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

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


Introduction
e 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][2][3] e 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. is 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 G be a graph with u and v being the vertices of G, then the distance d(u, v) is defined as the shortest length between u and v. e eccentricity ε(v) of a vertex v ∈ V(G) is defined as e minimum eccentricity in a graph G is known as the radius G rad(G), while the maximum eccentricity in a graph G is known as the diameter 2p − 2. 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 G, are known as eccentricity-based topological indices. e eccentric connectivity index [6] of G is defined by where d(v) is the degree of the vertex v and ε(v) is the eccentricity of v in G. e total eccentricity index [7] is defined as the summation of eccentricity of each vertex in graph G. In other words, when the vertex degrees are not considered in the eccentric connectivity index, then we obtain the total eccentricity index: where ε(v) is the eccentricity of u in G. Zagreb indices [8] have been introduced more than thirty years ago. Let G be a graph with u and v being the vertices of G; 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][11][12][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. e top most node is called the root. e vertex of the binary tree has three fields. e 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. e basic skeleton of the hypertreek-level is a complete binary tree HT(k). e nodes of the trees are labeled in such a way that root node has label 1. e 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 x are labeled as 2x and 2x + 1. Additional links in a hypertree are horizontal, and two nodes in the same level of the tree are joined if their label difference is 2 i− 2 [14]. e hypertree k-level HT(k) shown in Figure 2 has vertices 2 k+1 − 1 and edges 3(2 k − 1).

Main Results
In this section, we compute the close results for eccentricitybased 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 k-level by using the edge partition method. e molecular topological descriptors of fullerenes and several interconnection networks have been already computed in the literature [1][2][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 G � HT(k), 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  Journal of Chemistry e total eccentricity index of a binary tree is computed in the following theorem. □ Theorem 2. Consider the graph G � HT(k), then the total eccentricity index is equal to Proof. Let G 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 e first Zagreb eccentricity index of a binary tree is computed in the following theorem.
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 □ Theorem 4. Consider the graph G � HT(k), then the second Zagreb eccentricity index is equal to Proof. Let G be a graph of a hypertree (k-level). e 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 G � HT(k), 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 Table 1: Vertices partition of a hypertree (k-level) based on degree and eccentricity of each vertex with the existence of its frequencies.

Journal of Chemistry
After an easy calculation, we get e 1-rooted sibling tree ST 1 k shown in Figure 3 is obtained from the 1-rooted complete binary tree T 1 k by adding edges (sibling edges) between the left and right children of the same parent node [16].
An X-tree XT(k) shown in Figure 4 is obtained from a complete binary tree on 2 k+1 − 1 vertices of height 2 i − 1 and adding paths P i left to right through all the vertices at level i, is said to be isomorphic to the graph G 2 � (V 2 , E 2 ), if there is a one-to-one correspondence between the vertex sets V 1 and V 2 and a one-to-one correspondence between the edge sets E 1 and E 2 in such a way that if e 1 is an edge with end vertices u 1 and v 1 in G 1 , then the corresponding edge e 2 in G 2 has its end points in the vertices u 2 and v 2 in G 2 which correspond to u 1 and v 1 , respectively. Such a pair of correspondences is called the graph isomorphism. □ Remark 1. HT(k) ≇ ST 1 k , but their topological eccentricitybased indices are equal: Theorem 6. Consider the graph G � XT(k) and k ≥ 3, 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 After an easy calculation, we get (30) □ Theorem 7. Consider the graph G � XT(k) and k ≥ 3, then the total eccentricity index is equal to (31) Table 2: Edge partition of a hypertree (k-level) based on eccentricity of end vertices of each edge with existence of its frequencies.
(ε(u), ε(v)) Frequency Proof. Let G 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 G � XT(k) and k ≥ 3, 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 □ Theorem 9. Consider the graph G � XT(k) and k ≥ 3, then the second Zagreb eccentricity index is equal to Proof. Let G be a graph of an X-tree (k-level) and k ≥ 3. e 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 G � XT(k) and k ≥ 3, 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

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 k-level. ese results are useful in topological characterization of these important chemical networks.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.