# Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{0}, the reproduction number, and the level of spread, using the nested wreath product groups. Our plots of entropy and logarithms of topological indices of pandemic trees accentuate the underlying severity of COVID-19 over the 1918 Spanish flu pandemic.

## 1. Introduction

^{0}value close to 5.7 in some regions of a susceptible population. This dramatically contrasts with the previous Spanish flu virus which has a R

^{0}value close to 1.8. There are seven types of coronaviruses that infect humans [5], including the newly discovered SARS-CoV-2 virus. The level and duration of infectious virus replication are important factors in the risk assessments [6]. A study on interspecies transmission of the virus and its genetic diversity can provide mitigation strategies against the infection [7].

^{0}value, an epidemiological measure of the degree of infection, suggesting that an infected individual in turn infects R

^{0}others in a susceptible population pool. Consequently, quantitative measures of a pandemic network can provide significant new insights into the epidemic dynamics. Topological indices have been developed that aid in the prediction of chemical, physical, pharmaceutical, and biological properties [23,24,25,26,27,28,29]. The eccentric connectivity index is one such topological index that is currently being used for modeling biological activities of chemical compounds. In anticonvulsant, anti-inflammatory, and diuretic activities, this index exhibits high degree of predictability [30]. Ghorbani et al. [31] obtained various eccentricity–entropy-based topological measures of fullerenes.

## 2. Basic Concepts

_{u}(e) be the number of vertices lying closer to the vertex u than the vertex v and let n

_{v}(e) denote the number of vertices lying closer to the vertex v than the vertex u.

_{1}(G) [35] is defined as:

_{2}(G) [35] of graph G is defined as:

## 3. Main Results for the Topological Indices

^{0}, the reproduction number measures the severity of the pandemic, and it is defined as the number of individuals an infected person can in turn infect in a group of susceptible population set. For example, Figure 1 shows a pandemic tree for an epidemic with an R

^{0}value of 4. In this section, we have obtained the eccentricity-related indices for the pandemic tree network (also called a complete k-ary tree network), Cayley tree network, Christmas trees, and the corona product of Christmas tree and a path, all of which are defined in the respective sections.

#### 3.1. Topological Indices of Pandemic Trees

^{0}-ary tree, where R

^{0}is the reproduction number of a pandemic rounded to the nearest integer.

**Definition**

**1 ([39]).**

^{0}value of$k$. We denote this tree by${T}_{l}^{k}$, where$l$is the height of the tree,$k$, l ≥ 2.

**Theorem**

**1.**

- (i)
- $G{A}_{4}\left({T}_{l}^{k}\right)={\displaystyle \sum}_{i=1}^{l}{k}^{i}\times \frac{2\sqrt{\left(l+i-1\right)\left(l+i\right)}}{2\left(l+i-1\right)+1};$
- (ii)
- $Z{g}_{4}\left({T}_{l}^{k}\right)=\frac{k\left(k+2l-2lk+1\right)}{{\left(k-1\right)}^{2}}-\frac{{k}^{l+1}\left(k+4l-4lk+1\right)}{{\left(k-1\right)}^{2}};$
- (iii)
- ${\mathsf{\Pi}}_{4}^{*}\left({T}_{l}^{k}\right)={\displaystyle \prod}_{i=1}^{l}{\left(2\left(l+i\right)-1\right)}^{{k}^{i}};$
- (iv)
- $Z{g}_{6}\left({T}_{l}^{k}\right)=\frac{2{k}^{l+1}\left(2{k}^{2}{l}^{2}-l{k}^{2}-4k{l}^{2}+k+2{l}^{2}+l\right)}{{\left(k-1\right)}^{3}}-\frac{k\left({k}^{2}{l}^{2}-l{k}^{2}-2k{l}^{2}+2k+{l}^{2}+l\right)}{{\left(k-1\right)}^{3}};$
- (v)
- ${\mathsf{\Pi}}_{6}^{*}\left({T}_{l}^{k}\right)={\displaystyle \prod}_{i=1}^{l}{\left(\left(l+i-1\right)\left(l+i\right)\right)}^{{k}^{i}};$
- (vi)
- $Z{g}_{4}\left({T}_{l}^{k},x\right)={\displaystyle \sum}_{i=1}^{l}{k}^{i}\times {x}^{2\left(l+i-1\right)+1};$
- (vii)
- $Z{g}_{6}\left({T}_{l}^{k},x\right)={\displaystyle \sum}_{i=1}^{l}{k}^{i}\times {x}^{\left(l+i-1\right)\left(l+i\right)};$
- (viii)
- $AB{C}_{5}\left({T}_{l}^{k}\right)={\displaystyle \prod}_{i=1}^{l}{\left(\sqrt{\frac{2\left(l+i-1\right)-1}{\left(l+i-1\right)\left(l+i\right)}}\right)}^{{k}^{i}};$
- (ix)
- $PI\left({T}_{l}^{k}\right)=\frac{k\left({k}^{l}-1\right)\left({k}^{l+1}-1\right)}{{\left(k-1\right)}^{2}}$;
- (x)
- $Sz\left({T}_{l}^{k}\right)=\frac{{k}^{l+1}\left(2k-l-2k\times {k}^{l}+kl-kl\times {k}^{l}+{k}^{2+l}\times l\right)}{{\left(k-1\right)}^{3}}.$

**Proof.**

- ${E}_{l+i-1,l+i}=\left\{e=uv\in E\left({T}_{l}^{k}\right)|ec\left(u\right)=l+i-1\mathrm{and}ec\left(v\right)=l+i\right\}$ and ${n}_{l+i-1,l+i}=\left|{E}_{l+i-1,l+i}\right|={k}^{i},$ where $i\in \left\{1,\dots ,l\right\}:$and
- ${E}_{k+1,k}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=k+1\mathrm{and}d\left(v\right)=k,n\left(u\right)=\left(\frac{{k}^{l}-1}{k-1}\right)\mathrm{and}n\left(v\right)=\left(\frac{{k}^{l+1}-1}{k-1}\right)-\left(\frac{{k}^{l}-1}{k-1}\right)\}\mathrm{and}{n}_{k+1,k}=\left|{E}_{k+1,k}\right|=k;$
- ${E}_{1,k+1}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=1\mathrm{and}d\left(v\right)=k+1,n\left(u\right)=1\mathrm{and}n\left(v\right)=\left(\frac{{k}^{l+1}-1}{k-1}\right)-1\}\mathrm{and}\left|{E}_{1,k}\right|={k}^{l};$
- ${E}_{k+1,k+1}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=d\left(v\right)=k+1,n\left(u\right)={\displaystyle \sum}_{i=1}^{l-2}\left(\frac{{k}^{i+1}-1}{k-1}\right)\mathrm{and}n\left(v\right)=\left(\frac{{k}^{l+1}-1}{k-1}\right)-{\displaystyle \sum}_{i=1}^{l-2}\left(\frac{{k}^{i+1}-1}{k-1}\right)\}\mathrm{and}\left|{E}_{k,k}\right|={\displaystyle \sum}_{i=1}^{l-2}({k}^{l-i}).$

**Theorem**

**2.**

- (i)
- ${M}_{1}\left({T}_{l}^{k}\right)={k}^{2}+{\left(k+1\right)}^{2}\times \frac{{k}^{l}-k}{k-1}+{k}^{l};$
- (ii)
- ${M}_{2}\left({T}_{l}^{k}\right)={k}^{l}\left(k+1\right)+\frac{{k}^{l}-{k}^{2}}{k-1}{(k+1)}^{2}+{k}^{2}\left(k+1\right);$
- (iii)
- $ABC\left({T}_{l}^{k}\right)=k\sqrt{\frac{2k-1}{k\left(k+1\right)}}+{k}^{l}\sqrt{\frac{k}{k+1}}+\frac{{k}^{l}-{k}^{2}}{k-1}\times \frac{\sqrt{2k}}{k+1}.$

**Proof.**

- ${P}_{1}=\{v|d\left(v\right)=k\}\mathrm{and}\left|{P}_{1}\right|=1$;
- ${P}_{2}=\left\{v\right|d\left(v\right)=k+1\}\mathrm{and}\left|{P}_{2}\right|=\frac{{k}^{l}-k}{k-1};$
- ${P}_{3}=\{v|d\left(v\right)=1\}\mathrm{and}\left|{P}_{3}\right|={k}^{l}$;

- ${E}_{k,k+1}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=k\mathrm{and}d\left(v\right)=k+1\}\mathrm{and}\left|{E}_{k,k+1}\right|=k;$
- ${E}_{1,k+1}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=1\mathrm{and}d\left(v\right)=k+1\}\mathrm{and}\left|{E}_{1,k+1}\right|={k}^{l};$
- ${E}_{k+1,k+1}=\{e=uv\in E\left({T}_{l}^{k}\right)|d\left(u\right)=k+1\mathrm{and}d\left(v\right)=k+1\}\mathrm{and}\left|{E}_{k+1,k+1}\right|={\displaystyle \sum}_{i=2}^{l-1}{k}^{i}.$

#### 3.2. Topological Indices of Cayley Trees

**Definition**

**2.**

**Theorem**

**3.**

- (i)
- $G{A}_{4}\left(C\left(k,l\right)\right)={\displaystyle \sum}_{i=1}^{l}k{\left(k-1\right)}^{i-1}\times \frac{2\sqrt{\left(l+i-1\right)\left(l+i\right)}}{2\left(l+i-1\right)+1};$
- (ii)
- $Z{g}_{4}\left(C\left(k,l\right)\right)=\frac{(k(4l+k-2lk-8l{(k-1)}^{l}-k\left(k-1{)}^{l}+4lk\left(k-1{)}^{l}\right)\right)}{{\left(k-2\right)}^{2}};$
- (iii)
- ${\mathsf{\Pi}}_{4}^{*}\left(C\left(k,l\right)\right)={\displaystyle \prod}_{i=1}^{l}{\left(2\left(l+i\right)-1\right)}^{k{\left(k-1\right)}^{i-1}};$
- (iv)
- $Z{g}_{6}\left(C\left(k,l\right)\right)=k\left(\left(\frac{4{l}^{2}k-{l}^{2}{k}^{2}-4{l}^{2}+l{k}^{2}-2lk-2k+2}{{\left(k-2\right)}^{3}}\right)+\left(\frac{2{\left(k-1\right)}^{l}\left(2{l}^{2}{k}^{2}-8{l}^{2}k+8{l}^{2}-l{k}^{2}+2lk+k-1\right)}{{\left(k-2\right)}^{3}}\right)\right);$
- (v)
- ${\mathsf{\Pi}}_{6}^{*}\left(C\left(k,l\right)\right)={\displaystyle \prod}_{i=1}^{l}{\left(\left(l+i-1\right)\left(l+i\right)\right)}^{k{\left(k-1\right)}^{i-1}};$
- (vi)
- $Z{g}_{4}\left(C\left(k,l\right),x\right)={\displaystyle \sum}_{i=1}^{l}k{\left(k-1\right)}^{i-1}\times {x}^{2\left(l+i-1\right)+1};$
- (vii)
- $Z{g}_{6}\left(C\left(k,l\right),x\right)={\displaystyle \sum}_{i=1}^{l}k{\left(k-1\right)}^{i-1}\times {x}^{\left(l+i-1\right)\left(l+i\right)};$
- (viii)
- $AB{C}_{5}\left(C\left(k,l\right)\right)={\displaystyle \prod}_{i=1}^{l}{\left(\sqrt{\frac{2\left(l+i-1\right)-1}{\left(l+i-1\right)\left(l+i\right)}}\right)}^{k{\left(k-1\right)}^{i-1}}$;
- (ix)
- $PI\left(C\left(k,l\right)\right)=\frac{\left(k\left(k{\left(k-1\right)}^{l}-2\right)\left({\left(k-1\right)}^{l}-1\right)\right)}{{\left(k-2\right)}^{2}}$;
- (x)
- $Sz\left(C\left(k,l\right)\right)=\frac{\left(k\left(2k{\left(k-1\right)}^{l}-2k{\left(k-1\right)}^{2l}+{\left(k-1\right)}^{2l}-2kl{\left(k-1\right)}^{2l}+l{k}^{2}{\left(k-1\right)}^{2l}-1\right)\right)}{{\left(k-2\right)}^{3}}.$

**Proof.**

- ${E}_{l+i-1,l+i}=\left\{e=uv\in E\left(C\left(k,l\right)\right)|ec\left(u\right)=l+i-1\mathrm{and}ec\left(v\right)=l+i\right\}$ and ${n}_{l+i-1,l+i}$ = $\left|{E}_{l+i-1,l+i}\right|=k{\left(k-1\right)}^{i-1},$ where $i\in \left\{1,\dots ,l\right\}:$

- ${E}_{k,k}=\{e=uv\in E\left(C\left(k,l\right)\right)|d\left(u\right)=d\left(v\right)=k,n\left(u\right)=\left(\frac{{\left(k-1\right)}^{l}-1}{k-2}\right)\mathrm{and}n\left(v\right)=\left(\frac{k{\left(k-1\right)}^{l}-{\left(k-1\right)}^{l}-1}{k-2}\right)\}\mathrm{and}\left|{E}_{k,k}\right|=k;$
- ${E}_{1,k}=\{e=uv\in E\left(C\left(k,l\right)\right)|d\left(u\right)=1\mathrm{and}d\left(v\right)=k,n\left(u\right)=1\mathrm{and}n\left(v\right)=\left(\frac{k{\left(k-1\right)}^{l}-2}{k-2}-1\right)\}\mathrm{and}\left|{E}_{1,k}\right|=k{\left(k-1\right)}^{l-1};$
- ${E}_{k,k}=\{e=uv\in E\left(C\left(k,l\right)\right)|d\left(u\right)=d\left(v\right)=k,n\left(u\right)={\displaystyle \sum}_{i=1}^{l-2}\left(\frac{{\left(k-1\right)}^{i+1}-1}{k-2}\right)\mathrm{and}n\left(v\right)={\displaystyle \sum}_{i=1}^{l-2}\left(\frac{k{\left(k-1\right)}^{d}-2}{k-2}-\frac{{\left(k-1\right)}^{i+1}-1}{k-2}\right)\}\mathrm{and}\left|{E}_{k,k}\right|={\displaystyle \sum}_{i=1}^{l-2}k{\left(k-1\right)}^{l-i-1}.$

**Theorem**

**4.**

- (i)
- ${M}_{1}\left(C\left(k,l\right)\right)=\frac{k(k{\left(k-1\right)}^{l}-2k+2{\left(k-1\right)}^{l}}{k-2};$
- (ii)
- ${M}_{2}\left(C\left(k,l\right)\right)=\frac{{k}^{2}(2\left(k-1{)}^{l}-k\right)}{k-2};$
- (iii)
- $ABC\left(C\left(k,l\right)\right)=\left(\frac{k{\left(k-1\right)}^{l}-k+1}{{k}^{2}-3k+2}\right)\frac{\sqrt{2\left(k-1\right)}}{k}+k{\left(k-1\right)}^{l-1}\sqrt{\frac{k-1}{k}}.$

**Proof.**

- ${P}_{1}=\{v|d\left(v\right)=k\}\mathrm{and}\left|{P}_{1}\right|=\left(\frac{k{\left(k-1\right)}^{l-1}-2}{k-2}\right);$
- ${P}_{2}=\{v|d\left(v\right)=1\}\mathrm{and}\left|{P}_{2}\right|=k{\left(k-1\right)}^{l-1}$:

- ${E}_{k,k}=\{e=uv\in E\left(C\left(k,l\right)\right)|d\left(u\right)=k\mathrm{and}d\left(v\right)=k\}\mathrm{and}\left|{E}_{k,k}\right|=\left(\frac{k{\left(k-1\right)}^{l}-k+1}{{k}^{2}-3k+2}\right);$
- ${E}_{1,k}=\{e=uv\in E\left(C\left(k,l\right)\right)|d\left(u\right)=1\mathrm{and}d\left(v\right)=k\}\mathrm{and}\left|{E}_{1,k}\right|=k{\left(k-1\right)}^{l-1}.$

**Remark**

**2.**

**Remark**

**3.**

#### 3.3. Christmas Tree Network

**Definition**

**3 ([43]).**

^{th}slim tree ST(s) = (${V}_{1},{E}_{1},{u}_{1},{l}_{1},{r}_{1}$) and an (s + 1)

^{th}slim tree ST(s + 1) = (${V}_{2},{E}_{2},{u}_{2},{l}_{2},{r}_{2}$) together with the edges (${u}_{1},{u}_{2}$), (${l}_{1},{r}_{2}$), and (${l}_{2},{r}_{1}$), where ST(s) = (V, E, u, l, r), with V as the node set, E as the edge set, u ∈ V as the root node, l ∈ V as the left node, and r ∈ V as the right node defined below:

- 1
- ST(2) is the complete graph K
_{3}with its nodes labelled with u, l and r. - 2
- The s
^{th}slim tree ST(s), with s ≥ 3, is composed of a root node u and two disjoint copies of (s − 1)^{th}slim trees as the left subtree and right subtree, denoted by$S{T}^{l}\left(s-1\right)=\left({V}_{1},{E}_{1},{u}_{1},{l}_{1},{r}_{1}\right)$and$S{T}^{r}\left(s-1\right)=\left({V}_{2},{E}_{2},{u}_{2},{l}_{2},{r}_{2}\right)$, respectively and ST(s) = (V, E, u, l, r) is given by$V={V}_{1}{\displaystyle \cup}{V}_{2}{\displaystyle \cup}\left\{u\right\},E={E}_{1}{\displaystyle \cup}{E}_{2}{\displaystyle \cup}\left\{\left(u,{u}_{1}\right),\left(u,{u}_{2}\right),\left({r}_{1},{l}_{2}\right)\right\},l={l}_{1},r={r}_{2}.$

**Theorem**

**5.**

- (i)
- $G{A}_{4}\left(CT\left(s\right)\right)={\displaystyle \sum}_{i=1}^{s}3\left({2}^{i-1}\right)\frac{2\sqrt{\left(s+i-1\right)\left(s+i\right)}}{2\left(s+i-1\right)+1}+3\left({2}^{s-1}\right);$
- (ii)
- $Z{g}_{4}\left(CT\left(s\right)\right)=12s\left({2}^{s}\right)-6s-9\left({2}^{s}\right)+9+12s\left({2}^{s-1}\right);$
- (iii)
- ${\mathsf{\Pi}}_{4}^{*}\left(CT\left(s\right)\right)={\left(4s\right)}^{3\left({2}^{s-1}\right)}{\displaystyle \prod}_{i=1}^{s}{\left(2\left(s+i\right)-1\right)}^{3\left({2}^{i-1}\right)};$
- (iv)
- $Z{g}_{6}\left(CT\left(s\right)\right)=9s+{s}^{2}\left(12\times {2}^{s-1}-3\right)+3\times {2}^{s+1}\left(2{s}^{2}-3s+2\right)-12;$
- (v)
- ${\mathsf{\Pi}}_{6}^{*}\left(CT\left(s\right)\right)={\left(4{s}^{2}\right)}^{3\left({2}^{s-1}\right)}{\displaystyle \prod}_{i=1}^{s}{\left(\left(s+i-1\right)\left(s+i\right)\right)}^{3\left({2}^{i-1}\right)};$
- (vi)
- $Z{g}_{4}\left(CT\left(s\right),x\right)={\displaystyle \sum}_{i=1}^{s}3\left({2}^{i-1}\right){x}^{2\left(s+i\right)-1}+3\left({2}^{s-1}\right){x}^{4s};$
- (vii)
- $Z{g}_{6}\left(CT\left(s\right),x\right)={\displaystyle \sum}_{i=1}^{s}3\left({2}^{i-1}\right){x}^{\left(s+i-1\right)\left(s+i\right)}+3\left({2}^{s-1}\right){x}^{4{s}^{2}};$
- (viii)
- $AB{C}_{5}\mathsf{\Pi}\left(CT\left(s\right)\right)={\displaystyle \prod}_{i=1}^{s}{\left(\frac{2\left(s+i\right)-3}{\left(s+i-1\right)\left(s+i\right)}\right)}^{3\left({2}^{i-2}\right)}\times {\left(\frac{4s-2}{4{s}^{2}}\right)}^{3\left({2}^{s-2}\right)}.$

**Proof.**

- ${E}_{s+i-1,s+i}=\left\{e=uv\in E\left(CT\left(s\right)\right)|ec\left(u\right)=s+i-1\mathrm{and}ec\left(v\right)=s+i\right\}$ and ${n}_{s+i-1,s+i}=$ $\left|{E}_{s+i-1,s+i}\right|=3({2}^{i-1}),$ where $i\in \left\{1,\dots ,s\right\};$
- ${E}_{2s,2s}=\left\{e=uv\in E\left(CT\left(s\right)\right)|ec\left(u\right)=ec\left(v\right)=2s\right\}$ and ${n}_{2s,2s}=\left|{E}_{2s,2s}\right|=3({2}^{s-1})$.

**Theorem**

**6.**

- (i)
- ${M}_{1}\left(CT\left(s\right)\right)=9\left(3\times {2}^{s}-2\right);$
- (ii)
- ${M}_{2}\left(CT\left(s\right)\right)=$ 9$\left(\frac{9\times {2}^{s}-6}{2}\right)$;
- (iii)
- $ABC\left(CT\left(s\right)\right)=(3\times {2}^{s}-2).$

**Proof.**

- ${P}_{1}=\{v|d\left(v\right)=3\}\mathrm{and}\left|{P}_{1}\right|=\left({3.2}^{s}-2\right)$:

- ${E}_{3,3}=\left\{e=uv\in E\left(CT\left(s\right)\right)|d\left(u\right)=d\left(v\right)=3\right\}$ and $\left|{E}_{3,3}\right|=\frac{\left(9\times {2}^{s}-6\right)}{2}.$

#### 3.4. Corona Product of Christmas Tree and a Path

_{1}ʘG

_{2}of two graphs G

_{1}(with n

_{1}vertices and m

_{1}edges) and G

_{2}(with n

_{2}vertices and m

_{2}edges) is defined as the graph obtained by taking a copy of G

_{1}and n

_{1}copies of G

_{2}, and then joining the i

^{th}vertex of G

_{1}with edges to every vertex in the i

^{th}copy of G

_{2}. It follows from the definition of the corona product that G

_{1}ʘG

_{2}has n

_{1}+ n

_{1}n

_{2}vertices and m

_{1}+ n

_{1}m

_{2}+ n

_{1}n

_{2}edges. It is easy to see that G

_{1}ʘG

_{2}is not in general isomorphic to G

_{2}ʘG

_{1}[44].

_{3}is shown in Figure 5. The number of nodes and edges of CT(s)ʘP

_{n}are $3\times {2}^{s}-2+\left(3\times {2}^{s}-2\right)n$ and $\frac{9\times {2}^{s}-6}{2}+\left(\left(3\times {2}^{s}\right)-2\right)\left(2n-1\right)$, respectively. The eccentricity-based topological indices for CT(s)$\mathsf{\u0298}$P

_{n}, s, n ≥ 2 are given in Theorem 7.

**Theorem**

**7.**

- (i)
- $G{A}_{4}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)=\left(6+2n\right)\frac{\sqrt{\left(s+1\right)\left(s+2\right)}}{2s+3}+{{\displaystyle \sum}}_{i=2}^{s}3\left({2}^{i-2}\right)\left(2+n\right)\frac{2\sqrt{\left(s+i\right)\left(s+i+1\right)}}{2\left(s+i\right)+1}$$+6n\left({2}^{s-1}\right)\frac{\sqrt{\left(2s+1\right)\left(2s+2\right)}}{4s+3}+{{\displaystyle \sum}}_{i=1}^{s}3\left({2}^{i-1}\right)\left(n-1\right)\frac{2\left(s+i+2\right)}{2\left(s+i+1\right)+2}$$+3\left({2}^{s-1}\right)+n-1;$
- (ii)
- $Z{g}_{4}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)=n\left(9\times {2}^{s}-2+24s\times {2}^{s}\right)+s\left(6\times {2}^{s}-8n-2\right)-6\times {2}^{s}+5;$
- (iii)
- ${\mathsf{\Pi}}_{4}^{*}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)={\left(2s+3\right)}^{3+n}\times {{\displaystyle \prod}}_{i=2}^{s}{\left(2\left(s+i\right)+1\right)}^{\left(6+3n\right){2}^{i-2}}\times {\left(4s+3\right)}^{3n\left({2}^{s-1}\right)}$$\times {\displaystyle {\displaystyle \prod}_{i=1}^{s}}{\left(2\left(s+i+2\right)\right)}^{\left(3n-3\right){2}^{i-1}}\times {\left(4s+2\right)}^{3\left({2}^{s-1}\right)}\times {\left(2s+4\right)}^{n-1};$
- (iv)
- $Z{g}_{6}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)={s}^{2}\left(6\times {2}^{s}-4n+24n\times {2}^{s}-1\right)+s\left(5-2n-12\times {2}^{s}+18n\times {2}^{s}\right)+{2}^{s}\left(15n-\frac{3}{2}\right)-9n-1;$
- (v)
- ${\mathsf{\Pi}}_{6}^{*}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)={\left({s}^{2}+3s+2\right)}^{3+n}\times {{\displaystyle \prod}}_{i=2}^{s}{\left(\left(s+i\right)\left(s+i+1\right)\right)}^{\left(6+3n\right){2}^{i-2}}$$\times {\left(2\left(2{s}^{2}+3s+1\right)\right)}^{3n\left({2}^{s-1}\right)}\times {\displaystyle {\displaystyle \prod}_{i=1}^{s}}{\left(s+i+2\right)}^{\left(3n-3\right){2}^{i}}\times {\left(2s+1\right)}^{6\left({2}^{s-1}\right)}$$\times {\left(s+2\right)}^{2\left(n-1\right)};$
- (vi)
- $Z{g}_{4}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n},x\right)=\left(3+n\right){x}^{2s+3}+\left(6+3n\right){{\displaystyle \sum}}_{i=2}^{s}{2}^{i-2}{x}^{2\left(s+i\right)+1}+3n\left({2}^{s-1}\right){x}^{4s+3}$$+3\left(n-1\right){\displaystyle {\displaystyle \sum}_{i=1}^{s}}{2}^{i-1}{x}^{2\left(s+i+2\right)}+3\left({2}^{s-1}\right){x}^{4s+2}+\left(n-1\right){x}^{2\left(s+2\right)};$
- (vii)
- $Z{g}_{6}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n},x\right)=\left(3+n\right){x}^{{s}^{2}+3s+2}+\left(6+3n\right){{\displaystyle \sum}}_{i=2}^{s}{2}^{i-2}{x}^{\left(s+i\right)\left(s+i+1\right)}$$+3n\left({2}^{s-1}\right){x}^{\left(2s+1\right)\left(2s+2\right)}+3\left(n-1\right){\displaystyle {\displaystyle \sum}_{i=1}^{s}}{2}^{i-1}{x}^{{\left(s+i+2\right)}^{2}}$$+3\left({2}^{s-1}\right){x}^{{\left(2s+1\right)}^{2}}+\left(n-1\right){x}^{{\left(s+2\right)}^{2}};$
- (viii)
- $AB{C}_{5}\mathsf{\Pi}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)={\left(\sqrt{\frac{2s+1}{\left(s+1\right)\left(s+2\right)}}\right)}^{3+n}\times {{\displaystyle \prod}}_{i=2}^{s}{\left(\frac{2\left(s+i\right)-1}{\left(s+i+1\right)\left(s+i\right)}\right)}^{\left(6+3n\right)\left({2}^{i-3}\right)}$$\times {\left(\frac{4s+1}{\left(2s+1\right)\left(2s+2\right)}\right)}^{3n\left({2}^{s-2}\right)}\times {\displaystyle {\displaystyle \prod}_{i=1}^{s}}{\left(\frac{2\left(s+i+1\right)}{{\left(s+i+2\right)}^{2}}\right)}^{\left(3n-3\right)\left({2}^{i-2}\right)}$$\times {\left(\frac{\sqrt{4s}}{\left(2s+1\right)}\right)}^{3\left({2}^{s-1}\right)}\times {\left(\frac{\sqrt{2\left(s+1\right)}}{s+2}\right)}^{n-1}.$

**Proof.**

- ${E}_{s+1,s+2}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=s+1\mathrm{and}ec\left(v\right)=s+2\right\}$ and ${n}_{s+1,s+2}=\left|{E}_{s+1,s+2}\right|=3+n$;
- ${E}_{s+i,s+i+1}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=s+i\mathrm{and}ec\left(v\right)=s+i+1\right\}$ and ${n}_{s+i,s+i+1}=\left|{E}_{s+i,s+i+1}\right|=3({2}^{i-2})\left(2+n\right),$ where $i\in \left\{2,\dots ,s\right\};$
- ${E}_{2s+1,2s+2}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=2s+1\mathrm{and}ec\left(v\right)=2s+2\right\}$ and ${n}_{2s+1,2s+2}=\left|{E}_{2s+1,2s+2}\right|=3n({2}^{s-1})$;
- ${E}_{s+i+2,s+i+2}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=ec\left(v\right)=s+i+2\right\}$ and ${n}_{s+i+2,s+i+2}$ $=\left|{E}_{s+i+2,s+i+2}\right|=3({2}^{i-1})\left(n-1\right),$ where $i\in \left\{1,\dots ,s\right\};$
- ${E}_{2s+1,2s+1}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=ec\left(v\right)=2s+1\right\}$ and ${n}_{2s+1,2s+1}=\left|{E}_{2s+1,2s+1}\right|=3({2}^{s-1});$
- ${E}_{s+2,s+2}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|ec\left(u\right)=ec\left(v\right)=s+2\right\}$ and ${n}_{s+2,s+2}=\left|{E}_{s+2,s+2}\right|=n-1.$

_{3}are given as follows:

- ${E}_{3+1,3+2}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=3+1andec\left(v\right)=3+2\right\}$ and ${n}_{3+1,3+2}={n}_{4,5}=\left|{E}_{4,5}\right|=3+3=6;$
- ${E}_{s+i,s+i+1}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=s+i\mathrm{and}ec\left(v\right)=s+i+1\right\}$ and ${n}_{s+i,s+i+1}=\left|{E}_{s+i,s+i+1}\right|=3({2}^{i-2})\left(2+3\right)=15\left({2}^{i-2}\right),$ where $i\in \left\{2,3\right\};$
- ${E}_{6+1,6+2}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=6+1\mathrm{and}ec\left(v\right)=6+2\right\}$ and ${n}_{6+1,6+2}={n}_{7,8}=\left|{E}_{7,8}\right|=3(3({2}^{3-1}))=36$;
- ${E}_{s+i+2,s+i+2}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=ec\left(v\right)=s+i+2\right\}$ and ${n}_{s+i+2,s+i+2}=\left|{E}_{s+i+2,s+i+2}\right|=3({2}^{i-1})\left(n-1\right),$ where $i\in \left\{1,2,3\right\};$
- ${E}_{6+1,6+1}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=ec\left(v\right)=6+1\right\}$ and ${n}_{7,7}=\left|{E}_{7,7}\right|=3({2}^{3-1})=12;$
- ${E}_{3+2,3+2}=\left\{e=uv\in E\left(CT\left(3\right)\mathsf{\u0298}{P}_{3}\right)|ec\left(u\right)=ec\left(v\right)=3+2\right\}$ and ${n}_{3+2,3+2}={n}_{5,5}=\left|{E}_{5,5}\right|=3-1=2.$

**Theorem**

**8.**

- (i)
- ${M}_{1}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)=\left(3\times {2}^{s}-2\right)\left({n}^{2}+15n-1\right);$
- (ii)
- ${M}_{2}\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)=((3\times {2}^{s}-2)\left(9{n}^{2}+50n-15\right))/2;$
- (iii)
- $ABC\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)=\frac{(9\times {2}^{s}-6)}{2}\times \frac{\sqrt{2\left(n+2\right)}}{\left(n+3\right)}+2(3\times {2}^{s}-2)\sqrt{\frac{1}{2}}$$+\left(n-2\right)(3\times {2}^{s}-2)\sqrt{\frac{\left(n+4\right)}{\left(3\right)\left(n+3\right)}}+(3\times {2}^{s}-2)\sqrt{2}$$+\left(n-3\right)(3\times {2}^{s}-2)\left(2/3\right).$

**Proof.**

- ${P}_{1}=\{v|d\left(v\right)=2\}\mathrm{and}\left|{P}_{1}\right|=2\left(3\times {2}^{s}-2\right)$;
- ${P}_{2}=\{v|d\left(v\right)=3\}\mathrm{and}\left|{P}_{2}\right|=\left(n-2\right)\left(3\times {2}^{s}-2\right);$
- ${P}_{3}=\{v|d\left(v\right)=n+3\}\mathrm{and}\left|{P}_{3}\right|=\left(3\times {2}^{s}-2\right)$:

- ${E}_{n+3,n+3}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|d\left(u\right)=n+3\mathrm{and}d\left(v\right)=n+3\right\}$ and $\left|{E}_{n+3,n+3}\right|=(9\times {2}^{s}-6)/2$;
- ${E}_{2,n+3}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|d\left(u\right)=2\mathrm{and}d\left(v\right)=n+3\right\}$ and $\left|{E}_{2,n+3}\right|=2(3\times {2}^{s}-2)$;
- ${E}_{3,n+3}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|d\left(u\right)=3\mathrm{and}d\left(v\right)=n+3\right\}$ and $\left|{E}_{3,n+3}\right|=\left(n-2\right)(3\times {2}^{s}-2);$
- ${E}_{2,3}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|d\left(u\right)=2\mathrm{and}d\left(v\right)=3\right\}$ and $\left|{E}_{2,3}\right|=2(3\times {2}^{s}-2);$
- ${E}_{3,3}=\left\{e=uv\in E\left(CT\left(s\right)\mathsf{\u0298}{P}_{n}\right)|d\left(u\right)=d\left(v\right)=3\right\}$ and $\left|{E}_{3,3}\right|=\left(n-3\right)(3\times {2}^{s}-2)$.

## 4. Various Applications of Topological Indices for the COVID-19 Pandemic

^{−4}Sz + 9.210,

^{0}value close to 6 in a susceptible population set compared to k = 2 for the 1918 Spanish flu. As seen from Figure 7, we obtain a similar trend for the log of the fourth geometric-arithmetic indices for pandemic trees.

^{0}, which is a result of the illness itself and various interventions. It is a macroscopic rather than a microscopic measure. Thus, the comparison made here is by no means indicative of the total number of infections or deaths, as they depend on multiple factors such as the discovery of vaccines, adherence to social distancing, more advanced technological tools available today for diagnosis, contract tracing, treatments, mitigation measures, and prognosis compared to 1918. Hence, there are several substantial differences between the two pandemics, namely COVID 19 and the Spanish Flu. First, they are both caused by different types of viruses with different modes of infection. We have denser populations and a far more connected world with air travel which can contribute to the spread of COVID-19 globally. Specifically, the comparisons of Covid-19 and Spanish flu provided in this section do not take directly into consideration a number of factors enumerated below which could alter the assumed R

^{0}values dynamically and hence the computed indices:

- The difference in the awareness among all the countries and the general population on the severity of the disease and the necessary preventive actions that are needed as a result of internet and other forms of communications.
- The difference in the resources available for healthcare.
- The difference in number of qualified and trained virologists, doctors, nurses, government officials, government/private institutions, and other important frontline workers.
- The difference in information content available through research, innovations, and the data from the past pandemic, all of which can contribute to improved predictions. Researchers worldwide are working diligently to find a vaccine against the virus causing the COVID-19 pandemic. The WHO is working in collaboration with scientists, business, and global health organizations to accelerate the vaccine effectiveness and discovery [51].
- Other social, economic, and medical changes that occurred in the last 100 years.

^{0}for COVID-19, which itself could dynamically change with discoveries of vaccine, technological advances to effectively control the epidemic and the role of super spreaders. Furthermore, entropy measures from Section 5 suggest that the ongoing pandemic left to itself without such mitigating measures, poses an enormous and unique challenge globally, and the ongoing battle is nowhere near the end. Hence, significant interventions and countermeasures must be taken such as therapeutic interventions including vaccines, social distancing, vigorous quarantine measures, facial masks, improved oral and hand hygiene measures. The availability of preventive and therapeutic tools against viral infections and their related complications is an important factor for the disaster risk assessment [52].

## 5. Thermodynamic Entropy of Pandemic Trees

^{0}value of k. The R

^{0}epidemiological measure is exponential similar to other natural disasters such as an earthquake measured in a logarithmic Richter scale. We introduce here yet another measure of the chaos created by the COVID-19 pandemic which is based on the thermodynamic Boltzmann’s definition of entropy, a measure of the degree of disorder. At the very outset, it is pointed out that the thermodynamic entropy is different from the graph theoretical entropy discussed by Dehmer and co-workers [53,54,55] although in principle there should be a correspondence between statistical thermodynamic entropy and information theoretic entropy. The graph theoretical entropy concept invoked by these authors is based on Shannon’s information theoretic formulation as applied to graph theoretical invariants. On the other hand, the thermodynamic entropy is simply based on the celebrated Boltzmann’s definition and formula for the entropy. Consequently, in the present study, the term entropy refers to the thermodynamic entropy which is simply based on the number of distinct configurations or the number of inequivalent ways to label the vertices of a graph under the action of the automorphism group of the graph. The set R

^{D}of functions from the set D of vertices to the set R of colors would form equivalence classes under the action of the automorphism group. Among these classes, the number of classes where each vertex is colored with a unique color corresponds to the number of unique ways to label the graph, and hence, this number measures the number of distinct configurations a graph can take under the group action.

^{T}AP,

_{1}x H

_{2}x…H

_{n})^ G’, where x is a direct product of two groups, ^ is a semi-direct product and G′ is a group isomorphic with G acting on the whole set of vertices. The group is comprised of |H|

^{n}|G| operations. The order of each S

_{k}group composing the wreath product of a pandemic tree is k!. The order of a nested wreath product group was obtained by Balasubramanian [15] as:

_{Lab}is also referred to as the number of different configurations denoted by Ω in statistical mechanics. The entropy of any system, denoted by S, is defined by the Boltzmann equation and it measures the degree of disorder. It is given by:

_{B}ln(Ω),

_{B}is the Boltzmann constant.

_{B}as function of both k and l, and we can see that the order of S increases as a function of k and l. This shows the severity of the ongoing pandemic which has a k value of 6 and as l goes to 6 nearly the entire susceptible population is infected. Moreover, the entropy measure of the ongoing COVID-19 pandemic is dramatically larger than that of the 1918 Spanish flu virus, underscoring the severity on the novel coronavirus 2. Furthermore, entropy is measured in natural logarithmic scale analogous to the Richter scale of an earthquake and thus the ratios of the entropies of COVID-19 to Spanish flu of 1104.6 at level 6 would suggest about 10

^{3}greater severity as measured in a natural log scale. Thus, entropy provides yet another quantitative measure for the epidemiologists to measure the degree of disorder and hence the severity of the ongoing pandemic.

## 6. Stochasticity in Pandemic Tree Generation

^{0}other individuals, by studying a complete k-ary tree with k = R

^{0}as a particular case. This may not be the case always except in worst affected locations. In such cases, the pandemic trees will just be R

^{0}-ary trees. Stochasticity in the generation of such trees leads to considering the adjacency matrices of the trees for implementing distance-based graph algorithms. Degree and eccentricity-based topological indices can be easily computed from these matrices.

^{0}= 6 at level 0 and with every vertex of level i having R

^{0}-i children, $0\le i\le 5$, is considered in Figure 9.

- M
_{1}(T) = ${{\displaystyle \sum}}_{{v}_{i}\in V\left(T\right)}{d}_{i}^{2}$ = 1957l^{2}– 15,660l + 33,270; - ${M}_{2}$(T) = ${{\displaystyle \sum}}_{{v}_{i}{v}_{j}\in E\left(T\right)}{d}_{i}{d}_{j}$ = 1956l
^{2}– 13,710l+25,440; - ABC(T) =${{\displaystyle \sum}}_{{v}_{i}{v}_{j}\in E\left(T\right)}\sqrt{\frac{{d}_{i}+{d}_{j}-2}{{d}_{i}{d}_{j}}}$$=\left(\frac{\sqrt{10}}{6}\right)l+30\sqrt{\frac{2l-3}{{l}^{2}-l}}+120\sqrt{\frac{2l-5}{{l}^{2}-3l+2}}+360\sqrt{\frac{2l-7}{{l}^{2}-5l+6}+}120\sqrt{\frac{2l-9}{{l}^{2}-7l+12}}+720\sqrt{\frac{2l-11}{{l}^{2}-9l+20}};$
- PI(T) =${{\displaystyle \sum}}_{e=uv\in E\left(T\right)}\left[{n}_{u}\left(\mathrm{e}\right)+{n}_{v}(\mathrm{e}\right)]$ = 1957l + 3,816,150;
- Sz(T) =${{\displaystyle \sum}}_{e=uv\in E\left(T\right)}\left[{n}_{u}\left(\mathrm{e}\right)\times {n}_{v}(\mathrm{e}\right)]$= 531,706l + 15,153,306.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Whitelaw, S.; A Mamas, M.; Topol, E.; Van Spall, H.G. Applications of digital technology in COVID-19 pandemic planning and response. Lancet Digit. Health
**2020**, 2, e435–e440. [Google Scholar] [CrossRef] - Blackwood, J.C.; Childs, L.M. An introduction to compartmental modeling for the budding infectious disease modeler. Lett. Biomath.
**2018**, 5, 195–221. [Google Scholar] [CrossRef] - Zhou, P.; Yang, X.-L.; Wang, X.-G.; Hu, B.; Zhang, L.; Zhang, W.; Si, H.-R.; Zhu, Y.; Li, B.; Huang, C.-L.; et al. A pneumonia outbreak associated with a new coronavirus of probable bat origin. Nature
**2020**, 579, 270–273. [Google Scholar] [CrossRef] [Green Version] - Lai, C.-C.; Shih, T.-P.; Ko, W.-C.; Tang, H.-J.; Hsueh, P.-R. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and coronavirus disease-2019 (COVID-19): The epidemic and the challenges. Int. J. Antimicrob. Agents
**2020**, 55, 105924. [Google Scholar] [CrossRef] - Andersen, K.G.; Rambaut, A.; Lipkin, W.I.; Holmes, E.C.; Garry, R.F. The proximal origin of SARS-CoV-2. Nat. Med.
**2020**, 26, 450–452. [Google Scholar] [CrossRef] [Green Version] - Zhou, F.; Yu, T.; Du, R.; Fan, G.; Liu, Y.; Liu, Z.; Xiang, J.; Wang, Y.; Song, B.; Gu, X.; et al. Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: A retrospective cohort study. Lancet
**2020**, 395, 1054–1062. [Google Scholar] [CrossRef] - Cui, J.; Li, F.; Shi, Z.-L. Origin and evolution of pathogenic coronaviruses. Nat. Rev. Genet.
**2019**, 17, 181–192. [Google Scholar] [CrossRef] [Green Version] - Wu, J.T.; Leung, K.; Leung, G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study. Lancet
**2020**, 395, 689–697. [Google Scholar] [CrossRef] [Green Version] - Forster, P.; Forster, L.; Renfrew, C.; Forster, M. Phylogenetic network analysis of SARS-CoV-2 genomes. Proc. Natl. Acad. Sci. USA
**2020**, 117, 9241–9243. [Google Scholar] [CrossRef] [Green Version] - Balasubramanian, K.; Khokhani, K.; Basak, S.C. Complex Graph Matrix Representations and Characterizations of Proteomic Maps and Chemically Induced Changes to Proteomes. J. Proteome Res.
**2006**, 5, 1133–1142. [Google Scholar] [CrossRef] - Basak, S.C.; Grunwald, G.D.; Gute, B.D.; Balasubramanian, K.; Opitz, D. Use of statistical and neural net approaches in predicting toxicity of chemicals. J. Chem. Inf. Comput. Sci.
**2000**, 40, 885–890. [Google Scholar] [CrossRef] - Matsen, F.A. Phylogenetics and the Human Microbiome. Syst. Biol.
**2015**, 64, e26–e41. [Google Scholar] [CrossRef] - Delucchi, E. Nested set complexes of Dowling lattices and complexes of Dowling trees. J. Algebraic Comb.
**2007**, 26, 477–494. [Google Scholar] [CrossRef] [Green Version] - Balasubramanian, K. Tree pruning and lattice statistics on Bethe lattices. J. Math. Chem.
**1988**, 2, 69–82. [Google Scholar] [CrossRef] - Balasubramanian, K. Nested wreath groups and their applications to phylogeny in biology and Cayley trees in chemistry and physics. J. Math. Chem.
**2017**, 55, 195–222. [Google Scholar] [CrossRef] - Balasubramanian, K. Spectra of chemical trees. Int. J. Quantum Chem.
**1982**, 21, 581–590. [Google Scholar] [CrossRef] - Balasubramanian, K. Symmetry groups of chemical graphs. Int. J. Quantum Chem.
**1982**, 21, 411–418. [Google Scholar] [CrossRef] - Sellers, P.H. On the Theory and Computation of Evolutionary Distances. SIAM J. Appl. Math.
**1974**, 26, 787–793. [Google Scholar] [CrossRef] - Sellers, P.H. An algorithm for the distance between two finite sequences. J. Comb. Theory Ser. A
**1974**, 16, 253–258. [Google Scholar] [CrossRef] [Green Version] - Fischer, M.; Klaere, S.; Nguyen, M.A.T.; Von Haeseler, A. On the group theoretical background of assigning stepwise mutations onto phylogenies. Algorithms Mol. Biol.
**2012**, 7, 36. [Google Scholar] [CrossRef] [Green Version] - Yun, U.; Lee, G.; Kim, C.-H. The Smallest Valid Extension-Based Efficient, Rare Graph Pattern Mining, Considering Length-Decreasing Support Constraints and Symmetry Characteristics of Graphs. Symmetry
**2016**, 8, 32. [Google Scholar] [CrossRef] [Green Version] - Xue, L.; Jing, S.; Miller, J.C.; Sun, W.; Li, H.; Estrada-Franco, J.G.; Hyman, J.M.; Zhu, H. A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy. Math. Biosci.
**2020**, 326, 108391. [Google Scholar] [CrossRef] - Basavanagoud, B.; Desai, V.R.; Patil, S. (β,α)−Connectivity Index of Graphs. Appl. Math. Nonlinear Sci.
**2017**, 2, 21–30. [Google Scholar] [CrossRef] [Green Version] - Gao, W.; Wang, W. The Vertex Version of Weighted Wiener Number for Bicyclic Molecular Structures. Comput. Math. Methods Med.
**2015**, 2015, 1–10. [Google Scholar] [CrossRef] [Green Version] - Gao, W.; Farahani, M.R.; Shi, L. The forgotten topological index of some drug structures. Acta. Medica. Mediterr.
**2016**, 32, 579–585. [Google Scholar] - Gao, W.; Wang, W.; Farahani, M.R. Topological Indices Study of Molecular Structure in Anticancer Drugs. J. Chem.
**2016**, 2016, 1–8. [Google Scholar] [CrossRef] [Green Version] - Martínez-Pérez, Á.; Rodriguez, J.M. New Bounds for Topological Indices on Trees through Generalized Methods. Symmetry
**2020**, 12, 1097. [Google Scholar] [CrossRef] - Atanasov, R.; Furtula, B.; Škrekovski, R. Trees with Minimum Weighted Szeged Index Are of a Large Diameter. Symmetry
**2020**, 12, 793. [Google Scholar] [CrossRef] - Knor, M.; Imran, M.; Jamil, M.K.; Škrekovski, R. Remarks on Distance Based Topological Indices for ℓ-Apex Trees. Symmetry
**2020**, 12, 802. [Google Scholar] [CrossRef] - Liu, J.-B.; Shaker, H.; Nadeem, I.; Farahani, M.R. Eccentric Connectivity Index of t-Polyacenic Nanotubes. Adv. Mater. Sci. Eng.
**2019**, 2019, 1–9. [Google Scholar] [CrossRef] [Green Version] - Ghorbani, M.; Dehmer, M.; Emmert-Streib, F. Properties of Entropy-Based Topological Measures of Fullerenes. Mathematics
**2020**, 8, 740. [Google Scholar] [CrossRef] - Ghorbani, M.; Khaki, A. A note on the fourth version of geometric-arithmetic index. Optoelectron. Adv. Mat.
**2010**, 4, 2212–2215. [Google Scholar] - Gao, W.; Wu, H.; Siddiqui, M.K.; Baig, A.Q. Study of biological networks using graph theory. Saudi J. Biol. Sci.
**2018**, 25, 1212–1219. [Google Scholar] [CrossRef] - Farahani, M.R.; Kanna, R.M.R. Fourth zagreb index of circumcoronene series of benzenoid. Leonardo Electron. J. Pract. Technol.
**2015**, 27, 155–161. [Google Scholar] - Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem. Phys. Lett.
**1972**, 17, 535–538. [Google Scholar] [CrossRef] - Das, K.C.; Gutman, I.; Furtula, B. On atom-bond connectivity index. Chem. Phys. Lett.
**2011**, 511, 452–454. [Google Scholar] [CrossRef] [Green Version] - Khadikar, P.V.; Karmarkar, S.; Agrawal, V.K. A Novel PI Index and Its Applications to QSPR/QSAR Studies. J. Chem. Inf. Comput. Sci.
**2001**, 41, 934–949. [Google Scholar] [CrossRef] - Gutman, I. A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY
**1994**, 27, 9–15. [Google Scholar] - Song, C.; Hao, R.-X. Antimagic orientations for the complete k-ary trees. J. Comb. Optim.
**2019**, 38, 1077–1085. [Google Scholar] [CrossRef] - Balasubramanian, K. Topo-Chemie-2020 is a package of codes that computes numerous degree-based, distance-based, eccentricity-based, neighbour-based topological indices, characteristic polynomials, matching polynomials, distance polynomials, distance degree vector sequences, walks and self-returning walks, and automorphisms of graphs.
- Weisstein, E.W. Cayley Tree, MathWorld-A Wolfram Web Resource. Available online: https://mathworld.wolfram.com/CayleyTree.html (accessed on 17 September 2020).
- Gutman, I.; Dobrynin, A.A. The Szeged index—A success story. Graph Theory Notes NY
**1998**, 34, 37–44. [Google Scholar] - Hung, C.-N.; Hsu, L.-H.; Sung, T.-Y. Christmas tree: A versatile 1-fault-tolerant design for token rings. Inf. Process. Lett.
**1999**, 72, 55–63. [Google Scholar] [CrossRef] - Nada, S.; Elrokh, A.; Elsakhawi, E.; Sabra, D. The corona between cycles and paths. J. Egypt. Math. Soc.
**2017**, 25, 111–118. [Google Scholar] [CrossRef] - Khadikar, P.; Karmarkar, S.; Agrawal, V.; Singh, J.; Khadikar, P.V.; Lukovits, I.; Diudea, M.V. Szeged Index—Applications for Drug Modeling. Lett. Drug Des. Discov.
**2005**, 2, 606–624. [Google Scholar] [CrossRef] [Green Version] - Basak, S.C.; Mills, D.; Mumtaz, M.M.; Balasubramanian, K. Use of topological indices in predicting aryl hydrocarbon receptor binding potency of dibenzofurans: A hierarchical QSAR approach. Indian J. Chem.
**2003**, 42, 1385–1391. [Google Scholar] - Mondal, S.; De, N.; Pal, A. Topological Indices of Some Chemical Structures Applied for the Treatment of COVID-19 Patients. Polycycl. Aromat. Compd.
**2020**, 1–15. [Google Scholar] [CrossRef] - Balasubramanian, K.; Gupta, S.P. Quantum Molecular Dynamics, Topological, Group Theoretical and Graph Theoretical Studies of Protein-Protein Interactions. Curr. Top. Med. Chem.
**2019**, 19, 426–443. [Google Scholar] [CrossRef] - Balasubramanian, K. Mathematical and Computational Techniques for Drug Discovery: Promises and Developments. Curr. Top. Med. Chem.
**2019**, 18, 2774–2799. [Google Scholar] [CrossRef] - Patil, V.M.; Narkhede, R.R.; Masand, N.; Rameshwar, S.; Cheke, R.S.; Balasubramanian, K. Molecular insights into Resveratrol and its analogs as SARS-CoV-2 (COVID-19) protease inhibitors. Coronaviruses
**2020**, in press. [Google Scholar] - Available online: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/covid-19-vaccines (accessed on 25 November 2020).
- Javelle, E.; Raoult, D. COVID-19 pandemic more than a century after the Spanish flu. Lancet Infect. Dis.
**2020**. [Google Scholar] [CrossRef] - Dehmer, M.; Mowshowitz, A. A history of graph entropy measures. Inf. Sci.
**2011**, 181, 57–78. [Google Scholar] [CrossRef] - Mowshowitz, A.; Dehmer, M. Entropy and the Complexity of Graphs Revisited. Entropy
**2012**, 14, 559–570. [Google Scholar] [CrossRef] - Mowshowitz, A. Entropy and the complexity of graphs: I. An index of the relative complexity of a graph. Bull. Math. Biol.
**1968**, 30, 175–204. [Google Scholar] [CrossRef] - Ghorbani, M.; Dehmer, M.; Rahmani, S.; Rajabi-Parsa, M. A Survey on Symmetry Group of Polyhedral Graphs. Symmetry
**2020**, 12, 370. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**(

**a**): A 5-level pandemic tree ${T}_{5}^{3}$, with reproduction number 3. (

**b**): A 6-level pandemic tree ${T}_{6}^{6}$, with a reproduction number 6.

**Figure 3.**(

**a**): A 3-Cayley tree, C(3,6), of degree 3 with 6 levels. (

**b**): A 6-Cayley tree, C(6,6), of degree 6 with 6 levels.

**Figure 6.**log(Sz) values obtained for the Szeged index of pandemic trees ${T}_{l}^{k}$ demonstrate the underlying severity of COVID-19 as compared to the 1918 Spanish flu.

**Figure 7.**log(GA

_{4}) values obtained for the fourth geometric-arithmetic index of pandemic trees ${T}_{l}^{k}$ demonstrate the underlying severity of COVID-19 as compared to the 1918 Spanish flu.

**Figure 8.**Entropy values measured in the units of k

_{B}or S(l, k) obtained for pandemic tree ${T}_{l}^{k}$ accentuates the underlying severity of COVID-19 compared to the 1918 Spanish flu.

**Table 1.**Results obtained for the pandemic tree ${T}_{l}^{k}$ with the computer code, compared with the results from the expressions in Theorems 1 and 2.

Index | Dimension l, k | From Expressions 1 and 2 | Topo-Chemie-2020 [40] |
---|---|---|---|

GA_{4}$\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 38.8016 | 38.801700143357856 |

l = 3, k = 4 | 83.5947 | 83.59488962413361 | |

l = 4, k = 3 | 119.6838 | 119.6839561076481 | |

l = 4, k = 4 | 339.1496 | 339.1498266447425 | |

Zg_{4}$\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 399 | 399 |

l = 3, k = 4 | 876 | 876 | |

l = 4, k = 3 | 1692 | 1692 | |

l = 4, k = 4 | 4884 | 4884 | |

${\mathsf{\Pi}}_{4}^{*}\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 1.74212456 × 10^{39} | 1.742124563637115 × 10^{39} |

l = 3, k = 4 | 1.98337192 × 10^{85} | 1.9833719240465464 × 10^{85} | |

l = 4, k = 3 | 3.75944893 × 10^{137} | 3.759448938495563 × 10^{137} | |

l = 4, k = 4 | Infinity | +Inf | |

Zg_{6}$\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 1026 | 1026 |

l = 3, k = 4 | 2288 | 2288 | |

l = 4, k = 3 | 6000 | 6000 | |

l = 4, k = 4 | 17,584 | 17,584 | |

${\mathsf{\Pi}}_{6}^{*}\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 6.74664061 × 10^{54} | 6.746640616477462 × 10^{54} |

l = 3, k = 4 | 4.66622007 × 10^{119} | 4.666220065428966 × 10^{119} | |

l = 4, k = 3 | 4.24758724 × 10^{202} | 4.247587242244764 × 10^{202} | |

l = 4, k = 4 | Infinity | +Inf | |

ABC_{5}Π $\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 2.0850970590 × 10^{−10} | 2.0850970590395853 × 10^{−10} |

l = 3, k = 4 | 7.2443933360 × 10^{−22} | 7.2443933361227945 × 10^{−22} | |

l = 4, k = 3 | 2.6373891592 × 10^{−37} | 2.6373891591075294 × 10^{−37} | |

l = 4, k = 4 | 1.26583295 × 10^{−105} | 1.2658329528339406 × 10^{−105} | |

$PI\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 1560 | 1560 |

l = 3, k = 4 | 7140 | 7140 | |

l = 4, k = 3 | 14,520 | 14,520 | |

l = 4, k = 4 | 115,940 | 115,940 | |

$Sz\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 3402 | 3402 |

l = 3, k = 4 | 17,152 | 17,152 | |

l = 4, k = 3 | 44,712 | 44,712 | |

l = 4, k = 4 | 389,120 | 389,120 | |

${M}_{1}\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 228 | 228 |

l = 3, k = 4 | 580 | 580 | |

l = 4, k = 3 | 714 | 714 | |

l = 4, k = 4 | 2372 | 2372 | |

${M}_{2}\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 288 | 288 |

l = 3, k = 4 | 800 | 800 | |

l = 4, k = 3 | 936 | 936 | |

l = 4, k = 4 | 3360 | 3360 | |

$ABC\left({T}_{l}^{k}\right)$ | l = 3, k = 3 | 30.8308 | 30.83052949654569 |

l = 3, k = 4 | 68.6607 | 68.66073893642222 | |

l = 4, k = 3 | 94.13 | 94.12995706469174 | |

l = 4, k = 4 | 276.5956 | 276.59462680515827 |

**Table 2.**Results obtained for the Cayley Trees $C\left(k,l\right)$ with the computer code, compared with the results from the expressions in Theorems 3 and 4.

Index | Dimension k, l | Expressions 3 and 4 | Topo-Chemie-2020 [40] |
---|---|---|---|

GA_{4}$\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 20.8823882414 | 20.88238824139906 |

k = 3, l = 4 | 44.8676307844 | 44.86763078442802 | |

k = 4, l = 3 | 51.7356001911 | 51.73560019114378 | |

k = 4, l = 4 | 159.578608144 | 159.57860814353083 | |

Zg_{4}$\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 207 | 207 |

k = 3, l = 4 | 609 | 609 | |

k = 4, l = 3 | 532 | 532 | |

k = 4, l = 4 | 2256 | 2256 | |

${\mathsf{\Pi}}_{4}^{*}\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 5.72086104 × 10^{20} | 5.7208610362887386 × 10^{20} |

k = 3, l = 4 | 5.06517068 × 10^{50} | 5.065170680143067 × 10^{50} | |

k = 4, l = 3 | 2.09623003 × 10^{52} | 2.0962300336315637 × 10^{52} | |

k = 4, l = 4 | 2.71330692 × 10^{183} | 2.7133069091840815 × 10^{183} | |

Zg_{6}$\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 516 | 516 |

k = 3, l = 4 | 2088 | 2088 | |

k = 4, l = 3 | 1368 | 1368 | |

k = 4, l = 4 | 8000 | 8000 | |

${\mathsf{\Pi}}_{6}^{*}\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 5.87731231 × 10^{28} | 5.877312307199999 × 10^{28} |

k = 3, l = 4 | 1.5897237 × 10^{74} | 1.589723697730939 × 10^{74} | |

k = 4, l = 3 | 1.27482362 × 10^{73} | 1.2748236216396078 × 10^{73} | |

k = 4, l = 4 | 1.482028 × 10^{270} | 1.4820280048671849 × 10^{270} | |

ABC_{5}Π $\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 0.00000840649 | 0.000008406491899037542 |

k = 3, l = 4 | 4.4189082 × 10^{−14} | 4.419672639996076 × 10^{−14} | |

k = 4, l = 3 | 1.23642668 × 10^{−13} | 1.2364266794260217 × 10^{−13} | |

k = 4, l = 4 | 1.66132298 × 10^{−49} | 1.6913453312569212 × 10^{−49} | |

$PI\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 462 | 462 |

k = 3, l = 4 | 2070 | 2070 | |

k = 4, l = 3 | 2756 | 2756 | |

k = 4, l = 4 | 25,760 | 25,760 | |

$Sz\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 909 | 909 |

k = 3, l = 4 | 5661 | 5661 | |

k = 4, l = 3 | 6304 | 6304 | |

k = 4, l = 4 | 82,336 | 82,336 | |

${M}_{1}\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 102 | 102 |

k = 3, l = 4 | 222 | 222 | |

k = 4, l = 3 | 308 | 308 | |

k = 4, l = 4 | 956 | 956 | |

${M}_{2}\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 117 | 117 |

k = 3, l = 4 | 261 | 261 | |

k = 4, l = 3 | 400 | 400 | |

k = 4, l = 4 | 1264 | 1264 | |

$ABC\left(C\left(k,l\right)\right)$ | k = 3, l = 3 | 15.7979589714 | 15.797958971132715 |

k = 3, l = 4 | 33.5959179422 | 33.59591794226542 | |

k = 4, l = 3 | 40.9748735074 | 40.974873507372514 | |

k = 4, l = 4 | 125.374110265 | 125.37411026490041 |

**Table 3.**Results obtained for the Christmas trees, CT(s) with the computer code, compared with the results from the expressions in Theorems 5 and 6.

Index | Dimensions | From Expressions 5 and 6 | Topo-Chemie-2020 [40] |
---|---|---|---|

GA_{4}$\left(CT\left(s\right)\right)$ | s = 3 | 32.8823 | 32.882388241399056 |

s = 4 | 68.8671 | 68.86763078442803 | |

s = 5 | 140.8350 | 140.83501683597572 | |

Zg_{4}$\left(CT\left(s\right)\right)$ | s = 3 | 351 | 351 |

s = 4 | 993 | 993 | |

s = 5 | 2571 | 2571 | |

${\mathsf{\Pi}}_{4}^{*}\left(CT\left(s\right)\right)$ | s = 3 | 5.10073079 × 10^{33} | 5.10077716500643 × 10^{33} |

s = 4 | 4.01304165 × 10^{79} | 4.0130416580886176 × 10^{79} | |

s = 5 | 1.91126114 × 10^{177} | 1.9112611351919165 × 10^{177} | |

Zg_{6}$\left(CT\left(s\right)\right)$ | s = 3 | 948 | 948 |

s = 4 | 3624 | 3624 | |

s = 5 | 11,862 | 11,862 | |

${\mathsf{\Pi}}_{6}^{*}\left(CT\left(s\right)\right)$ | s = 3 | 2.7848947 × 10^{47} | 2.7848946955924445 × 10^{47} |

s = 4 | 3.54520226 × 10^{117} | 3.5452023119163584 × 10^{117} | |

s = 5 | 3.37859549 × 10^{269} | 3.3785954904846475 × 10^{269} | |

ABC_{5}Π $\left(CT\left(s\right)\right)$ | s = 3 | 3.8618890813 × 10^{−9} | 3.861889064427591 × 10^{−9} |

s = 4 | 5.3059950832 × 10^{−22} | 5.305995082939164 × 10^{−22} | |

s = 5 | 1.7589082206 × 10^{−50} | 1.7589082206095107 × 10^{−50} | |

${M}_{1}\left(CT\left(s\right)\right)$ | s = 3 | 198 | 198 |

s = 4 | 414 | 414 | |

s = 5 | 846 | 846 | |

${M}_{2}\left(CT\left(s\right)\right)$ | s = 3 | 297 | 297 |

s = 4 | 621 | 621 | |

s = 5 | 1269 | 1269 | |

$ABC\left(CT\left(s\right)\right)$ | s = 3 | 22 | 22.000000000000004 |

s = 4 | 46 | 45.99999999999997 | |

s = 5 | 94 | 94.00000000000011 |

**Table 4.**Results obtained for the corona product CT(s)ʘP

_{n}of a Christmas tree CT(s) and a path P

_{n}with the computer code, compared with the results from the expressions in Theorems 7 and 8.

Index | Dimension s, n | From Expressions 7 and 8 | Topo-Chemie-2020 [40] |
---|---|---|---|

GA_{4}(CT(s)ʘP_{n}) | s = 3, n = 3 | 142.7317 | 142.73175770071703 |

s = 4, n = 4 | 390.5798 | 390.5798602154981 | |

Zg_{4}(CT(s)ʘP_{n}) | s = 3, n = 3 | 1961 | 1961 |

s = 4, n = 4 | 6869 | 6869 | |

${\mathsf{\Pi}}_{4}^{*}$(CT(s)ʘP_{n}) | s = 3, n = 3 | 9.65986697 × 10^{161} | 9.659866968127555 × 10^{161} |

s = 4, n = 4 | Infinity | +Inf | |

Zg_{6}(CT(s)ʘP_{n}) | s = 3, n = 3 | 6824 | 6824 |

s = 4, n = 4 | 30,567 | 30,567 | |

${\mathsf{\Pi}}_{6}^{*}$(CT(s)ʘP_{n}) | s = 3, n = 3 | 4.3849717 × 10^{237} | 4.384971723725398 × 10^{237} |

s = 4, n = 4 | Infinity | +Inf | |

ABC_{5}Π(CT(s)ʘP_{n}) | s = 3,n = 3 | 1.4118921672 × 10^{−43} | 1.411892167070263 × 10^{−43} |

s = 4, n = 4 | 6.08702627 × 10^{−136} | 6.087026262790365 × 10^{−136} | |

${M}_{1}\left(CT\left(s\right)\mathsf{\u0298}Pn\right)$ | s = 3, n = 3 | 1166 | 1166 |

s = 4, n = 4 | 3450 | 3450 | |

${M}_{2}\left(CT\left(s\right)\mathsf{\u0298}Pn\right)$ | s = 3, n = 3 | 2376 | 2376 |

s = 4, n = 4 | 7567 | 7567 | |

$ABC\left(CT\left(s\right)\mathsf{\u0298}Pn\right)$ | s = 3, n = 3 | 93.33733429352 | 93.33733429351358 |

s = 4, n = 4 | 251.70409168311 | 251.70409168311375 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nandini, G.K.; Rajan, R.S.; Shantrinal, A.A.; Rajalaxmi, T.M.; Rajasingh, I.; Balasubramanian, K.
Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory. *Symmetry* **2020**, *12*, 1992.
https://doi.org/10.3390/sym12121992

**AMA Style**

Nandini GK, Rajan RS, Shantrinal AA, Rajalaxmi TM, Rajasingh I, Balasubramanian K.
Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory. *Symmetry*. 2020; 12(12):1992.
https://doi.org/10.3390/sym12121992

**Chicago/Turabian Style**

Nandini, G. Kirithiga, R. Sundara Rajan, A. Arul Shantrinal, T. M. Rajalaxmi, Indra Rajasingh, and Krishnan Balasubramanian.
2020. "Topological and Thermodynamic Entropy Measures for COVID-19 Pandemic through Graph Theory" *Symmetry* 12, no. 12: 1992.
https://doi.org/10.3390/sym12121992