Hybrid Graph Theory and Network Analysis

Novak, Ladislav A.; Gibbons, Alan

doi:10.1017/cbo9780511666391

Cited by 15 publications

(10 citation statements)

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“…is solved that all things are easy. Next we conduct the same matrix transformation used in (11) to (7) and (8). Applying P m to ( 7) and ( 8) on the left-hand sides, we are led to…”

Section: Approach Of the Matrix Transformationmentioning

confidence: 99%

“…Modeling the resistor network has become a basic method to solve a series of complicated problems in the fields of engineering, science and technology. Such as the computation of resistances is relevant to a wide range of problems ranging from classical transport in disordered media [2], first-passage processes [3], lattice Greens functions [4], random walks [5], resistance distance [6,7], to hybrid graph theory [8].…”

Section: Introductionmentioning

confidence: 99%

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Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

Tan

2015

Phys. Rev. E

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We develop a general recursion-transform (R-T) method for a two-dimensional resistor network with a zero resistor boundary. As applications of the R-T method, we consider a significant example to illuminate the usefulness for calculating resistance of a rectangular m×n resistor network with a null resistor and three arbitrary boundaries, a problem never solved before, since Green's function techniques and Laplacian matrix approaches are invalid in this case. Looking for the exact calculation of the resistance of a binary resistor network is important but difficult in the case of an arbitrary boundary since the boundary is like a wall or trap which affects the behavior of finite network. In this paper we obtain several general formulas of resistance between any two nodes in a nonregular m×n resistor network in both finite and infinite cases. In particular, 12 special cases are given by reducing one of the general formulas to understand its applications and meanings, and an integral identity is found when we compare the equivalent resistance of two different structures of the same problem in a resistor network.

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“…is solved that all things are easy. Next we conduct the same matrix transformation used in (11) to (7) and (8). Applying P m to ( 7) and ( 8) on the left-hand sides, we are led to…”

Section: Approach Of the Matrix Transformationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

Tan

2015

Phys. Rev. E

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“…The computation of two-vertex resistances in electrical networks is a very old problem considered by many researchers over many years [2]. The computation of resistance is pertinent to a wide selection of problems extending from random walks [4], opinion formation [12], classical transport in disordered media [3], robustness of coupled oscillators network [5]- [7], first-passage processes [8], identifying the influential spreader node in a network [11], lattice Greens functions [9], [10], resistance distance [13], [14], to graph theory [10], [15]. There are numbers of techniques and formulae have been developed for calculating the resistance distance, i.e., algebraic formulae [17], [19]- [24], series and parallel rules, combinatorial formula [4], deltawye transformation [16], sum rules [17], [18], star-triangle transformation [16], probabilistic formulae [4], [25], starmesh transformation, the principle of elimination, recursion formula [26], the principle of substitution and so forth.…”

Section: Introductionmentioning

confidence: 99%

A Novel and Efficient Method for Computing the Resistance Distance

et al. 2021

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The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to G, with each edge being replaced by a unit resistor. Hypercube Q n is one of the most efficient and versatile topological structure of the interconnection networks which received much attention over the past few years. The folded n-cube graph is obtained from hypercube Q n by merging vertices of the hypercube Q n that are antipodal, i.e., lie at a distance n. Folded n-cube graph have been studied in parallel computing as a potential network topology. The folded n-cube has the same number of vertices but the half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. Our intention is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded n-cube by using the symmetry method and classic Kirchhoff's equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in Biggs-Smith graph, which shows the competency of the proposed method.

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“…The computation of the two-point resistance in a resistor network is a classical problem in circuit theory and graph theory, which has been researched for more than 160 years. The computation of resistance is relevant to a wide range of problems ranging from classical transport in disordered media 1 , first-passage processes 2 , lattice Greens functions 3 4 , random walks 5 , resistance distance 6 7 , to graph theory 4 8 . However, it is usually very difficult to obtain the explicit expression of resistance in a non-regular resistor networks (it means different resistors are arranged on the network, which differs from a normal resistor network), since the non-regular condition is like a wall or trap which affects the behavior of finite network.…”

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confidence: 99%

Recursion-Transform method to a non-regular m×n cobweb with an arbitrary longitude

Tan

2015

Sci Rep

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A general Recursion-Transform method is put forward and is applied to resolving a difficult problem of the two-point resistance in a non-regular m × n cobweb network with an arbitrary longitude (or call radial), which has never been solved before as the Green’s function technique and the Laplacian matrix approach are difficult in this case. Looking for the explicit solutions of non-regular lattices is important but difficult, since the non-regular condition is like a wall or trap which affects the behavior of finite network. This paper gives several general formulae of the resistance between any two nodes in a non-regular cobweb network in both finite and infinite cases by the R-T method which, is mainly composed of the characteristic roots, is simpler and can be easier to use in practice. As applications, several interesting results are deduced from a general formula, and a globe network is generalized.

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Hybrid Graph Theory and Network Analysis

Cited by 15 publications

References 0 publications

Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries

A Novel and Efficient Method for Computing the Resistance Distance

Recursion-Transform method to a non-regular m×n cobweb with an arbitrary longitude

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