Analysis of COVID-19 by Means of Graph Theory

Graph theory is a powerful computational method used in biological mathematics that deals with different biological issues. In the field of microbiology, graphs can communicate the sub-atomic structure where cell quality or protein can be indicated as the vertex and the associate component can be viewed as the edge. Thus, the properties of the biological activity of the cell can be measured via a topological index by comparing six graphs. The current article focuses on certain topological lists for the corona virus graph. Initially, a general type of M-polynomial was explored from the M-polynomial, we recouped eight well-known degree-based topological lists, for example, Randic´ Index, First and Second Zagreb Indices, General Randic´ Index, Second Modified Zagreb Index, Symmetric Division Index, Harmonic Index, Inverse Sum Index, and Augmented Zagreb Index. The results /conformed to the findings of the previous studies.


Introduction
Coronaviruses are a group of large, enveloped, positive standard RNA viruses that enters the windpipe, digestive system and central nervous system in humans and other animals 16 [1].The spread of coronavirus in humans causes mild respiratory disease in humans [2].The causative agent is the novel coronavirus which is recognized 24 and separated from a solitary patient towards the beginning of January and accordingly was confirmed 25 in 16 extra patients [3].Specifically a live animal and seafood whole sale 26 market in Wuhan, was regarded as the source of this novel coronavirus.As it is discovered 27 55 % cases were connected to the market place [4].In the interim ongoing correlation of the genetic 28 sequences of this virus and bat coronavirus both show 96% similarity [5].This virus rapidly spread in 29 China and subsequently all over the world.

Infinite Graph
If we have n (Hemagglutinin+ Spikes+ RNA) and 1 (Envelop or Viruses) then CoV(n,m) graph 41 will be of form.The information which is concealed in the symmetry of molecular graphs of various compounds can be studied by tools mathematical chemistry tools such as functions and polynomials.These help in predicting the properties of compounds without employing quantum mechanics.Topology of a graph can be described by topological index which is a numerical parameter.Topological indices give a numerical description of the molecular structure.This helps in developing a qualitative structure activity relationships (QSARs).Degree-based topological indices are the most well-known invariants of this type.These numerical values correlate the molecular structure with different aspects of properties such as chemical reactivity, physical properties and biological activities.It is a proven fact that the graphical structure of a molecule is correlated to its various properties such as boiling point, heat formation, rigidity and strain energy and fracture toughness [6].
A Graph is number of distinct dots and lines.These dots are called vertices and the lines/paths that connects these dot are called edges.The path between the two dots (vertices) like u and v are 58 known as length between the two vertices.Number of edges connected to a vertex are called degree 59 of the vertex.Degree of a vertex is a key point of finding an M-polynomial of our desired graph.60 M-polynomial is use to find variations by changing our variables.
The tables of partition of a generalized CoV(n,m) graph consists of vertices, edges and loops given as Where, • m denotes number of viruses.
• n denotes number of vertices.

M-Polynomial and Topological indices
Definition 1. Suppose G=(V,E) is a graph and v∈V , then dv (G) denotes the degree of v. Let mi,j(G), i, j=1, be the number of edges uv of G such that dv(G), du(G)= i, j.The generalized graph G M-polynomial [7] can be given as : This polynomial has better computational characteristics of materials.Topological indices can be computed by utilizing this Mpolynomial.  () where degree of vertices are represented by u and v.
Definition 5.The GRI is known as General Randic´ Index of G which was introduced by Ballobas, Erdos [12] and Amic [13] in 1998.This index was equally popular in mathematics and chemistry [14].
where α is an any real number, αeR.[7] Definition 6.Out of 148 discrete Adriatic indices, the total surface area for polychlorobiphenyls is predicted well by the Symmetric Division Index (SDI) [15].For a connected graph G, SDI can be defined as given: Furtula et al [17].It stated as: It is valuable for computing heat of formation of alkanes.The calculation of residence of molecules (chemical and physical) can be viably categorized by these indices.Previously, Munir et al have calculated the M-polynomials.They also calculated corresponding topological indices for Titania Nanotubes in and Nanostar Dendrimers [17].
A few topological indices are degree based which can be determined from M-polynomial [12].

M-Polynomial of CoV(n,m)
The    Proof.As we know that the M-polynomial of wheel CoVn,m is defined in Eq(1), then General Randic´ is, ,,  Proof.As we know that the M-polynomial of CoVn,m CoVn,m is defined in Eq (1), then Inverse Sum Index is ,,  topological indices help in understanding the chemical reactivity, physical features and biological activities.Hence, it can be said that the topological indices are a core function.Every molecular structure can be mapped to a real number with its help.It can also be used asdescriptors of the molecules under testing.In this paper the Mpolynomial of coronavirus is proposed.From M-polynomial we find some degree based topological indices such as Modified Second Zagreb Index, First and Second Zagreb Indices, Augmented Zagreb and Symmetric Division Index.
During 2002-2004 17 SARS-CoV (Severe Acute Respiratory Syndrome) first rose in China and quickly spread to 18 parts of the world causing 8000 contaminations and as per a rough estimation passed around some 8000 related cases around 19 the world (WHO-2004).Further study reveals that SARS-CoV is transmitted from civet cats to humans.20 In 2012 MERS-CoV (Middle East Respiratory Syndrome) was first recognized Scientific Inquiry and Review Volume 4 Issue 2, 2020 in the Middle East and 21 afterwards spread to different nations.MERS-CoV transferred from dromedary camel to a human.In December 2019 that third Zoonotic human Coronavirus emerged in Wuhan, China after (SARS-CoV) 23 in 2002 and (MERS-CoV) in 2012.

Definition 2 .Definition 3 .Definition 4 .
First and Second Zagreb indices was introduced by Gutman and Trinajstic [8, 9, 10] in 1972 and 1975 respectively.The 1st and 2nd Zagreb indices are stated as:M1(G) = ∑ (du + dv) The 2nd modified Zagreb index is stated as: The RI is known as Randic´ Index which was introduced by Milan Randic´ in 1975.It is also called connectivity index of graph.[11]

Definition 7 .Definition 8 .Definition 9 .
The HI which is also known as Randic´ Index (H(G)) .It is alternate variant of Randic´ index which was introduced by Fajtlowicz[16] in 1987.It is stated as: The Inverse Sum Index (ISI) stated as: The AZI known as Augmented Zagreb Index of G which was introduced by Boris

Figure 10 .Figure 11 .Theorem 14 .
Figure 10.2st Zagreb index Theorem 13.Let CoVn,m be generalized COVID − 19 Graph.Then Second Modified Zagreb of COVID − 19 graph is given by, Zagreb is,    2 , ,1 ( ) ,, m n m x y n m xy M CoV S S M CoV x y

Figure 17 .
Figure 17.Augmented Zagreb index6.ConclusionThe graph theory is the discipline of mathematics which reinforces the investigation of complex networks in biological application or in any other uses.It has been effectively used in the investigation of the biological network topology and different biomolecules.Thus