On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain

The Moore–Penrose inverse: a hundred years on a frontline of physics research

“…Further papers demonstrating applications of the Moore-Penrose matrix inverse within the scope of theoretical physics include, among other, articles: Beylkin et al (2008), where the formulae for the inverse of modified matrices were exploited in a Green's function iteration algorithm introduced to solve the timeindependent, multiparticle Schrödinger equation, He et al (2012), which introduced phase-entanglement and phase-squeezing criteria for two bosonic fields that are robust against a number of fluctuations using the inverse to normalize the particle number operator, Huang and Li (2020), where formulae for the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain were derived by means of the inverses of Laplacian matrices, Kametaka et al (2015), where the inverse of singular discrete Laplacian was used to solve difference equations to estimate a maximal deviation of a carbon atom from the steady state in C60 fullerene buckyball, Kirkland (2015) dealing with a quantum state transfer in a quantum walk on a graph, with the inverse used to derive expressions for the first and second partial derivatives of the fidelity of the transfer with respect to a weight of an edge, Kougioumtzoglou et al (2017), where an inverse based frequency response function was introduced to generalize frequency domain random vibration solution methodologies to account for linear and nonlinear structural systems with singular matrices, Lian et al (2019), where the inverse was exploited for calculating charge density distribution through Hartree potential to disclose the physical mechanism of electrostatic potential anomaly in 2D Janus transition metal dichalcogenides, McCartin (2009) reexpressing the Rayleigh-Schrödinger perturbation theory procedure in terms of the inverse, Meister et al (2014), where the inverse was used to formulate an optimal control algorithm with a control subspace defined by a superposition of arbitrary waveforms, Pignier et al (2017), where a model of an aeroacoustic sound source was created based on compressible flow simulations, with the inverse used to compute the sound source strengths, Ranjan and Zhang (2013) exploring the geometry of complex networks in terms of an Euclidean embedding represented by the inverse of its graph Laplacian, Yang et al (2018), where the inverse was used to solve an equilibrium equation originating in an empirical mode decomposition method combining the static and dynamic information for structural damage detection, and Yang et al (2020), where an expression for the inverse of Laplacian matrices of two connected weighted graphs was established and utilized to derive a recursion formula for the resistance distance.…”

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

“…In 2007, the resistance distance and Kirchhoff index in circulant graphs are discussed in [17]. Huang and Li [18] computed the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain. The tenure resistance was considered due to the physical construction: replace each edge in G by a resistance of one ohm, and consider the resistance distance between two vertices u and v of G, denoted by R uv is an effective resistance among them.…”

Resistance distance of generalized wheel and dumbbell graph using symmetric {1}-inverse of Laplacian matrix

“…Resistance distances have been computed for a variety of interesting classes of graphs so far, with a focus on electrical networks and chemical graphs. Resistance distances have been obtained for some particular classes of graphs, for example, regular graphs [22], circulant graphs [11], distance regular networks [12], wheels and fans [31], Cayley graphs [40], complete graph minus N edges [27], complete n-partite graphs [34], Cayley graphs on symmetric groups [24], some class of graphs [37], pseudo-distance regular [13], almost complete bipartite graphs [23], ring clique network [35], and so on.…”