Journal of Chemistry

Volume 2019, Article ID 4291943, 11 pages

https://doi.org/10.1155/2019/4291943

## Topology-Based Analysis of OTIS (Swapped) Networks and

Correspondence should be addressed to Iftikhar Ahmad; moc.liamg@1036iffi

Received 2 September 2019; Accepted 10 October 2019; Published 7 November 2019

Guest Editor: Shaohui Wang

Copyright © 2019 Hai-Xia Li 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 the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In this paper, M-polynomial and networks are computed. The M-polynomial is rich in information about degree-based topological indices. By applying the basic rules of calculus on M-polynomials, the first and second Zagreb indices, modified second Zagreb index, general Randić index, inverse Randić index, symmetric division index, harmonic index, inverse sum index, and augmented Zagreb index are recovered.

#### 1. Introduction

Cheminformatics is a new branch of science which relates chemistry, mathematics, and computer sciences. Quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) are the main components of cheminformatics which are helpful to study physicochemical properties of chemical compounds [1–3].

A topological index is a numeric quantity associated with the graph of chemical compound, which characterizes its topology and is invariant under graph automorphism [4–6]. There are numerous applications of graph theory in the field of structural chemistry. The first well-known use of a topological index in chemistry was by Wiener in the study of paraffin boiling points [7–9]. After that, in order to explain physicochemical properties, various topological indices have been introduced and studied [10, 11].

A computer network is a digital telecommunications network which allows nodes to share resources. In computer networks, computing devices exchange data with each other using connections (data links) between nodes. These data links are established over cable media such as wires or optic cables, or wireless media such as WiFi. Optical transpose interconnection system (OTIS) networks were initially contrived to give productive network to new optoelectronic computer models that profit by both optical and electronic advancements [12]. In OTIS networks, processors are orchestrated into groups. Electronic inter-connects are used between processors within the same cluster, while optical links are used for intercluster communication. Various algorithms have been produced for directing, determination/arranging, certain numerical calculations, Fourier transformation [13], matrix multiplication [14], image processing [15], and so on [16, 17]. The structure of an interconnection system can be scientifically modeled by a graph. The vertices of this graph are the processor nodes and the edges are the connections between the processors. The topology of a graph decides the manner by which vertices are associated by edges. From the topology of a system, certain properties can be decided. The diameter of a graph is the maximum distance between any two vertices of the graph.

*Definition 1. *(OTIS (swapped) network ). The OTIS (swapped) network is derived from the graph , which is a graph with vertex set and edge set and .

The graph of OTIS (swapped) network given in Figure 1 has edges and vertices.