Journal of Chemistry
Volume 2020 (2020), Article ID 3794592, 7 pages
https://doi.org/10.1155/2020/3794592

Research Article

Computing Eccentricity-Based Topological Indices of 2-Power Interconnection Networks

Muhammad Imran,¹ Muhammad Azhar Iqbal,² Yun Liu,³ Abdul Qudair Baig,⁴ Waqas Khalid,⁵ and Muhammad Asad Zaighum²

¹Department of Mathematical Sciences, College of Science, United Arab Emirates University, P. O. Box 15551, Al Ain, UAE
²Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan
³School of Electronic Engineering, Huainan Normal University, Huainan 232038, China
⁴Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan
⁵Punjab College of Science and Commerce, Attock Campus, Pakistan

Correspondence should be addressed to Muhammad Imran; imrandhab@gmail.com

Received 11 October 2019; Accepted 11 May 2020; Published 13 June 2020

Academic Editor: Juan L. G. Guirao

Copyright © 2020 Muhammad Imran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In a connected graph with a vertex , the eccentricity of is the distance between and a vertex farthest from in the graph . Among eccentricity-based topological indices, the eccentric connectivity index, the total eccentricity index, and the Zagreb index are of vital importance. The eccentric connectivity index of is defined by  = , where is the degree of the vertex and is the eccentricity of in . The topological structure of an interconnected network can be modeled by using graph explanation as a tool. This fact has been universally accepted and used by computer scientists and engineers. More than that, practically, it has been shown that graph theory is a very powerful tool for designing and analyzing the topological structure of interconnection networks. The topological properties of the interconnection network have been computed by Hayat and Imran (2014), Haynes et al. (2002), and Imran et al. (2015). In this paper, we compute the close results for eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index of a hypertree, sibling tree, and -tree for -level by using the edge partition method.

1. Introduction

The advancement of large-scale integrated circuit technology has enabled the construction of complex interconnection networks. Graph theory provides a fundamental tool for designing and analyzing such networks. Graph theory and interconnection networks provide a thorough understanding of these interrelated topics [1–3] The architecture of an interconnected network is represented by a graph, where nodes represent the processors and edges represent the links between processors. Electric power companies need to continually monitor the state of their systems as in the case of the voltage magnitude at loads and the machine phase angle at generators. In the electric power system, a vertex represents an electric node and an edge represents a transmission line joining two electrical nodes [4, 5]. Chemical graph theory is a branch of mathematical chemistry in which we apply tools of graph theory to model the chemical phenomenon mathematically. This theory contributes a prominent role in the fields of chemical sciences. A molecular or chemical graph is a simple finite graph in which vertices denote the atoms and edges denote the chemical bonds in the underlying chemical structure. A topological index is actually a numeric quantity associated with chemical constitution purporting for correlation of the chemical structure with many physiochemical properties, chemical reactivity, and biological activities. Let be a graph with and being the vertices of , then the distance is defined as the shortest length between and . The eccentricity of a vertex is defined as

The minimum eccentricity in a graph is known as the radius, while the maximum eccentricity in a graph is known as the diameter. Topological descriptors play an important role in the quantitative structure-activity (QSAR) and structure-property (QSPR) study. Topological indices, which are based on eccentricity of the vertices in a graph , are known as eccentricity-based topological indices. The eccentric connectivity index [6] of is defined by

where is the degree of the vertex and is the eccentricity of in .

The total eccentricity index [7] is defined as the summation of eccentricity of each vertex in graph . In other words, when the vertex degrees are not considered in the eccentric connectivity index, then we obtain the total eccentricity index:

where is the eccentricity of in .

Zagreb indices [8] have been introduced more than thirty years ago. Let be a graph with and being the vertices of ; then the Zagreb indices are defined as follows:

Some new modified versions of Zagreb indices [9] are expressed in terms of eccentricity as follows:

On further results for certain degree-based topological indices of networks and nanostructures, consult [1, 10–13]. A tree is a connected acyclic graph, and the common type of a tree is a binary tree. A binary tree is made of nodes, where each node contains a left reference, a right reference, and a data element. The top most node is called the root. The vertex of the binary tree has three fields. The first field represents data, while the second and third contain information of the left and right sons of the vertex. If each internal vertex/node has exactly two descendents, then the binary tree is said to be a complete binary tree as shown in Figure 1. The basic skeleton of the hypertree-level is a complete binary tree . The nodes of the trees are labeled in such a way that root node has label 1. The root is at level 0. Labels of the left and right children are formed by appending 0 and 1, respectively, to the labels of the parent node. Here, the children of the nodes are labeled as and . Additional links in a hypertree are horizontal, and two nodes in the same level of the tree are joined if their label difference is [14]. The hypertree k-level shown in Figure 2 has vertices and edges .

Figure 1: Binary tree.

Figure 2: -level hypertree .

2. Main Results

In this section, we compute the close results for eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index of a hypertree, sibling tree, and X-tree for -level by using the edge partition method. The molecular topological descriptors of fullerenes and several interconnection networks have been already computed in the literature [1–3, 15].

In the next theorem, an exact expression for the eccentric connectivity index for a binary tree is computed.

Theorem 1. Consider the graph , then the eccentric connectivity index is equal to

Proof. In order to prove the above result, we use the formula of the eccentric connectivity index:

By using Table 1, we have

After an easy calculation, we get

The total eccentricity index of a binary tree is computed in the following theorem.

Table 1: Vertices partition of a hypertree (k-level) based on degree and eccentricity of each vertex with the existence of its frequencies.

Frequency
Range of i
Range of p

2

4

2

Theorem 2. Consider the graph , then the total eccentricity index is equal to

Proof. Let be a graph of a hypertree (k-level). To prove (10), we use the total eccentricity index formula:

By using Table 1, we get

After an easy calculation, we get

The first Zagreb eccentricity index of a binary tree is computed in the following theorem.

Theorem 3. Consider the graph , then the first Zagreb eccentricity index is equal to

Proof. In order to proof (14), we use the first Zagreb eccentricity index formula:

By using Table 2, we get

After an easy calculation, we get

Table 2: Edge partition of a hypertree (k-level) based on eccentricity of end vertices of each edge with existence of its frequencies.

Frequency
Range of i
Range of p

Theorem 4. Consider the graph , then the second Zagreb eccentricity index is equal to

Proof. Let G be a graph of a hypertree (k-level). The formula of the second Zagreb eccentricity index is given by

By using Table 1, we get

After an easy calculation, we get

Theorem 5. Consider the graph , then the third Zagreb eccentricity index is equal to

Proof. In order to prove (22), we use the formula of the third Zagreb eccentricity index:

By using Table 2, we get

After an easy calculation, we get

The 1-rooted sibling tree shown in Figure 3 is obtained from the 1-rooted complete binary tree by adding edges (sibling edges) between the left and right children of the same parent node [16].
An X-tree shown in Figure 4 is obtained from a complete binary tree on vertices of height and adding paths left to right through all the vertices at level i, .
A graph is said to be isomorphic to the graph , if there is a one-to-one correspondence between the vertex sets and and a one-to-one correspondence between the edge sets and in such a way that if is an edge with end vertices and in , then the corresponding edge in has its end points in the vertices and in which correspond to and , respectively. Such a pair of correspondences is called the graph isomorphism.

Figure 3: 1-rooted sibling tree .

Figure 4: k-level X-tree .

Remark 1. , but their topological eccentricity-based indices are equal:

Theorem 6. Consider the graph and , then the eccentric connectivity index is equal to

Proof. In order to prove the above result, we use the formula of the eccentric connectivity index:

By using Table 3, we have

After an easy calculation, we get

Table 3: Vertices partition of an X-tree (-level) based on degree and eccentricity of each vertex with the existence of its frequencies.

Frequency
Range of i
Range of

2

4

2

5

5

,
,

3

3

Theorem 7. Consider the graph and , then the total eccentricity index is equal to

Proof. Let be a graph of an X-tree (k-level). In order to prove, we use the total eccentricity index formula:

By using Table 3, we get

After an easy calculation, we get

Theorem 8. Consider the graph and , then the first Zagreb eccentricity index is equal to

Proof. In order to prove the above result, we use the first Zagreb eccentricity index formula:

By using Table 4, we get

After an easy calculation, we get

Table 4: Edge partition of an X-tree (-level) based on eccentricity of end vertices of each edge with existence of its frequencies.

Frequency
Range of i
Range of

Theorem 9. Consider the graph and , then the second Zagreb eccentricity index is equal to

Proof. Let be a graph of an X-tree (k-level) and . The formula of the second Zagreb eccentricity index is given by

By using Table 3, we get the following expression:

After an easy calculation, we get

Theorem 10. Consider the graph and , then the third Zagreb eccentricity index is equal to

Proof. In order to prove the above result, we use the formula of the third Zagreb eccentricity index:

By using Table 4, we get

After an easy calculation, we get

3. Conclusion

In this paper, we have computed the eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index for certain interconnection networks such as a hypertree, sibling tree, and X-tree -level. These results are useful in topological characterization of these important chemical networks.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the research foundation of Huainan Normal University (Grant no. 2019XJZD05) and UPAR grants of United Arab Emirates University (Grant nos. G00002590 and G00003271).

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