum of resistance distances between all pairs of vertices. The multiplicative eccentricity resistance-distance (MERD) of a connected graph G is defined as , where is the set of vertices of , is the resistance-distance between the vertices and , and are the eccentricity of the vertices and , respectively. The MERD of the G can be obtained by using Kirchhoff index. In this paper, we characterize the bipartite graphs which have the smallest and largest MERD with domination number 2 are given. We also characterize the bipartite graphs which have the smallest MERD with the domination number 3.

Topological index analysis of a class of networks analogous to alicyclic hydrocarbons and their derivatives

AbstractAlicyclic hydrocarbons and their derivatives have been studied extensively by researchers in recent years. Actually, they are a subclass of the toroidal networks. The kind of networks can be characterized as the toroidal networks obtained from associating lots of identical basic sub‐networks into a cycle. In this paper, we study topological indices correlated with the toroidal networks, such as resistance distance, the Kirchhoff index, and the degree‐Kirchhoff index by creating n‐separations which can compute resistance distance simply. Furthermore, we also derive and analyze lots of topological indices based on degree of the toroidal networks. For instance, the Zagreb index, the general Randić index, the Forgotten index, and so on. These characteristic quantities can be utilized to evaluate the properties of alicyclic hydrocarbons as well as can be extensively applied to many fields, like mathematics, physics, chemistry, biology, or other fields.

On unitary Cayley graphs of matrix rings

Let Fq be a finite field of order q and let R be a finite commutative local ring which is not a field. Recently, three (resp. four) distinct eigenvalues of the unitary Cayley graph CMn(Fq) (resp. CMn(R)) have been determined in Rattanakangwanwong and Meemark (2020) [20]. In this paper, completely explicit closed formulas for all the eigenvalues of CMn(Fq) and CMn(R) are obtained by using a new approach. As applications, the energy, the Kirchhoff index and the number of spanning trees of CMn(Fq) and CMn(R) are derived, respectively.

Hubs-biased resistance distances on graphs and networks

We define and study two new kinds of “effective resistances” based on hubs-biased – hubs-repelling and hubs-attracting – models of navigating a graph/network. We prove that these effective resistances are squared Euclidean distances between the vertices of a graph. They can be expressed in terms of the Moore–Penrose pseudoinverse of the hubs-biased Laplacian matrices of the graph. We define the analogous of the Kirchhoff indices of the graph based of these resistance distances. We prove several results for the new resistance distances and the Kirchhoff indices based on spectral properties of the corresponding Laplacians. After an intensive computational search we conjecture that the Kirchhoff index based on the hubs-repelling resistance distance is not smaller than that based on the standard resistance distance, and that the last is not smaller than the one based on the hubs-attracting resistance distance. We also observe that in real-world brain and neural systems the efficiency of standard random walk processes is as high as that of hubs-attracting schemes. On the contrary, infrastructures and modular software networks seem to be designed to be navigated by using their hubs.

Resistance Distance-Based Indices and Spanning Trees of Linear Pentagonal-Quadrilateral Networks

Let Qn be the linear pentagonal-quadrilateral chain containing 2n pentagons and n squares. In this paper, we determined the (normalized) Laplacian spectra of Qn based on the decomposition techniques for the corresponding matrices. Further, we obtained explicit expressions for (multiplicative degree-) Kirchhoff index and the number of spanning trees of linear pentagonal-quadrilateral chains which are based on relationships between the coefficients and roots.

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.

The Hermitian Kirchhoff Index and Robustness of Mixed Graph

Large-scale social graph data poses significant challenges for social analytic tools to monitor and analyze social networks. The information-theoretic distance measure, namely, resistance distance, is a vital parameter for ranking influential nodes or community detection. The superiority of resistance distance and Kirchhoff index is that it can reflect the global properties of the graph fairly, and they are widely used in assessment of graph connectivity and robustness. There are various measures of network criticality which have been investigated for underlying networks, while little is known about the corresponding metrics for mixed networks. In this paper, we propose the positive walk algorithm to construct the Hermitian matrix for the mixed graph and then introduce the Hermitian resistance matrix and the Hermitian Kirchhoff index which are based on the eigenvalues and eigenvectors of the Hermitian Laplacian matrix. Meanwhile, we also propose a modified algorithm, the directed traversal algorithm, to select the edges whose removal will maximize the Hermitian Kirchhoff index in the general mixed graph. Finally, we compare the results with the algebraic connectivity to verify the superiority of the proposed strategy.