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

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.

the concise expression of resistance of an arbitrary m × n resistor networks except for a series of simple resistor networks (perhaps this method will also lead to solving this problem in the future).
For the arbitrary resistor networks of various topologies, various new results of the resistor networks have been obtained by applying the R-T method. For example, a cobweb model 17 , a globe network 18 , a fan network 19 , a cobweb network with 2r boundary 20 , a hammock network 15 , especially, very recently this method was further generalized in Ref. 21, which can applied to the resistor network with an arbitrary boundary.
However, there are still some non-regular resistor networks unresolved because of the complexity of the boundary conditions of the network in real life. Thanks to the R-T method provides us with new technique to deal with the complex networks of various topologies. In this paper we will study the resistances of a complex network with arbitrary resistors on the longitude line. Figure. 1 is called a non-regular m × n resistor cobweb which has arbitrary resistor r 2 on the boundary and an arbitrary longitude with arbitrary resistors r 1 (in order to facilitate our study, we refer the name in globe and call the radial as longitude, and call the circle line as latitude). This paper focus on the computation of the two-point resistance between any two nodes in the non-regular m × n cobweb network, which has never been solved before, the Green's function technique and the Laplacian matrix approach are difficult in this case because they depend on the two matrices along two directions (one may refer the discussion at the end). Figure. 1 is a multipurpose network model, when the boundary resistor r 2 = 0, the non-regular cobweb degrades into a nearly globe network as shown in Fig. 2.

Results
The boundary resistor r 2 is a key parameter since the different parameter can represent different geometric structure, such as a globe is from a nearly cobweb with r 2 = 0. In this paper we will consider three cases of r 2 = {0, r, 2r}, and give several results in three cases. We first define several variables , and t i for later uses by  Noticing that θ i has three kinds different values with three different resistors of The results in the case of r 2 = r. Consider a nearly m × n resistor cobweb with resistances r and r 0 in the respective latitudes and longitudes except for a longitude resistors r 1 , where m and n are, respectively, the numbers of grids along longitude and latitude directions as shown in Fig. 1. Assuming the center node O is the origin of the rectangular coordinate system, and a longitude with r 1 act as Y axis. Denote nodes of the network by coordinate {x, y}. The equivalent resistance between any two nodes d 1 (x 1 , y 1 ) and d 2 (x 2 , y 2 ) in a nearly m × n cobweb network with free boundary can be written as (1) and (2) In particular, from (4) we have the special cases: Case 1. When h 1 = 1, the network degrades into a normal cobweb, we have  where Case 6. When d 1 = (x, y 1 ) and d 2 = (x, y 2 ) are both on the same longitude, we have Especially, when d 1 = (0, y 1 ) and d 2 = (0, y 2 ) are both on the Y axis, we have The result in the case of r 2 = 2r. When r 2 = 2r, the equivalent resistance between any two nodes d 1 (x 1 , y 1 ) and d 2 (x 2 , y 2 ) in a nearly m × n cobweb network with 2r boundary is  with Δ x = x 2 − x 1 .
The result in the case of r 2 = 0. When r 2 = 0, the non-regular cobweb network degrades into a nearly globe network with an arbitrary longitude as shown in Fig. 2. The equivalent resistance between any two nodes d 1 (x 1 , y 1 ) and d 2 (x 2 , y 2 ) in a nearly m × n globe network with an arbitrary longitude can be written as where θ k = (k − 1)π/m. Especially, when h 1 = 1, the nearly globe degrades into a normal globe network, from (15) we have

Method
Calculating resistance by Ohm's law. Assuming the electric current J is constant and goes from the input d 1 (x 1 , y 1 ) to the output d 2 (x 2 , y 2 ) as shown in Fig. 1. Denote the currents in all segments of the network as shown in Fig. 3 where [ ] T denote matrix transposes, and (H k ) i is the elements of H x with the injection of current J at d 1 (x 1 , y 1 ) and the exit of current J at d 2 (x 2 , y 2 ), and A m is an m × m matrix, where h = r/r 0 , h 2 = r 2 /r 0 . We consider the bound conditions of a specific longitude with resistors r 1 . Applying Kirchhoff 's laws to two meshes adjacent to the specific longitude we obtain three matrix equations to model the bound currents, where h 1 = r 1 /r 0 and E is an m-dimensional identity matrix, and matrix A m is given by (21). Above Eqs.(18) ~ (22) are all equations we need to calculate the equivalent resistance of the non-regular m × n cobweb network with an arbitrary longitude.
Approach of the matrix transform. To solve the matrix equation (18) (17), we therefore obtain the resistance R m × n (d 1 , d 2 ) in three cases of r 2 = {0, r, 2r}.

Discussion
Considerable progress has recently been made in the development of techniques to exactly determine two-point resistances in the networks of various topologies. In this paper the R-T method is applied to computation of the two-point resistance in a non-regular m × n cobweb network with an arbitrary boundary and an arbitrary longitude, which has never been solved before. The resistance formulae are given in the form of a single summation, and several previous research results have become our special cases. Such as formula (5) Fig. 1 is a multipurpose network model which contains several different cases.
The reason why the Green's function technique and the Laplacian matrix approach are difficult to resolve the non-regular network in Fig. 1 is that they depend on the two matrices along two directions, besides matrix (21), there is another matrix with an arbitrary element of r 1 when we set up a matrix along the latitudes directions, which is impossible for us to obtain the explicit eigenvalues and eigenvectors of a matrix with arbitrary element.
The reason why we just consider r 2 = {0, r, 2r} is that, at present, we cannot obtain the explicit eigenvalues and eigenvectors of matrix (21) when r 2 ≠ {0, r, 2r}, although the R-T method is feasible. Of course, we can obtain the resistance of the non-regular network as soon as we obtain the explicit eigenvalues and eigenvectors of matrix (21).
The R-T method splits the derivation into three parts. The first creates a recursion relation between the current distributions on three successive longitude lines. The second part derives a recursion relation between the current distributions on the bound longitude. The third part implements the matrix transform of diagonalization to produce a recurrence relation involving only variables on the same axis. Basically, the method reduces the problem from two dimensions to one dimension.
In addition, an important usefulness is that the R-T method can be extended to impedance networks, since the Ohm's law based on which the method is formulated is applicable to impedances. Such as the grid elements r and r 0 can be either a resistors or impedances, we can therefore study the m × n complex impedance networks by the R-T method.