Complexity

Complexity / 2020 / Article
Special Issue

Complexity in Deep Neural Networks

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 5917098 | https://doi.org/10.1155/2020/5917098

Jing Zhao, Jia-Bao Liu, Ali Zafari, "Complete Characterization of Resistance Distance for Linear Octagonal Networks", Complexity, vol. 2020, Article ID 5917098, 13 pages, 2020. https://doi.org/10.1155/2020/5917098

Complete Characterization of Resistance Distance for Linear Octagonal Networks

Academic Editor: M. Irfan Uddin
Received26 Jul 2020
Revised17 Aug 2020
Accepted26 Aug 2020
Published15 Sep 2020

Abstract

Computing the resistance distance of a network is a fundamental and classical topic. In the aspects of considering the resistances between any two points of the lattice networks, there are many studies associated with the ladder networks and ladderlike networks. But the resistances between any two points for more complex structures than ladder networks or ladderlike networks are still unknown. In this paper, a rather complicated structure which is named linear octagonal network is considered. Treelike octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. A linear octagonal network is a cata-condensed octagonal system with no branchings. Moreover, the resistances between any two points of a linear octagonal network are first determined. One finds that the effective resistances between new inserted points and others points of a linear octagonal network can be given by the effective resistances between two initial points which are inherited from the linear polyomino network.

1. Introduction

Our convention in this paper follows [1] for notations that we omit here. The resistance distance between vertices and was introduced in 1993 by Klein and Randić [2]. Based on the electrical circuit theory, one can obtain the resistance distance between two points by replacing each edge of the network with a unit resistor. In the past two decades, the resistance distance has attracted experts’ much attention. The most important reason for this heightened activity is that it goes beyond the practical applications of the traditional electrical ones, such as in chemistry and network science. The resistance distance is a good indicator to distinguish isomer or molecular networks with similar structures.

We say the electrical network is simple. It means that the resistance distance of each edge in an electrical network is the unit resistor [3]. Denoting as the resistance distance of the edge , then the conductance of the edge of is . The effective resistance between any points can be represented by the resistance distance between any points on a network. For a general network, it is hard enough to determine the effective resistance between any points, unless the network is very small or one knows the complete information of the network, such as degree of each vertex and distances between any two vertices. Hence, such methods are proposed which only need some local information; examples include star-triangle transformation [4, 5], resistance distance local sum rules [6], and effective resistance sum rules [3]. For more information associated with the effective resistance or resistance distance, see [715].

For path and its copy , let and . If ( are all positive integers), we connect to for and denote this graph by . Obviously, if , is the linear polymino network; if , is the linear hexagonal network; and if , is the linear octagonal network. Polyomino systems are widely studied in organic chemistry, especially in polycyclic aromatic compounds [16]. Treelike octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. A linear octagonal network is a cata-condensed octagonal system with no branchings [17]. Also, a linear octagonal network can be constructed from a linear polyomino network by inserting some new points in the line with certain rules (see Figure 1). These networks are very crucial in theoretical chemistry because they are natural graph representations of some hydrocarbons.

The Kirchhoff index and multiplicative degree-Kirchhoff index are, respectively, defined as [2] and [18], where is the degree of vertex , which is equal to the number of edges connected to vertex . These two indices are found to be well associated with the Laplacian eigenvalues and normalized Laplacian eigenvalues . Putting them in other ways, [2] and [19]. In most recent years, the study of Kirchhoff index and multiplicative degree-Kirchhoff index caught some scholars’ eyes, especially for linear systems. The Laplacian and normalized Laplacian spectra of linear systems come from much symmetric tridiagonal matrices, like how Zhu [20] determined the Laplacian and normalized Laplacian spectra of linear octagonal networks and those two indices. The computation of the Laplacian and normalized Laplacian spectra of the corresponding symmetric tridiagonal matrices seems interesting but rather complicated. Hence, from the point of view of the definitions for the Kirchhoff index and multiplicative degree-Kirchhoff index, we aim to compute the effective resistances between any points of linear octagonal networks in this paper via circuit reduction theory and effective resistance sum rule. The effective resistances of other linear systems and circuit reduction theory or star-triangle transformation can be referred in [4, 8, 2123].

In this paper, some preliminaries are proposed in Section 2. The effective resistances between two initial points which are inherited from a linear polyomino network are determined in Section 3. The effective resistances between new inserted points and other points in a linear octagonal network are given in Section 4. The whole paper is concluded in Section 5.

2. Preliminaries

In this section, we put a lemma that is used in computing the effective resistances between new inserted points and other points in a linear octagonal network.

Lemma 1 (see [3]). Suppose that is a electrical network. For any two points and . Then,where is the neighbor set of point .

For the sake of convenience, we here define three functions:

According to the structures of the linear octagonal network, we can divide its effective resistances between any points into two cases. For any , one has the following cases:Case 1. Effective resistances between two initial points areCase 2. Effective resistances between new inserted points and other points are

3. Effective Resistances between Two Initial Points

In this section, the effective resistances between two initial points which are inherited from a linear polyomino network are determined. Let the linear octagonal network be a simple electrical network. Putting it in another way, the resistance of each edge in a linear octagonal network is 1. One gets the linear polyomino network from the linear octagonal network by using circuit reductions (see Figure 2).

Assume that . One obtains the network by making a circuit reduction for the linear polyomino network with respect to points and (see Figure 3).

According to the network , one obtains

By computing equation (5), we obtain

In what follows, we first calculate . One obtains the network from the network through a circuit reduction or star-triangle transformation about points , and (see Figure 4).

On the one hand, one has based on the network , namely,

On the other hand, we have

Combining equations (7) and (8) yieldswhere .

Subtracting equation (10) from equation (9) leads to

According to equations (6) and (11), one arrives at

Substituting equations (6) and (12) into equation (10), one obtains

Hence,

By equations (12) and (14), we obtain

Moreover,

Next, we will compute , and , . We make a circuit reduction of the network with respect to points , and (see Figure 5).

It is easy to check that the upper and lower parts of the network are, respectively, the networks and in line with Figure 5. Also, one has .

According to equation (16), we arrive at

By equation (17), we have

Substituting equations (6) and (17) into (18) yields

It is not hard to verify that equation (19) holds for .

At this place, we are going to give the explicit formulas for and . One makes a circuit reduction for the network with respect to points , and , as illustrated in Figure 6.

Based on Figure 6, one has

Thus,

Computing equation (21), we have

Combining equations (6), (19), (21), and (22), one obtains

One can straightforwardly verify that the formulas of and also hold for . Moreover, we have

4. Effective Resistances between New Inserted Points and Others

In Section 3, the formulas of effective resistances between the initial points of the linear octagonal network are determined. At this point, we will give the complete information of effective resistances between new inserted points and other points in the linear octagonal network by dividing them into 10 subcases. One obtains the network by making a circuit reduction for the linear octagonal network with respect to point sets and (see Figure 7).

In what follows, we will concentrate on computing and . According to Lemma 1 and Figure 7, we obtain

On the one hand, subtracting equation (27) from the equation which is obtained using equation (26) times 4 leads to

Moreover, . Hence,

Substituting equation (29) into equation (28), one obtains

On the other hand, subtracting equation (26) from the equation which is obtained using equation (27) times 5/2 yields

Substituting equation (29) into equation (31), one has

For the resistance distance between the point and other points, we can obtain

By calculations, one arrives at

One can easily check that and . Thus, we have

Putting them in another way,

It turns out that the formulas for , , , , and hold for any . Besides, we have

One obtains the network by making a circuit reduction for the linear octagonal network with respect to point sets and (see Figure 8).

In what follows, we will concentrate on computing and . According to Lemma 1 and Figure 8, we obtain

On the one hand, subtracting equation (39) from the equation which is obtained using equation (38) times 5/2 leads to

On the other hand, subtracting equation (38) from the equation which is obtained using equation (39) times 4 yields

Remark 1. Combining equations (32) and (40), we find thatfor any .
For the resistance distance between the point and other points, we can obtainBy calculations, one arrives atOne can easily check that and . Thus, we haveImmediately,In what follows, we will devote to calculate the values of and . Then,Namely,In the same way, one can obtainThe further form of equation (49) is as follows:

Remark 2. Combining equations (30), (41), and (48), we find thatfor any .
For and , one hasBy solving equations (51) and (52), one arrives atIt turns out that the formulas for , , , , , and hold for any . Besides, we haveBy using equations (24) and (25), one obtains that the effective resistances between new inserted points and others points can be given by the effective resistances between two initial points which are inherited from thelinear polyomino network.

Theorem 1. For linear octagonal network , we have the following:(i)Effective resistances between two initial points are(ii)Effective resistances between new inserted points and others are