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.
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 Figure 1. This kind of system presented in [1] is a consistent time network in the form of an rectangular matrix array having rows and columns (see Figures 1ā3 for some values of and ).
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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).
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A cell , where and with its th neighborhood, can be presented as and is described as the set of cells , where and , such that and . The cells in lth neighborhood of a cell are directly interconnected with cell through , , , and , where and are known as the feedback weights and and known as the feedforward weights. The index pair describes the direction of signal from to . The cell is connected directly with its adjacent cells . Since every has its adjacent cells, the cell can also be linked with all other cells indirectly as shown in Figure 5.
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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 [4ā13].
Thoroughly, we take the graph which does not contain loops and multiple or directed edges, where the sets and are of vertices and edges, respectively. The length of the shortest path from to (denoted by ) is called its distance and is known by the degree of , where . 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 [14ā16]). In particular, the degree-based topological properties for the CNN are studied in [17].
The first Zagreb index is studied for the total -electron energy [18], and the second Zagreb index appeared to compute molecular branching [19]; they are denoted by and , respectively:
In relation to the above equations, the first and second Zagreb connection indices (ZCIs) have been put forward in [20, 21] independently:where denotes the number of vertices such that . It has been proved by Ali and Trinajstic [20] that the topological index can be written as
In [17], the authors checked the chemical applicability of these three Zagreb connection indices on the set of octane isomers, and they found that 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 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 , 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 -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
Let denote the number of vertices in with connection number and denote the number of edges in whose vertices have connection numbers and .
The following formulas for the ZCIs are equivalent to the previous definitions:
From Figure 1 and definition of the ZCIs, we have the following:(1)For and ,(a)(b)(c)(2)For and ,(a)(b)(c)(3)For and ,(a)(b)(c)
Theorem 1. Let and be the CNN. Then,(1)(2)(3)
Proof. In order to prove our result, we will compute , the number of vertices of connection number , and is the edge of whose vertices have connection numbers and . It is easy to see from the structure of that , and . Thus, from equation (4), we have the following:The edge set of can be partitioned into different classes depending upon the edge types of as listed in Table 1.
From the definition of the second ZCI and substitution of from Table 1 in (5), it follows thatSimilarly, from substitution of from Table 1 in (6), we have
Theorem 2. Let and be the CNN. Then,(1)(2)(3)
Proof. It is easy to see from the structure of that , and . Thus, from equation (4), we have the following:
The edge set of can be partitioned into different classes depending upon the edge types of as listed in Table 2.
From the definition of the second ZCI and substitution of from Table 2 in (5), it follows that
Similarly, from substitution of from Table 2 in (6). we have(1)(2)(3)
Theorem 3. Let and be the CNN. Then,
Proof. In order to prove our result, we will compute , the number of vertices of connection number , and is the edge of whose vertices have connection numbers and . It is easy to see from the structure of that , and . Thus, from equation (4), we have the following:The edge set of can be partitioned into different classes depending upon the edge types of as listed in Table 3.
From the definition of the second ZCI and substitution of from Table 3 in (5), it follows thatSimilarly, from substitution of from Table 3 in (6), we have
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 (Figures 5 and 6). The numerical comparison is given in Tables 4ā6. We can see from the 3D plots and numerical tables that the second Zagreb index is always greater than the other two indices.
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.
Data Availability
All the data are included within this paper. However, the reader may contact the corresponding author for more details of the data.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
Zahid Raza has been funded during this work by the University of Sharjah under Project #1802144068 and MASEP Research Group.