# Zagreb Connection Numbers for Cellular Neural Networks

Authors: Jia-Bao Liu; Zahid Raza; Muhammad Javaid
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Neural networks in which communication works only among the neighboring units are called cellular neural networks (CNNs). These are used in analyzing 3D surfaces, image processing, modeling biological vision, and reducing nonvisual problems of geometric maps and sensory-motor organs. Topological indices (TIs) are mathematical models of the (molecular) networks or structures which are presented in the form of numerical values, constitutional formulas, or numerical functions. These models predict the various chemical or structural properties of the under-study networks. We now consider analogous graph invariants, based on the second connection number of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for the cellular neural networks (CNNs). In order to find their efficiency, a comparison among the obtained indices of CNN is also performed in the form of numerical tables and 3D plots.


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## Body

## 1. Introduction

A neural system that consists of a multidimensional cluster of neurons and neighborhood-connected associations between the cells is called a cellular neural network (CNN) as shown in Figure1. This kind of system presented in [1] is a consistent time network in the form of an n×m rectangular matrix array having n rows and m columns (see Figures 1–3 for some values of m and n).Figure 1
Cellular neural networkCNN3,3 (a) and CNN4,3 (b).
(a)(b)Figure 2
Cellular neural networkCNNm,3 (a) and CNNm,4 (b).
(a)(b)Figure 3
Cellular neural networkCNNm,n.A component of the rectangular array corresponds to a cell in a neural arrangement. But it is noted that the geometry exhibited requires not only to be rectangular, but also such shapes can be triangles or hexagons [2]. Multiple clusters can be represented with a proper interconnected structure to construct a multilayered cell neural system (Figure4).Figure 4
Cellular neural network.
(a)(b)A cellCh;k, where 1≤h≤n and 1≤j≤m with its lth neighborhood, can be presented as Nrh;k and is described as the set of cells Cp;q, where 1≤p≤n and 1≤q≤m, such that p−h≤l and q−k≤l. The cells in lth neighborhood of a cell Ch,k are directly interconnected with cell Ch,k through Ap,q,h,k, Ah,k,p,q, Bp,q,h,k, and Bh,k,p,q, where Ap,q,h,k and Ah,k,p,q are known as the feedback weights and Bp,q,h,k and Bh,k,p,q known as the feedforward weights. The index pair p,q,h,k describes the direction of signal from Ch,k to Cp,q. The cell Ch,k is connected directly with its adjacent cells Cp,q∈Nrh,k. Since every Cp,q has its adjacent cells, the cell Ch,k can also be linked with all other cells indirectly as shown in Figure 5.Figure 5
Differences of indices forCNNm,3 (a) and for CNNm,4 (b).
(a)(b)The CNN has a lot of applications that are indicated by their spatial dynamics. The filtering image processing is one of the good applications of CNN [3]. For more related works about CNN and PNN, one can consult the references [4–13].Thoroughly, we take the graphG=V,E which does not contain loops and multiple or directed edges, where the sets V and E⊆V×V are of vertices and edges, respectively. The length of the shortest path from u to v (denoted by du,v) is called its distance and du=N is known by the degree of u, where N=v∈V:dv,u=1. A topological index (TI) defined with the help of the degrees of nodes of the (molecular) network is a class of indices which are used to find out and model the certain properties of the chemical compounds of the (molecular) networks (see [14–16]). In particular, the degree-based topological properties for the CNN are studied in [17].The first Zagreb index is studied for the totalπ-electron energy [18], and the second Zagreb index appeared to compute molecular branching [19]; they are denoted by M1 and M2, respectively:(1)M1G=∑u∈VGdu2,M2G=∑u∈EGdudv.In relation to the above equations, the first and second Zagreb connection indices (ZCIs) have been put forward in [20, 21] independently:(2)ZC1G=∑u∈VGτu2,ZC2G=∑uv∈EGτuτv,ZC1∗G=∑u∈VGduτu,where τu denotes the number of vertices v∈G such that du,v=2. It has been proved by Ali and Trinajstic [20] that the topological index ZC1∗ can be written as(3)ZC1∗G=∑uv∈EGτu+τv.In [17], the authors checked the chemical applicability of these three Zagreb connection indices on the set of octane isomers, and they found that ZC1∗G has better correlating ability than the other two Zagreb connection indices in the cases of entropy, enthalpy of vaporization, standard enthalpy of vaporization, and acentric factor. Basavanagoud and Jakkannavar checked the chemical applicability of ZC1 and found that the index has a very good correlation with physical properties of chemical compounds such as boiling point, entropy, enthalpy of evaporation, standard enthalpy of vaporization, and acentric factor (see [23]).Ali and Trinajstic [20] checked the chemical applicability of ZC1∗, and they found that this TI correlates well with the entropy and acentric factor of octane isomers. A large number of networks has been studied with the help of connection number-based TIs such as T-sum networks [24], resultant networks [25, 26], connected networks [27, 28], alkanes [22, 29, 30], dendrimer nanostars [31], trees, and unicyclic networks [32] and subdivided and semitotal point networks [33].


## 2. Main Results and Discussion

LetckG denote the number of vertices in G with connection number k and mk,lG denote the number of edges in G whose vertices have connection numbers k and l.The following formulas for the ZCIs are equivalent to the previous definitions:(4)ZC1G=∑0≤k≤n−2ckGk2,(5)ZC2G=∑0≤l≤k≤n−2mk,lGk.l,(6)ZC1∗G=∑0≤l≤k≤n−2mk,lGk+l.From Figure1 and definition of the ZCIs, we have the following:(1)
Form=3 and n=3,(a)
ZC1CNN3,3=100(b)
ZC2CNN3,3=156(c)
ZC1∗CNN3,3=88(2)
Form=4 and n=3,(a)
ZC1CNN4,3=280(b)
ZC2CNN4,3=555(c)
ZC1∗CNN4,3=258(3)
Form=4 and n=4,(a)
ZC1CNN4,4=584(b)
ZC2CNN4,4=1634(c)
ZC1∗CNN4,4=524Theorem 1.
Letm≥5 and CNNm,3 be the CNN. Then,(1)
ZC1CNNm,3=198m−512(2)
ZC2CNNm,3=360m−840(3)
ZC1∗CNNm,3=102m−144Proof.
In order to prove our result, we will computeck, the number of vertices of connection number k, and yk,lG is the edge of CNNm,3 whose vertices have connection numbers k and l. It is easy to see from the structure of CNNm,n that c3=4,c5=4,c6=m, and c9=2m−8. Thus, from equation (4), we have the following:(7)ZC1CNNm,3=∑0≤k≤n−2ckGk2=c332+c552+c662+c992=36+100+36m+812m−8=198m−512.
The edge set ofCNNm,3 can be partitioned into different classes depending upon the edge types of yk,lG as listed in Table 1.
From the definition of the second ZCI and substitution ofyk,l from Table 1 in (5), it follows that(8)ZC2CNNm,3=y3,53×5+y3,63×6+y3,93×9+y5,65×6+y6,66×6+y6,96×9=815+818+427+430+m−136+6m−2454=360m−840.
Similarly, from substitution ofyk,lG from Table 1 in (6), we have(9)ZC1∗CNNm,3=y3,53+5+y3,63+6+y3,93+9+y5,65+6+y6,66+6+y6,96+9=88+89+412+411+m−112+6m−2415=102m−144.Table 1
Partition of edge set ofCNNm,3.
Edges of typeyk,lNumber of edgesy3,58y3,68y3,94y5,64y6,6m−1y6,96m−24Theorem 2.
Letm≥5 and CNNm,4 be the CNN. Then,(1)
ZC1CNNm,4=404m−1032(2)
ZC2CNNm,4=1361m−3940(3)
ZC1∗CNNm,4=266m−540Proof.
It is easy to see from the structure ofCNNm,4 that c5=4,c6=8,c7=4,c9=2m−8, and c11=2m−8. Thus, from equation (4), we have the following:(10)ZC1CNNm,4=∑0≤k≤n−2ckGk2=c552+c662+c772+c992+c11112=425+836+449+2m−881+2m−8121=404m−1032.The edge set ofCNNm,4 can be partitioned into different classes depending upon the edge types of yk,lG as listed in Table 2.From the definition of the second ZCI and substitution ofyk,l from Table 2 in (5), it follows that(11)ZC2CNNm,4=y5,65×6+y5,75×7+y6,66×6+y6,76×7+y6,96×9+y6,116×11+y7,77×7+y7,97×9+y7,117×11+y9,99×9+y9,119×11+y11,1111×11=830+435+636+1242+454+466+249+463+877+2m−1081+6m−2899+5m−24121=1361m−3940.Similarly, from substitution ofyk,lG from Table 2 in (6). we have(12)ZC1∗CNNm,4=y5,65+6+y5,75+7+y6,66+6+y6,76+7+y6,96+9+y6,116+11+y7,77+7+y7,97+9+y7,117+11+y9,99+9+y9,119+11+y11,1111+11=811+412+612+1213+415+417+214+416+818+2m−1018+6m−2820+5m−2422=266m−540.(1)
ZC1CNNm,n=256mn−700m−700n+1448(2)
ZC2CNNm,n=1024mn−2808m−2808n+7474(3)
ZC1∗CNNm,n=128mn−246m−246n−100Theorem 3.
Letm,n≥5 and CNNm,n be the CNN. Then,Table 2
Partition of edge set ofCNNm,4.
Edges of typeyk,lNumber of edgesy5,68y5,74y6,66y6,712y6,94y6,114y7,72y7,94y7,118y9,92m−10y9,116m−28y11,115m−24Proof.
In order to prove our result, we will computeck, the number of vertices of connection number k, and yk,lG is the edge of CNNm,n whose vertices have connection numbers k and l. It is easy to see from the structure of CNNm,n that c5=4,c6=8,c7=4,c9=2m+2n−16,c11=2m+2n−16, and c16=m−4n−4. Thus, from equation (4), we have the following:(13)ZC1CNNm,n=c552+c662+c772+c992+c11112+c16162=425+836+449+2m+2n−1681+m−4n−4256+2m+2n−16121=256mn−700m−700n+1448.
The edge set ofCNNm,n can be partitioned into different classes depending upon the edge types of yk,lG as listed in Table 3.
From the definition of the second ZCI and substitution ofyk,l from Table 3 in (5), it follows that(14)Z2CNNm,n=y5,65×6+y5,75×7+y6,66×6+y6,76×7+y6,96×9+y6,116×11+y7,97×9+y7,117×11+y7,167×16+y9,99×9+y9,119×11+y11,1111×11y11,1611×16+y16,1616×16=830+435+436+842+854+866+863+877+4112+2m+2n−2081+996m+6n−56+1212m+2n−16+6m+6n−56176+4mn−19m−19n+90256=1024mn−2808m−2808n+7474.
Similarly, from substitution ofyk,lG from Table 3 in (6), we have(15)Z1∗CNNm,n=y5,65+6+y5,75+7+y6,66+6+y6,76+7+y6,96+9+y6,116+11+y7,97+9+y7,117+11+y7,167+16+y9,99+9+y9,119+11+y11,1111+11+y11,1611+16+y16,1616+16=811+412+412+813+815+817+816+818+423+2m+2n−2018+206m+6n−56+222m+2n−16+276m+6n−56+324mn−19m−19n+90=128mn−246m−246n−100.Table 3
Partition of edge set ofCNNm,n.
Edges of typeyk,lNumber of edgesy5,68y5,74y6,64y6,78y6,98y6,118y7,98y7,118y7,164y9,92m+2n−20y9,116m+6n−56y11,112m+2n−16y11,166m+6n−56y16,164mn−19m−19n+90


## 3. Numerical and Graphical Comparisons

In this section, we will give numerical and graphical comparisons of the Zagreb connection indices with respect to the cellular neural network. Maple software is used to construct a simple comparison of the Zagreb connection indices related to the cellular neural network into 3D plots (Figures5 and 6). The numerical comparison is given in Tables 4–6. We can see from the 3D plots and numerical tables that the second Zagreb index is always greater than the other two indices.Figure 6
Differences of indices forCNNm,n.Table 4
Differences of indices forCNNm,3.
mZC1ZC2ZC1∗547896036666761320468787416805708107220406729127024007741014682760876111666312097812186434801080132062384011821422604200128415245845601386Table 5
Differences of indices forCNNm,4.
mZC1ZC2ZC1∗59882865790613924226105671796558713228220069481588926048309185410300896702120113412110312386123816123922652134220137532918144624151143184155028164753450Table 6
Differences of indices forCNNm,n.
m,nZC1ZC2ZC1∗(5, 5)8484994640(6, 5)142873061034(7, 5)200896181428(8, 5)2588119301822(9, 5)3168142422216(10, 5)3748165542610(6, 6)2264106421556(7, 6)3100139782078(8, 6)3936173142600(9, 6)4772206503122(10, 6)5608239863644


## 4. Conclusion

The Zagreb connection indices for the cellular neural system on a rectangular grid have been computed. Later on, the obtained results for the Zagreb connection indices, has an application; with the help of numerical tables and 3D plots, the determination of detailed comparisons among these indices of CNN has been outlined. It is notable that the obtained results for these networks are all quadratic in terms of the order of the network, which showed that one can build efficient graph algorithms to compute the indices within polynomial time.



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