The equivalent resistance of a 3 ×ncobweb network and its conjecture of anm×ncobweb network

Abstract

We investigate the equivalent resistance of a 3 × n cobweb network. The difference equations of the model are constructed by network analysis and their general solution is obtained by matrix transformations. It is found that the equivalent resistance can be expressed by the trigonometric function of kπ/7, which decreases with the increase of the order n. By analyzing and comparing the equivalent resistances of the 1 × n, 2 × n and 3 × n cobweb networks, a conjecture on the equivalent resistance of the m × n cobweb network is proposed.

Two-point resistance of a resistor network embedded on a globe

“…Early discussions along this line are due to Tan et al [6,[8][9][10]. A similar analysis for a fan network has been given recently in [11].…”

“…An alternative direct approach of computing resistances had been developed by Tan et al [6,[8][9][10] that, when applied to the cobweb and globe networks, gives the result in terms of a single summation. This offers a direct and somewhat simpler approach.…”

“…1. Tan, Zhou, and Yang [6] proposed a conjecture on the resistance between two nodes on the cobweb. It is difficult to adopt the Laplacian approach directly to the problem due to the special geometry of the cobweb.…”

“…Studies of the resistance problem have also been carried out independently by Tan et al along a route that we shall refer to as the method of direct evaluation [6,[8][9][10]. The direct method is useful in cases when there exists a special node such as a pole of the globe or the center of the cobweb, with all other nodes connected to it equally along longitudes of a globe or the radii of a cobweb.…”

“…One thus circumvents the need of diagonalizing a nonregular Laplacian matrix. The direct method of computing resistances had been applied successively to the cobweb network for fixed values of m up to m = 4 [6,8,9]. It has also been used recently to compute the resistances in a fan network [11].…”