# Complete Characterization of Resistance Distance for Linear Octagonal Networks

Authors: Jing Zhao; Jia-Bao Liu; Ali Zafari
Journal: Complexity
 (2020)

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## 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.


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## Body

## 1. Introduction

Our convention in this paper follows [1] for notations that we omit here. The resistance distance Ωij between vertices i and j 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 Ωe as the resistance distance of the edge e, then the conductance of the edge of e is ce=1/Ωe. 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 [7–15].For pathPk and its copy Pk′, let VPk=x1,x2,…,xk and VPk′=x1′,x2′,…,xk′. If k=Th+1 (T,h are all positive integers), we connect xTi+1 to xTi+1′ for i=0,1,…,h and denote this graph by Lk. Obviously, if T=1, Lk is the linear polymino network; if T=2, Lk is the linear hexagonal network; and if T=3, Lk 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.Figure 1
Linear polyomino networkQn and linear octagonal network Ln.The Kirchhoff index and multiplicative degree-Kirchhoff index are, respectively, defined asKfG=∑i<jΩij [2] and Kf∗G=∑i<jdidjΩij [18], where di is the degree of vertex i, which is equal to the number of edges connected to vertex i. These two indices are found to be well associated with the Laplacian eigenvalues λi and normalized Laplacian eigenvalues υi. Putting them in other ways, KfG=n⋅∑i=2n1/λi [2] and Kf∗G=2m⋅∑i=2n1/υi [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, 21–23].In this paper, some preliminaries are proposed in Section2. 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 thatN=G,C is a electrical network. For any two points i and j,i≠j. Then,(1)cirij+∑k∈Nicikrik−rjk=2,ci=∑k∈Nicik,where Ni is the neighbor set of point i.For the sake of convenience, we here define three functions:(2)α=4−15,Hx,y,n=31−αx−y2−αx+y−1+α2y−1+α2n−2x+11−αx−y−2αx+y−14151−α2n,Gx,y,n=31+αx−y2+αx+y−1+α2y−1+α2n−2x+11+αx−y+2αx+y−14151−α2n.According to the structures of the linear octagonal network, we can divide its effective resistances between any points into two cases. For anyi,j∈1,2,...,n, one has the following cases:Case 1. Effective resistances between two initial points are(3)rLnpi,pj,rLnqi,pj.Case 2. Effective resistances between new inserted points and other points are(4)rLnsi,sj,rLnsi,tj,rLnsi,pj,rLnsi,qj,rLnui,uj,rLnui,vj,rLnsi,uj,rLnui,pj,rLnui,qj,rLnsi,vj.


## 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 networkQn∗ from the linear octagonal network Ln by using circuit reductions (see Figure 2).Figure 2
Linear polyomino networkQn∗.Assume thatxn=rQn∗pn,p1,yn=rQn∗pn,q1,andzn=rQn∗pn,qn. One obtains the network Qn−1∗ by making a circuit reduction for the linear polyomino network Qn∗ with respect to points pn−1 and qn−1 (see Figure 3).Figure 3
The networkQn−1∗.According to the networkQn−1∗, one obtains(5)1zn=1+16+zn−1=7+zn−16+zn−1,z1=1.By computing equation (5), we obtain(6)zn=−3+15−4−15n+3+15−4+15n−4−15n−−4+15n=−3−15+2154+154+15n−4−15n.In what follows, we first calculaterQn∗pn,p1andrQn∗pn,q1. One obtains the network Qn+1∗ from the network Qn∗ through a circuit reduction or star-triangle transformation about points pn,qn, and p1 (see Figure 4).Figure 4
The networkQn+1∗.On the one hand, one hasK+N=xn,M+N=yn,andK+M=zn based on the network Qn+1∗, namely,(7)K=xn−yn+zn2,M=−xn+yn+zn2,N=xn+yn−zn2.On the other hand, we have(8)xn+1=3+K4+M7+zn+N,yn+1=3+M4+K7+zn+N.Combining equations (7) and (8) yields(9)xn+1=6+xn−yn+zn8−xn+yn+zn47+zn+xn+yn−zn2,n≥1,(10)yn+1=6−xn+yn+zn8+xn−yn+zn47+zn+xn+yn−zn2,n≥1,where x1=0andy1=1.Subtracting equation (10) from equation (9) leads to(11)xn+1−yn+1=−∏k=1n1zk+7.According to equations (6) and (11), one arrives at(12)xn−yn=2154−15n−4+15n.Substituting equations (6) and (12) into equation (10), one obtains(13)yn+1=yn+151−4−15n+1−151−4−15n+32,y1=1.Hence,(14)yn=151−4−15n+3n−2−152,n≥1.By equations (12) and (14), we obtain(15)xn=151+4−15n+3n−2−152,n≥1.Moreover,(16)xn−yn+zn=−3−15+151+4−15n,−xn+yn+zn=−3−15+151−4−15n,xn+yn−zn=3n−1.Next, we will computerQn∗pn,pi,rQn∗pn,qi, and rQn∗pi,qi, n>i>1. We make a circuit reduction of the network Qn∗ with respect to points pn,pi+1, and qi+1 (see Figure 5).Figure 5
The networkQn∗ by a circuit reduction about points pn,pi+1, and qi+1.It is easy to check that the upper and lower parts of the networkQn∗ are, respectively, the networks Qn−i∗ and Qi∗ in line with Figure 5. Also, one has A+B=xn−i,A+C=yn−i,andB+C=zn−i.According to equation (16), we arrive at(17)A=xn−i+yn−i−zn−i2=3n−i−12,B=xn−i−yn−i+zn−i2=−3−152+152+24−15n−i,C=−xn−i+yn−i+zn−i2=−3−152+152−24−15n−i.By equation (17), we have(18)rQn∗pn,pi=A+3+BC+3+zi6+zn−i+zi,rQn∗pn,qi=A+3+CB+3+zi6+zn−i+zi,rQn∗pi,qi=zizn−i+66+zn−i+zi.Substituting equations (6) and (17) into (18) yields(19)rQn∗pn,pi=3n−i2+31−αn−i2−2αn+i−αn+i−1−αn−i+1+α2i−1+α4151−α2n,rQn∗pn,qi=3n−i2+31+αn−i2+2αn+i+αn+i−1+αn−i+1+α2i−1+α4151−α2n,rQn∗pi,qi=1+α2i−1+α2n−2i+1+α2n151−α2n.It is not hard to verify that equation (19) holds for n≥i≥1.At this place, we are going to give the explicit formulas forrLnpi,pj and rLnpi,qj. One makes a circuit reduction for the network Qn∗ with respect to points pi,qi, and pj, as illustrated in Figure 6.Figure 6
The networkQn∗ by a circuit reduction about points pi,qi, and pj.Based on Figure6, one has(20)rQn∗pi,pj=D+E,rQn∗qi,pj=E+F,rQn∗pi,qi=D+F=zi.Thus,(21)D=rQn∗pi,pj+zi−rQn∗qi,pj2,E=rQn∗pi,pj−zi+rQn∗qi,pj2,F=−rQn∗pi,pj+zi+rQn∗qi,pj2.Computing equation (21), we have(22)rQn∗pi,pj=DF+zn−i+6zi+zn−i+6+E,rQn∗qi,pj=FD+zn−i+6zi+zn−i+6+E.Combining equations (6), (19), (21), and (22), one obtains(23)rQn∗pi,pj=3i−j2+Hi,j,n,rQn∗qi,pj=3i−j2+Gi,j,n.One can straightforwardly verify that the formulas ofrLnpi,pj and rLnqi,pj also hold for n≥i≥j≥1. Moreover, we have(24)rLnpi,pj=rQn∗pi,pj=3i−j2+Hi,j,n,(25)rLnqi,pj=rQn∗qi,pj=3i−j2+Gi,j,n.


## 4. Effective Resistances between New Inserted Points and Others

In Section3, 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 Wn∗ by making a circuit reduction for the linear octagonal network Ln with respect to point sets v1,…,vn−1 and u1,…,un−1 (see Figure 7).Figure 7
The networkWn∗.In what follows, we will concentrate on computingrWn∗si,pi and rWn∗si,pi+1. According to Lemma 1 and Figure 7, we obtain(26)32rWn∗si,pi+rWn∗si,pi−rWn∗pi,pi+12rWn∗si,pi+1−12rWn∗pi,pi+1=2,(27)32rWn∗si,pi+1+rWn∗si,pi−rWn∗pi+1,pi+12rWn∗si,pi+1−12rWn∗pi+1,pi+1=2.On the one hand, subtracting equation (27) from the equation which is obtained using equation (26) times 4 leads to(28)rWn∗si,pi=196+rWn∗pi,pi+1.Moreover,rWn∗pi,pi+1=rWn∗pi+1,pi. Hence,(29)rWn∗pi+1,pi=32+Hi+1,i,n.Substituting equation (29) into equation (28), one obtains(30)rWn∗si,pi=56+19Hi+1,i,n.On the other hand, subtracting equation (26) from the equation which is obtained using equation (27) times 5/2 yields(31)rWn∗si,pi+1=293+2rWn∗pi+1,pi.Substituting equation (29) into equation (31), one has(32)rWn∗si,pi+1=43+49Hi+1,i,n.For the resistance distance between the pointsi and other points, we can obtain(33)32rWn∗si,pj+rWn∗si,pi−rWn∗pj,pi+12rWn∗si,pi+1−12rWn∗pj,pi+1=2,32rWn∗si,qj+rWn∗si,pi−rWn∗qj,pi+12rWn∗si,pi+1−12rWn∗qj,pi+1=2,32rWn∗si,sj+rWn∗si,pi−rWn∗sj,pi+12rWn∗si,pi+1−12rWn∗sj,pi+1=2,32rWn∗si,tj+rWn∗si,pi−rWn∗tj,pi+12rWn∗si,pi+1−12rWn∗tj,pi+1=2.By calculations, one arrives at(34)rWn∗si,pj=56+32i−j−29Hi+1,i,n+23Hi,j,n+13Hi+1,j,n,rWn∗si,qj=56+32i−j−29Hi+1,i,n+23Gi,j,n+13Gi+1,j,n,rWn∗si,sj=23+32i−j−29Hi+1,i,n−29Hj+1,j,n+49Hi,j,n+29Hi,j+1,n+29Hi+1,j,n+19Hi+1,j+1,n,rWn∗si,tj=23+32i−j−29Hi+1,i,n−29Hj+1,j,n+49Hi,j,n+29Gi,j+1,n+29Gi+1,j,n+19Gi+1,j+1,n.One can easily check thatrWn∗si,pj≠rWn∗sj,pi and rWn∗si,qj≠rWn∗sj,qi. Thus, we have(35)32rWn∗sj,pi+rWn∗sj,pj−rWn∗pi,pj+12rWn∗sj,pj+1−12rWn∗pi,pj+1=2,32rWn∗sj,qi+rWn∗sj,pj−rWn∗qi,pj+12rWn∗sj,pj+1−12rWn∗qi,pj+1=2.Putting them in another way,(36)rWn∗sj,pi=−16+32i−j−29Hj+1,j,n+23Hi,j,n+13Hi,j+1,n,rWn∗sj,qi=−16+32i−j−29Hj+1,j,n+23Gi,j,n+13Gi,j+1,n.It turns out that the formulas forrWn∗si,pj, rWn∗sj,pi, rWn∗si,qj, rWn∗sj,qi, rWn∗si,sj, and rWn∗si,tj hold for any n≥i≥j≥1. Besides, we have(37)rLnsj,si=rLnsi,sj=rWn∗sj,si=rWn∗si,sj,rLnsi,tj=rLnsj,ti=rWn∗si,tj=rWn∗sj,ti,rLnsi,pj=rWn∗si,pj,rLnsj,pi=rWn∗sj,pi,rLnsi,qj=rWn∗si,qj,rLnsj,qi=rWn∗sj,qi.One obtains the networkWn# by making a circuit reduction for the linear octagonal network Ln with respect to point sets s1,…,sn−1 and t1,…,tn−1 (see Figure 8).Figure 8
The networkWn#.In what follows, we will concentrate on computingrWn#ui,pi and rWn#ui,pi+1. According to Lemma 1 and Figure 8, we obtain(38)32rWn#ui,pi+12rWn#ui,pi−12rWn#pi,pi+rWn#ui,pi+1−rWn#pi,pi+1=2,(39)32rWn#ui,pi+1+12rWn#ui,pi−12rWn#pi+1,pi+rWn#ui,pi+1−rWn#pi+1,pi+1=2.On the one hand, subtracting equation (39) from the equation which is obtained using equation (38) times 5/2 leads to(40)rWn#ui,pi=43+49Hi+1,i,n.On the other hand, subtracting equation (38) from the equation which is obtained using equation (39) times 4 yields(41)rWn#ui,pi+1=56+19Hi+1,i,n.Remark 1.
Combining equations (32) and (40), we find that(42)rLnpi,ui=rLnsi,pi+1,for any i<n.
For the resistance distance between the pointui and other points, we can obtain(43)32rWn#ui,pj+12rWn#ui,pi−12rWn#pj,pi+rWn#ui,pi+1−rWn#pj,pi+1=2,32rWn#ui,qj+12rWn#ui,pi−12rWn#qj,pi+rWn#ui,pi+1−rWn#qj,pi+1=2,32rWn#ui,uj+12rWn#ui,pi−12rWn#uj,pi+rWn#ui,pi+1−rWn#uj,pi+1=2,32rWn#ui,vj+12rWn#ui,pi−12rWn#vj,pi+rWn#ui,pi+1−rWn#vj,pi+1=2.
By calculations, one arrives at(44)rWn#ui,pj=43+32i−j−29Hi+1,i,n+13Hi,j,n+23Hi+1,j,n,rWn#ui,qj=43+32i−j−29Hi+1,i,n+13Gi,j,n+23Gi+1,j,n,rWn#ui,uj=23+32i−j−29Hi+1,i,n−29Hj+1,j,n+19Hi,j,n+29Hi,j+1,n+29Hi+1,j,n+49Hi+1,j+1,n,rWn#ui,vj=23+32i−j−29Hi+1,i,n−29Hj+1,j,n+19Hi,j,n+29Gi,j+1,n+29Gi+1,j,n+49Gi+1,j+1,n.
One can easily check thatrWn#ui,pj≠rWn∗uj,pi and rWn#ui,qj≠rWn∗uj,qi. Thus, we have(45)32rWn#uj,pi+12rWn#uj,pj−12rWn#pi,pj+rWn#uj,pj+1−rWn#pi,pj+1=2,32rWn#uj,qi+12rWn#uj,pj−12rWn#qi,pj+rWn#uj,pj+1−rWn#qi,pj+1=2.
Immediately,(46)rWn#uj,pi=−23+32i−j−29Hj+1,j,n+13Hi,j,n+23Hi,j+1,n,rWn#uj,qi=−23+32i−j−29Hj+1,j,n+13Gi,j,n+23Gi,j+1,n.
In what follows, we will devote to calculate the values ofrLnsi,uj and rLnsj,ui. Then,(47)2rLnsi,uj+rLnsi,pi−rLnuj,pi+rLnsi,ui−rLnuj,ui=2,2rLnsi,ui+rLnsi,pi−rLnui,pi+rLnsi,ui−rLnui,ui=2.
Namely,(48)rLnsi,ui=56+19Hi+1,i,n,rLnsi,uj=16+32i−j−29Hi+1,i,n−29Hj+1,j,n+29Hi,j,n+49Hi,j+1,n+19Hi+1,j,n+29Hi+1,j+1,n.
In the same way, one can obtain(49)2rLnsj,ui+rLnsj,pj−rLnui,pj+rLnsj,uj−rLnui,uj=2.The further form of equation (49) is as follows:(50)rLnsj,ui=76+32i−j−29Hj+1,j,n−29Hi+1,i,n+29Hi,j,n+49Hi+1,j,n+19Hi,j+1,n+29Hi+1,j+1,n.Remark 2.
Combining equations (30), (41), and (48), we find that(51)rLnpi,si=rLnsi,ui=rLnui,pi+1,for any i<n.
ForrLnsi,vj and rLnsj,vi, one has(52)2rLnsi,vj+rLnsi,pi−rLnvj,pi+rLnsi,ui−rLnvj,ui=2,2rLnsj,vi+rLnsj,pj−rLnvi,pj+rLnsj,uj−rLnvi,uj=2.
By solving equations (51) and (52), one arrives at(53)rLnsi,vj=16+32i−j−29Hj+1,j,n−29Hi+1,i,n+118Hi,j,n+16Gi,j,n+49Gi,j+1,n+19Gi+1,j,n+29Gi+1,j+1,n,rLnsi,vj=76+32i−j−29Hj+1,j,n−29Hi+1,i,n+118Hi,j,n+16Gi,j,n+49Gi+1,j,n+19Gi,j+1,n+29Gi+1,j+1,n.
It turns out that the formulas forrWn#ui,pj, rWn#uj,pi, rWn#ui,qj, rWn#uj,qi, rWn#ui,uj, and rWn#si,vj hold for any n≥i≥j≥1. Besides, we have(54)rLnuj,ui=rLnui,uj=rWn#uj,ui=rWn#ui,uj,rLnui,vj=rLnuj,vi=rWn#ui,vj=rWn#uj,vi,rLnui,pj=rWn#ui,pj,rLnui,qj=rWn#ui,qj,rLnuj,pi=rWn#uj,pi,rLnuj,qi=rWn#uj,qi.
By 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 networkn≥i≥j≥1, we have the following:(i)
Effective resistances between two initial points are(55)rLnpi,pj=rQn∗pi,pj=3i−j2+Hi,j,n,rLnqi,pj=rQn∗qi,pj=3i−j2+Gi,j,n.(ii)
Effective resistances between new inserted points and others are(56)rLnsi,sj=43−29rLnpi,pi+1−29rLnpj,pj+1+49rLnpi,pj+29rLnpi,pj+1+29rLnpi+1,pj+19rLnpi+1,pj+1,rLnsi,tj=43−29rLnpi,pi+1−29rLnpj,pj+1+49rLnpi,pj+29rLnpi,qj+1+29rLnpi+1,qj+19rLnpi+1,qj+1,rLnsi,pj=23−29rLnpi,pi+1+23rLnpi,pj+13rLnpi+1,pj,rLnsj,pi=23−29rLnpj,pj+1+23rLnpi,pj+13rLnpi,pj+1,rLnsi,qj=23−29rLnpi,pi+1+23rLnpi,qj+13rLnpi+1,qj,rLnsj,qi=23−29rLnpj,pj+1+23rLnpi,qj+13rLnpi,qj+1,rLnui,uj=43−29rLnpi,pi+1−29rLnpj,pj+1+19rLnpi,pj+29rLnpi,pj+1+29rLnpi+1,pj+49rLnpi+1,pj+1,rLnui,vj=43−29rLnpi,pi+1−29rLnpj,pj+1+19rLnpi,qj+29rLnpi,qj+1+29rLnpi+1,qj+49rLnpi+1,qj+1,rLnsi,uj=43−29rLnpi,pi+1−29rLnpj,pj+1+29rLnpi,pj+49rLnpi,pj+1+19rLnpi+1,pj+29rLnpi+1,pj+1,rLnsj,ui=43−29rLnpi,pi+1−29rLnpj,pj+1+29rLnpi,pj+49rLnpi+1,pj+19rLnpi,pj+1+29rLnpi+1,pj+1,rLnui,pj=43−29rLnpi,pi+1+13rLnpi,pj+23rLnpi,pj+1,rLnuj,pi=43−29rLnpj,pj+1+13rLnpi,pj+23rLnpi+1,pj,rLnui,qj=43−29rLnpi,pi+1+13rLnpi,qj+23rLnpi+1,qj,rLnuj,qi=43−29rLnpj,pj+1+13rLnpi,qj+23rLnpi,pj+1,rLnsi,vj=43−29rLnpi,pi+1−29rLnpj,pj+1+118rLnpi,pj+16rLnpi,qj+49rLnpi,qj+1+19rLnpi+1,qj+29rLnpi+1,qj+1,rLnsj,vi=43−29rLnpi,pi+1−29rLnpj,pj+1+118rLnpi,pj+16rLnpi,qj+49rLnpi+1,qj+19rLnpi,qj+1+29rLnpi+1,qj+1.


## 5. Conclusion

In this paper, we have determined the effective resistances between any points of a linear octagonal network. Our methods used in this paper are standard. On the one hand, one can use circuit reductions and effective resistance sum rules to study other linear networks, such as linear hexagonal networks, octagonal-quadrilateral networks, and generalized plenylenes networks. On the other hand, the resistance distance-based parameters, namely, Kirchhoff index, multiplicative degree-Kirchhoff index, and additive degree-Kirchhoff index, are well defined; we may compute those resistance distance-based parameters in terms of the results obtained in this paper.



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