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*Symmetry*
**2018**,
*10*(7),
237;
https://doi.org/10.3390/sym10070237

Article

On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers

^{1}

Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea

^{2}

Center for General Education, China Medical University, Taichung 40402, Taiwan

^{3}

School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad 44000, Pakistan

^{4}

University of Engineering and Technology, Lahore 54000, Pakistan (RCET)

^{5}

Department of Mathematics, Division of Science and Technology, University of Education, Lahore 54000, Pakistan

^{*}

Author to whom correspondence should be addressed.

Received: 26 May 2018 / Accepted: 21 June 2018 / Published: 24 June 2018

## Abstract

**:**

In the study of the quantitative structure–activity relationship and quantitative structure-property relationships, the eccentric-connectivity index has a very important place among the other topological descriptors due to its high degree of predictability for pharmaceutical properties. In this paper, we compute the exact formulas of the eccentric-connectivity index and its corresponding polynomial, the total eccentric-connectivity index and its corresponding polynomial, the first Zagreb eccentricity index, the augmented eccentric-connectivity index, and the modified eccentric-connectivity index and its corresponding polynomial for a class of phosphorus containing dendrimers.

Keywords:

eccentric-connectivity index; augmented eccentric-connectivity index; molecular graph; phosphorus containing dendrimersMSC:

05C90## 1. Introduction

Dendrimers are synthetic polymers with highly branched structures, consisting of multiple branched monomers radiating from a central core. Layers of monomers are attached stepwise during synthesis, with the number of branch points defining the generation of a dendrimer [1]. Different kinds of experiments have proved that these polymers with well-defined dimensional structures and topological architectures have an array of applications in medicine [2]. Nowadays, dendrimers are currently attracting the interest of a great number of scientists because of their unusual physical and chemical properties and their wide range of potential applications in different fields, such as physics, biology, chemistry, engineering, and medicine [3]. A topological index, sometimes known as a graph theoretic index, is a numerical invariant of a chemical graph. Topological indices are the mathematical measures associated with molecular graph structures that correlate a chemical structure with various physical properties, biological activities or chemical reactivities. A topological index is an invariant of a graph, ${G}_{1}$; that is, if $Top\left({G}_{1}\right)$ denotes a topological index of graph ${G}_{1}$, and if ${G}_{2}$ is another graph such that ${G}_{1}\cong {G}_{2}$, then $Top\left({G}_{1}\right)=Top\left({G}_{2}\right)$. In chemistry, biochemistry and nanotechnology, distance-based topological indices of graphs are useful in isomer discrimination, structure–property relationships and structure–activity relationships.

## 2. Definitions and Notations

Let G be a connected and simple molecular graph with vertex set, $V\left(G\right)$, and edge set, $E\left(G\right)$. The vertices of G correspond to atoms, and an edge between two vertices corresponds to the chemical bond between these vertices. In graph G, two vertices, u and v, are adjacent, if and only if, they are the end vertices of an edge, $e\in E\left(G\right)$, and we write $e=uv$ or $e=vu$. For a vertex, u, the set of neighbor vertices is denoted by ${N}_{u}$ and is defined as ${N}_{u}=\{v\in V\left(G\right):uv\in E\left(G\right)\}$. The degree of vertex $u\in V\left(G\right)$ is denoted by ${d}_{u}$ and is defined as ${d}_{u}=\left|{N}_{u}\right|$. Let ${S}_{u}$ denote the sum of the degrees of all neighbors of vertex u, that is ${S}_{u}={\sum}_{v\in {N}_{u}}{d}_{v}$. A $({u}_{1},{u}_{n})$-path on n vertices is defined as a graph with vertex set, $\{{u}_{i}:1\le i\le n\}$, and edge set, $\{{u}_{i}{u}_{i+1}:1\le i\le n-1\}$. The distance, $d(u,v)$, between two vertices, $u,v\in V\left(G\right)$, is defined as the length of the shortest $(u,v)$-path in G. For a given vertex, $v\in V\left(G\right)$, the eccentricity, $\epsilon \left(v\right)$, is defined as the largest distance between v and any other vertex, u in G. In 1947, Harold Wiener published a paper entitled “Structural Determination of Paraffin Boiling Points” [4]. In this work, the quantity, ${W}_{e}$, eventually named the Wiener index or Wiener number, was introduced for the first time, and he showed that there are excellent correlations between ${W}_{e}$ and a variety of physico-chemical properties of organic compounds. Another distance-based topological index of the graph G is the eccentric-connectivity index, $\xi \left(G\right)$, which is defined as [5]

$$\xi \left(G\right)=\sum _{u\in V\left(G\right)}^{}\epsilon \left(u\right){d}_{u}.$$

Different applications and mathematical properties of this index were discussed in [6,7,8,9]. For a graph, G, the eccentric-connectivity polynomial in variable y is defined as [10]

$$ECP(G,y)=\sum _{u\in V\left(G\right)}{d}_{u}{y}^{\epsilon \left(u\right)}.$$

The total eccentricity index of a graph, G, is expressed as follows:

$$\varsigma \left(G\right)=\sum _{u\in V\left(G\right)}\epsilon \left(u\right).$$

The total eccentric-connectivity polynomial in variable y of a graph, G, is defined as [10]

$$TECP(G,y)=\sum _{u\in V\left(G\right)}{y}^{\epsilon \left(u\right)}.$$

The first Zagreb index of a graph, G, in terms of eccentricity was given by Ghorbani and Hosseinzadeh [11], as follows:

$${M}_{1}^{**}\left(G\right)=\sum _{u\in V\left(G\right)}{\left(\epsilon \left(u\right)\right)}^{2}.$$

Gupta and his co-authors [12] introduced the augmented eccentric-connectivity index of a graph, G, and it is defined as
where $M\left(u\right)$ denotes the product of degrees of all neighbors of vertex u. Various properties of this index have been studied in [13,14]. For a graph, G, the modified versions of the eccentric-connectivity index and its polynomial are defined as follows

$${}^{A}\epsilon \left(G\right)=\sum _{u\in V\left(G\right)}\frac{M\left(u\right)}{\epsilon \left(u\right)},$$

$$\mathsf{\Lambda}\left(G\right)=\sum _{u\in V\left(G\right)}{S}_{u}\epsilon \left(u\right),$$

$$MECP(G,y)=\sum _{u\in V\left(G\right)}{S}_{u}{y}^{\epsilon \left(u\right)}.$$

Several mathematical and chemical properties of the modified eccentric-connectivity index and its polynomial were studied in [10,15]. Some major types of topological indices of graphs are degree-based, distance-based, and counting-related. Some degree-based topological indices have been computed for some classes of dendrimers, see for instance [16,17,18]. For a study of distance-based topological indices, see [19,20,21]. In this paper, we compute several distance-based indices, namely, the eccentric-connectivity index, the total eccentric-connectivity index, and the modified eccentric-connectivity index for the phosphorus-containing dendrimer Cyclotriphosphazene $\left({N}_{3}{P}_{3}\right)$ [22]. We also compute the corresponding polynomials of these indices for the same dendrimer. We also compute the first Zagreb eccentricity index and the augmented eccentric-connectivity index for the said dendrimer.

## 3. The Eccentricity-Based Indices and Polynomials for the Molecular Graph

Let the molecular graph of this dendrimer be $D\left(n\right)$, where the generation stage of $D\left(n\right)$ is represented by n. The first and second generations are shown in Figure 1 and Figure 2 respectively.

The size and order of graph $D\left(n\right)$ are $6(9\times {2}^{n+2}-13)$ and $9(-8+11\times {2}^{n})$, respectively. To compute the eccentricity-based indices and polynomials of $D\left(n\right)$, it is enough to compute the required information for a set of representatives of $V\left(D\right(n\left)\right)$. We will compute the required information by using computational arguments. We make three sets of representatives of $V\left(D\right(n\left)\right)$, say $A=\{{\alpha}_{1},{\alpha}_{2}\},$$B=\{{\beta}_{1},{\beta}_{2},\cdots ,{\beta}_{13}\}$ and $C=\{{a}_{i},{b}_{i},{c}_{i},{d}_{i},{e}_{i},{f}_{i},{g}_{i},{h}_{i},{j}_{i},{k}_{i},{l}_{i}\}$ where $1\le i\le n$, as shown in Figure 1 and Figure 2. The degree, ${S}_{u}$, $M\left(u\right)$, and eccentricity for each u for the sets A, B, and C are shown in Table 1 and Table 2. For simplicity, we assume $\gamma =9n+9i$ throughout the paper. By using Table 1 and Table 2, we calculate the different eccentricity-based indices and their corresponding polynomials. In the following theorem, we determine the eccentric-connectivity index of $D\left(n\right)$.

**Theorem**

**1.**

For graph $D\left(n\right)$, the eccentric-connectivity index is given by

$$\xi \left(D\left(n\right)\right)=18({2}^{n+2}\times 79-78n+{2}^{n}\times 303n+1).$$

**Proof.**

By putting the values of Table 1 and Table 2 into Equation (1), the eccentric-connectivity index of $D\left(n\right)$ can be written as follows:

$$\begin{array}{cc}\hfill \xi \left(D\right(n\left)\right)& =\xi \left(A\right)+\xi \left(B\right)+\xi \left(C\right)=\sum _{u\in A}\epsilon \left(u\right){d}_{u}+\sum _{u\in B}\epsilon \left(u\right){d}_{u}+\sum _{u\in C}\epsilon \left(u\right){d}_{u}\hfill \\ & =(2\times 3)(9n+15)+(3\times 4)(9n+14)+(3\times {2}^{n+1}\times 2)(9n+15)\hfill \\ & +(3\times {2}^{n+1}\times 3)(9n+16)+(2\times {2}^{n+2}\times 3)(9n+17)+(2\times {2}^{n+2}\times 3)(9n+18)\hfill \\ & +(3\times {2}^{n+1}\times 3)(9n+19)+(2\times {2}^{n+1}\times 3)(9n+20)+(2\times {2}^{n+1}\times 3)(9n+21)\hfill \\ & +(2\times {2}^{n+2}\times 3)(9n+23)+(4\times {2}^{n+2}\times 3)(9n+24)+(1\times {2}^{n+3}\times 3)(9n+25)\hfill \\ & +(3\times {2}^{n+1}\times 3)(9n+22)+(3\times {2}^{n+1}\times 3)(9n+25)+(1\times {2}^{n+1}\times 3)(9n+26)\hfill \\ & +\sum _{i=1}^{n}((2\times {2}^{i}\times 3)(\gamma +6)+(3\times {2}^{i}\times 3)(\gamma +7)+(2\times {2}^{i+1}\times 3)(\gamma +8)\hfill \\ & +({2}^{i+2}\times 3)(\gamma +9)+(3\times {2}^{i}\times 3)(\gamma +10)+(3\times {2}^{i+1})(\gamma +11)+({2}^{i+1}\times 3)(\gamma +12)\hfill \\ & +(3\times {2}^{i}\times 3)(\gamma +13)+({2}^{i}\times 3)(\gamma +14)+(4\times {2}^{i}\times 3)(\gamma +14)+({2}^{i}\times 3)(\gamma +15)).\hfill \end{array}$$

After some calculations, we get
which completes the theorem. ☐

$$\xi \left(D\left(n\right)\right)=18({2}^{n+2}\times 79-78n+{2}^{n}\times 303n+1),$$

When the degrees of vertices are not taken into account, then by using the values of Table 1 and Table 2 in (3), we have the following result.

**Corollary**

**1.**

For graph $D\left(n\right)$, the total eccentric-connectivity index is given by

$$\varsigma \left(D\left(n\right)\right)=9({2}^{n+2}\times 69n+{2}^{n+1}\times 149-72n-3).$$

In the next theorem, the eccentric-connectivity polynomial for the molecular graph is derived.

**Theorem**

**2.**

For graph $D\left(n\right)$, the eccentric-connectivity polynomial is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill ECP\left(D\right(n),y)& =6{y}^{9n+14}(y+2)+3\times {2}^{n+1}{y}^{9n+15}({y}^{11}+7{y}^{10}+8{y}^{9}+4{y}^{8}+3{y}^{7}+2{y}^{6}+2{y}^{5}\hfill \\ & +3{y}^{4}+4{y}^{3}+4{y}^{2}+3y+2)+\frac{6({y}^{3}+5{y}^{2}+3y+2)\times {y}^{9n+21}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}\hfill \\ & +\frac{6(2{y}^{5}+3{y}^{4}+4{y}^{3}+4{y}^{2}+3y+2)\times {y}^{9n+15}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}.\hfill \end{array}\end{array}$$

**Proof.**

By using Table 1 and Table 2 in (2), we have

$$\begin{array}{cc}\hfill ECP\left(D\right(n),y)& =ECP(A,y)+ECP(B,y)+ECP(C,y)\hfill \\ & =\sum _{u\in A}{d}_{u}{y}^{\epsilon \left(u\right)}+\sum _{u\in B}{d}_{u}{y}^{\epsilon \left(u\right)}+\sum _{u\in C}{d}_{u}{y}^{\epsilon \left(u\right)}\hfill \\ & =(2\times 3){y}^{9n+15}+(4\times 3){y}^{9n+14}+(3\times {2}^{n+2}){y}^{9n+15}+(3\times 3\times {2}^{n+1}){y}^{9n+16}\hfill \\ & +(2\times 3\times {2}^{n+2}){y}^{9n+17}+(2\times 3\times {2}^{n+2}){y}^{9n+18}+(3\times 3\times {2}^{n+1}){y}^{9n+19}\hfill \\ & +(2\times 3\times {2}^{n+1}){y}^{9n+20}+(2\times 3\times {2}^{n+1}){y}^{9n+21}+(3\times 3\times {2}^{n+1}){y}^{9n+22}\hfill \\ & +(2\times 3\times {2}^{n+2}){y}^{9n+23}+(4\times 3\times {2}^{n+2}){y}^{9n+24}+(1\times 3\times {2}^{n+3}){y}^{9n+25}\hfill \\ & +(3\times 3\times {2}^{n+1}){y}^{9n+25}+(1\times 3\times {2}^{n+1}){y}^{9n+26}+\sum _{i=1}^{n}((2\times 3\times {2}^{i}){y}^{\gamma +6}\hfill \\ & +(2\times 3\times {2}^{i+1}){y}^{\gamma +8}+(2\times 3\times {2}^{i+1}){y}^{\gamma +9}+(3\times 3\times {2}^{i}){y}^{\gamma +10}\hfill \\ & +(2\times 3\times {2}^{i}){y}^{\gamma +11}+(3\times 3\times {2}^{i}){y}^{\gamma +7}+(2\times 3\times {2}^{i}){y}^{\gamma +12}\hfill \\ & +(3\times 3\times {2}^{i}){y}^{\gamma +13}+(3\times {2}^{i}){y}^{\gamma +14}+(4\times 3\times {2}^{i}){y}^{\gamma +14}+(3\times {2}^{i}){y}^{\gamma +15}).\hfill \end{array}$$

After some calculations, we get the required result. ☐

**Corollary**

**2.**

For graph $D\left(n\right)$, the total eccentric-connectivity polynomial is given by

$$\begin{array}{cc}\hfill TECP\left(D\right(n),y)& =3{y}^{9n+14}(y+1)+3\times {2}^{n+1}{y}^{9n+15}({y}^{11}+5{y}^{10}+2{y}^{9}+2{y}^{8}+{y}^{7}+{y}^{6}+{y}^{5}\hfill \\ & +{y}^{4}+2{y}^{3}+2{y}^{2}+y+1)+\frac{6({y}^{3}+2{y}^{2}+y+1)\times {y}^{9n+21}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}\hfill \\ & +\frac{6(y+1){({y}^{2}+1)}^{2}\times {y}^{9n+15}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}.\hfill \end{array}$$

In the next theorem, we compute the closed formula for the first Zagreb eccentricity index.

**Theorem**

**3.**

For graph $D\left(n\right)$, the first Zagreb eccentricity index is given by

$${M}_{1}^{**}\left(D\left(n\right)\right)=3({2}^{n+4}\times 7295{n}^{2}+{2}^{n+3}\times 2097n-1944{n}^{2}-162n+{2}^{n+1}\times 11641-4053).$$

**Proof.**

By using the values of Table 1 and Table 2 in (5), we compute the first Zagreb eccentricity index of $D\left(n\right)$ as follows:

$$\begin{array}{cc}\hfill {M}_{1}^{**}\left(D\left(n\right)\right)& ={M}_{1}^{**}\left(A\right)+{M}_{1}^{**}\left(B\right)+{M}_{1}^{**}\left(C\right)=\sum _{v\in A}{\left[\epsilon \left(v\right)\right]}^{2}+\sum _{v\in B}{\left[\epsilon \left(v\right)\right]}^{2}+\sum _{v\in C}{\left[\epsilon \left(v\right)\right]}^{2}\hfill \\ & =3{(9n+15)}^{2}+3{(9n+14)}^{2}+(3\times {2}^{n+1}){(9n+15)}^{2}+(3\times {2}^{n+1}){(9n+16)}^{2}\hfill \\ & +(3\times {2}^{n+2}){(9n+17)}^{2}+(3\times {2}^{n+2}){(9n+18)}^{2}+(3\times {2}^{n+1}){(9n+19)}^{2}\hfill \\ & +(3\times {2}^{n+1}){(9n+20)}^{2}+(3\times {2}^{n+1}){(9n+21)}^{2}+(3\times {2}^{n+1}){(9n+22)}^{2}\hfill \\ & +(3\times {2}^{n+2}){(9n+23)}^{2}+(3\times {2}^{n+2}){(9n+24)}^{2}+(3\times {2}^{n+3}){(9n+25)}^{2}\hfill \\ & +(3\times {2}^{n+1}){(9n+25)}^{2}+(3\times {2}^{n+1}){(9n+26)}^{2}+\sum _{i=1}^{n}((3\times {2}^{i}){(\gamma +6)}^{2}\hfill \\ & +(3\times {2}^{i}){(\gamma +7)}^{2}+(3\times {2}^{i+1}){(\gamma +8)}^{2}+(3\times {2}^{i+1}){(\gamma +9)}^{2}+(3\times {2}^{i}){(\gamma +10)}^{2}\hfill \\ & +(3\times {2}^{i}){(\gamma +11)}^{2}+(3\times {2}^{i}){(\gamma +12)}^{2}+(3\times {2}^{i}){(\gamma +13)}^{2}+(3\times {2}^{i}){(\gamma +14)}^{2}\hfill \\ & +(3\times {2}^{i}){(\gamma +14)}^{2}+(3\times {2}^{i}){(\gamma +15)}^{2}).\hfill \end{array}$$

After some calculations, we obtain
which finishes the theorem. ☐

$${M}_{1}^{**}\left(D\left(n\right)\right)=3({2}^{n+4}\times 7295{n}^{2}+{2}^{n+3}\times 2097n-1944{n}^{2}-162n+{2}^{n+1}\times 11,641-4053),$$

We determine the augmented eccentric-connectivity index in the next theorem.

**Theorem**

**4.**

For graph $D\left(n\right)$, the augmented eccentric-connectivity index is given by

$$\begin{array}{cc}\hfill {}^{A}\epsilon \left(D\left(n\right)\right)& =\frac{48}{9n+15}+\frac{48}{9n+14}+\frac{36\times {2}^{n+1}}{9n+15}+\frac{24\times {2}^{n+1}}{9n+16}+\frac{18\times {2}^{n+2}}{9n+17}+\frac{18\times {2}^{n+2}}{9n+18}\hfill \\ & +\frac{24\times {2}^{n+1}}{9n+19}+\frac{18\times {2}^{n+1}}{9n+20}+\frac{18\times {2}^{n+1}}{9n+21}+\frac{24\times {2}^{n+1}}{9n+22}+\frac{36\times {2}^{n+2}}{9n+23}+\frac{18\times {2}^{n+2}}{9n+24}\hfill \\ & +\frac{12\times {2}^{n+3}}{9n+25}+\frac{48\times {2}^{n+1}}{9n+25}+\frac{9\times {2}^{n+1}}{9n+26}+\left(\frac{72}{9n+15}+\cdots +\frac{36\times {2}^{n}}{18n+6}\right)\hfill \\ & +\left(\frac{48}{9n+16}+\cdots +\frac{24\times {2}^{n}}{18n+7}\right)+\left(\frac{72}{9n+17}+\cdots +\frac{18\times {2}^{n+1}}{18n+8}\right)\hfill \\ & +\left(\frac{72}{9n+18}+\cdots +\frac{18\times {2}^{n+1}}{18n+9}\right)+\left(\frac{48}{9n+19}+\cdots +\frac{24\times {2}^{n}}{18n+10}\right)\hfill \\ & +\left(\frac{36}{9n+20}+\cdots +\frac{18\times {2}^{n}}{18n+11}\right)+\left(\frac{36}{9n+21}+\cdots +\frac{18\times {2}^{n}}{18n+12}\right)\hfill \\ & +\left(\frac{48}{9n+22}+\cdots +\frac{24\times {2}^{n}}{18n+13}\right)+\left(\frac{18}{9n+23}+\cdots +\frac{9\times {2}^{n}}{18n+14}\right)\hfill \\ & +\left(\frac{72}{9n+23}+\cdots +\frac{36\times {2}^{n}}{18n+14}\right)+\left(\frac{24}{9n+24}+\cdots +\frac{12\times {2}^{n}}{18n+15}\right).\hfill \end{array}$$

**Proof.**

By using the values of Table 1 and Table 2 in (6), we compute the augumented eccentric-connectivity index of $D\left(n\right)$ in the following way:

$$\begin{array}{cc}\hfill {}^{A}\epsilon \left(D\left(n\right)\right)& {=}^{A}\epsilon \left(A\right){+}^{A}\epsilon \left(B\right){+}^{A}\epsilon \left(C\right)=\sum _{u\in A}\frac{M\left(u\right)}{\epsilon \left(u\right)}+\sum _{u\in B}\frac{M\left(u\right)}{\epsilon \left(u\right)}+\sum _{u\in C}\frac{M\left(u\right)}{\epsilon \left(u\right)}\hfill \\ & =\frac{3\times 16}{9n+15}+\frac{3\times 16}{9n+14}+\frac{3\times {2}^{n+1}\times 12}{9n+15}+\frac{3\times {2}^{n+1}\times 8}{9n+16}+\frac{3\times {2}^{n+2}\times 6}{9n+17}\hfill \\ & +\frac{3\times {2}^{n+2}\times 6}{9n+18}+\frac{3\times {2}^{n+1}\times 8}{9n+19}+\frac{3\times {2}^{n+1}\times 6}{9n+20}+\frac{3\times {2}^{n+1}\times 6}{9n+21}\hfill \\ & +\frac{3\times {2}^{n+1}\times 8}{9n+22}+\frac{3\times {2}^{n+2}\times 12}{9n+23}+\frac{3\times {2}^{n+2}\times 6}{9n+24}+\frac{3\times {2}^{n+3}\times 4}{9n+25}\hfill \\ & +\frac{3\times {2}^{n+1}\times 16}{9n+25}+\frac{3\times {2}^{n+1}\times 3}{9n+26}+\sum _{i=1}^{n}(\frac{3\times {2}^{i}\times 12}{\gamma +6}+\frac{3\times {2}^{i}\times 8}{\gamma +7}\hfill \\ & +\frac{3\times {2}^{i+1}\times 6}{\gamma +8}+\frac{3\times {2}^{i+1}\times 6}{\gamma +9}+\frac{3\times {2}^{i}\times 8}{\gamma +10}+\frac{3\times {2}^{i}\times 6}{\gamma +11}+\frac{3\times {2}^{i}\times 6}{\gamma +12}\hfill \\ & +\frac{3\times {2}^{i}\times 8}{\gamma +13}+\frac{3\times {2}^{i}\times 3}{\gamma +14}+\frac{3\times {2}^{i}\times 12}{\gamma +14}+\frac{3\times {2}^{i}\times 4}{\gamma +15}).\hfill \end{array}$$

After some calculations, we obtain the required result. ☐

Now, we compute the closed formula for the modified eccentric-connectivity index.

**Theorem**

**5.**

For graph $D\left(n\right)$, the modified eccentric-connectivity index is given by

$$\mathsf{\Lambda}\left(D\left(n\right)\right)=6({2}^{n}\times 2277n-567n+{2}^{n+1}\times 1229+21).$$

**Proof.**

By using the values of Table 1 and Table 2 in (7), we compute the modified eccentric-connectivity index of $D\left(n\right)$ in the following way:

$$\begin{array}{cc}\hfill \mathsf{\Lambda}\left(D\right(n\left)\right)& =\mathsf{\Lambda}\left(A\right)+\mathsf{\Lambda}\left(B\right)+\mathsf{\Lambda}\left(C\right)=\sum _{u\in A}{S}_{u}\epsilon \left(u\right)+\sum _{u\in B}{S}_{u}\epsilon \left(u\right)+\sum _{u\in C}{S}_{u}\epsilon \left(u\right)\hfill \\ & =(8\times 3)(9n+15)+(8\times 3)(9n+14)+(7\times 3\times {2}^{n+1})(9n+15)\hfill \\ & +(5\times 3\times {2}^{n+2})(9n+17)+(5\times 3\times {2}^{n+2})(9n+18)+(6\times 3\times {2}^{n+1})(9n+19)\hfill \\ & +(5\times 3\times {2}^{n+1})(9n+20)+(5\times 3\times {2}^{n+1})(9n+21)+(6\times 3\times {2}^{n+1})(9n+22)\hfill \\ & +(7\times 3\times {2}^{n+2})(9n+23)+(7\times 3\times {2}^{n+2})(9n+24)+(4\times 3\times {2}^{n+3})(9n+25)\hfill \\ & +(9\times 3\times {2}^{n+1})(9n+25)+(3\times 3\times {2}^{n+1})(9n+26)+(6\times 3\times {2}^{n+1})(9n+16)\hfill \\ & +\sum _{i=1}^{n}((7\times 3\times {2}^{i})(\gamma +6)+(6\times 3\times {2}^{i})(\gamma +7)+(5\times 3\times {2}^{i+1})(\gamma +8)\hfill \\ & +(5\times 3\times {2}^{i+1})(\gamma +9)+(6\times 3\times {2}^{i})(\gamma +10)+(5\times 3\times {2}^{i})(\gamma +11)\hfill \\ & +(5\times 3\times {2}^{i})(\gamma +12)+(7\times 3\times {2}^{i})(\gamma +13)+(3\times 3\times {2}^{i})(\gamma +14)\hfill \\ & +(8\times 3\times {2}^{i})(\gamma +14)+(4\times 3\times {2}^{i})(\gamma +15)).\hfill \end{array}$$

After some calculations, we obtain
which completes the proof. ☐

$$\mathsf{\Lambda}\left(D\left(n\right)\right)=6({2}^{n}\times 2277n-567n+{2}^{n+1}\times 1229+21),$$

Finally, we compute the closed formula for the modified eccentric-connectivity polynomial.

**Theorem**

**6.**

For graph $D\left(n\right)$, the modified eccentric-connectivity polynomial is given by

$$\begin{array}{cc}\hfill MECP\left(D\right(n),y)& =24{y}^{9n+14}(y+1)+{2}^{n+1}\times {y}^{9n+15}(9{y}^{11}+75{y}^{10}+42{y}^{9}+42{y}^{8}\hfill \\ & +18{y}^{7}+15{y}^{6}+15{y}^{5}+18{y}^{4}+30{y}^{3}+30{y}^{2}+18y+21)\hfill \\ & +\frac{6(5{y}^{5}+6{y}^{4}+10{y}^{3}+10{y}^{2}+6y+7){y}^{9n+15}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}\hfill \\ & +\frac{6(4{y}^{3}+11{y}^{2}+7y+5){y}^{9n+21}({2}^{n}{y}^{9n}-1)}{2{y}^{9}-1}.\hfill \end{array}$$

**Proof.**

By using the values of Table 1 and Table 2 in (8), we compute the modified eccentric-connectivity polynomial of $D\left(n\right)$ in the following way:

$$\begin{array}{cc}\hfill MECP\left(D\right(n),y)& =MECP(A,y)+MECP(B,y)+MECP(C,y)\hfill \\ & =\sum _{u\in A}{S}_{u}{y}^{\epsilon \left(u\right)}+\sum _{u\in B}{S}_{u}{y}^{\epsilon \left(u\right)}+\sum _{u\in C}{S}_{u}{y}^{\epsilon \left(u\right)}\hfill \\ & =(8\times 3){y}^{9n+15}+(8\times 3){y}^{9n+14}+(7\times 3\times {2}^{n+1}){y}^{9n+15}\hfill \\ & +(6\times 3\times {2}^{n+1}){y}^{9n+16}+(5\times 3\times {2}^{n+2}){y}^{9n+17}+(5\times 3\times {2}^{n+2}){y}^{9n+18}\hfill \\ & +(6\times 3\times {2}^{n+1}){y}^{9n+19}+(5\times 3\times {2}^{n+1}){y}^{9n+20}+(5\times 3\times {2}^{n+1}){y}^{9n+21}\hfill \\ & +(6\times 3\times {2}^{n+1}){y}^{9n+22}+(7\times 3\times {2}^{n+2}){y}^{9n+23}+(7\times 3\times {2}^{n+2}){y}^{9n+24}\hfill \\ & +(4\times 3\times {2}^{n+3}){y}^{9n+25}+(9\times 3\times {2}^{n+1}){y}^{9n+25}+(3\times 3\times {2}^{n+1}){y}^{9n+26}\hfill \\ & +\sum _{i=1}^{n}((7\times 3\times {2}^{i})\left({y}^{\gamma +6}\right)+(6\times 3\times {2}^{i})\left({y}^{\gamma +7}\right)+(5\times 3\times {2}^{i+1})\left({y}^{\gamma +8}\right)\hfill \\ & +(5\times 3\times {2}^{i+1})\left({y}^{\gamma +9}\right)+(6\times 3\times {2}^{i})\left({y}^{\gamma +10}\right)+(5\times 3\times {2}^{i})\left({y}^{\gamma +11}\right)\hfill \\ & +(5\times 3\times {2}^{i})\left({y}^{\gamma +12}\right)+(7\times 3\times {2}^{i})\left({y}^{\gamma +13}\right)+(3\times 3\times {2}^{i})\left({y}^{\gamma +14}\right)\hfill \\ & +(8\times 3\times {2}^{i})\left({y}^{\gamma +14}\right)+(4\times 3\times {2}^{i})\left({y}^{\gamma +15}\right)).\hfill \end{array}$$

After some calculations, we obtain the required result. ☐

## 4. Conclusions

In this paper we discussed the theoretical topics in molecular science and computed the eccentric topological indices for a class of phosphorus-containing dendrimers in regard to their molecular structure analysis, distance computing and mathematical derivation. Phosphorus-containing dendrimers have various applications in nanomedicine and materials science; therefore, these theoretical results could have applications in medical science.

## Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

## Funding

This research received no external funding.

## Acknowledgments

This work was supported by Higher Education Commission Pakistan.

## Conflicts of Interest

The authors declare no conflict of interest. We are thankful to both reviewers and editor for positive suggestions that improve the quality of this paper.

## References

- Adronov, A.; Frechet, J.M.J. Light-harvesting dendrimers. Chem. Commun.
**2000**, 33, 1701–1710. [Google Scholar] [CrossRef] - Naka, K.; Tanaka, Y.; Chujo, Y. Effect of anionic starburst dendrimers on the crystallization of CaCO
_{3}in aqueous solution, Size control of spherical vaterite particles. Langmuir**2002**, 18, 3655–3658. [Google Scholar] [CrossRef] - Suresh, R.; Singh, C.; Rewar, P. Dendrimers as carriers and its application in therapy. Int. J. Anal. Pharm. Biomed. Sci.
**2015**, 4, 15–23. [Google Scholar] - Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc.
**1947**, 69, 17–20. [Google Scholar] [CrossRef] [PubMed] - Sharma, V.; Goswami, R.; Madan, A.K. Eccentric-connectivity index: A novel highly discriminating topological descriptor for structure–property and structure–activity studies. J. Chem. Inf. Comput. Sci.
**1997**, 37, 273–282. [Google Scholar] [CrossRef] - Dureja, H.; Madan, A.K. Topochemical models for prediction of cyclin-dependent kinase 2 inhibitory activity of indole-2-ones. J. Mol. Model.
**2005**, 11, 525–531. [Google Scholar] [CrossRef] [PubMed] - Ilic, A.; Gutman, I. Eccentric-connectivity index of chemical trees. MATCH Commun. Math. Comput. Chem.
**2011**, 65, 731–744. [Google Scholar] - Kumar, V.; Madan, A.K. Application of graph theory: Prediction of cytosolic phospholipase A(2) inhibitory activity of propan-2-ones. J. Math. Chem.
**2006**, 39, 511–521. [Google Scholar] [CrossRef] - Zhou, B. On eccentric-connectivity index. MATCH Commun. Math. Comput. Chem.
**2010**, 63, 181–198. [Google Scholar] - Ashrafi, A.R.; Ghorbani, M.; Hossein-Zadeh, M.A. The eccentric-connectivity polynomial of some graph operations. Serdica J. Comput.
**2011**, 5, 101–116. [Google Scholar] - Ghorbani, M.; Hosseinzadeh, M.A. A new version of Zagreb indices. Filomat
**2012**, 26, 93–100. [Google Scholar] [CrossRef] - Gupta, S.; Singh, M.; Madan, A. K Connective eccentricity index: A novel topological descriptor for predicting biological activity. J. Mol. Graph. Model.
**2000**, 18, 18–25. [Google Scholar] [CrossRef] - De, N. Relationship between augmented eccentric-connectivity index and some other graph invariants. Int. J. Adv. Math.
**2013**, 1, 26–32. [Google Scholar] [CrossRef] - Doślić, T.; Saheli, M. Augmented eccentric-connectivity index. Miskolc Math. Notes
**2011**, 12, 149–157. [Google Scholar] - Alaeiyan, M.; Asadpour, J.; Mojarad, R. A numerical method for MEC polynomial and MEC index of one-pentagonal carbon nanocones. Fuller. Nanotub. Carbon Nanostruct.
**2013**, 21, 825–835. [Google Scholar] [CrossRef] - Aslam, A.; Jamil, M.K.; Gao, W.; Nazeer, W. Topological aspects of some dendrimer structures. Nanotechnol. Rev.
**2018**, 7, 123–129. [Google Scholar] [CrossRef] - Aslam, A.; Bashir, Y.; Ahmad, S.; Gao, W. On Topological Indices of Certain Dendrimer Structures. Z. Naturforschung A
**2017**, 72, 559–566. [Google Scholar] [CrossRef] - Bashir, Y.; Aslam, A.; Kamran, M.; Qureshi, M.I.; Jahangir, A.; Rafiq, M.; Bibi, N.; Muhammad, N. On forgotten topological indices of some dendrimers structure. Molecules
**2017**, 22, 867. [Google Scholar] [CrossRef] [PubMed] - Soleimania, N.; Bahnamirib, S.B.; Nikmehr, M.J. Study of dendrimers by topological indices. ACTA CHEMICA IASI
**2017**, 25, 145–162. [Google Scholar] [CrossRef] - Wu, H.; Zhao, B.; Gao, W. Distance indices calculating for two classes of dendrimer. Geol. Ecol. Landsc.
**2017**, 1, 133–142. [Google Scholar] [CrossRef] - Yang, J.; Xia, F. The eccentric connectivity index of dendrimers. Int. J. Contemp. Math. Sci.
**2010**, 5, 2231–2236. [Google Scholar] - Badetti, E.; Lloveras, V.; Muñoz-Gómez, J.L.; Sebastián, R.M.; Camimade, A.M.; Majoral, J.P.; Veciana, J.; Vidal-Gancedo, J. Radical dendrimers: A family of five generations of phosphorus dendrimers functionalized with TEMPO radicals. Macromolecules
**2014**, 47, 7717–7724. [Google Scholar] [CrossRef]

**Table 1.**Sets A and B with their degrees, ${S}_{u}$, $M\left(u\right)$, eccentricities, and frequencies.

Representative | Degree | ${\mathit{S}}_{\mathit{u}}$ | $\mathit{M}\left(\mathit{u}\right)$ | Eccentricity | Frequency |
---|---|---|---|---|---|

${\alpha}_{1}$ | 2 | 8 | 16 | $9n+15$ | 3 |

${\alpha}_{2}$ | 4 | 8 | 16 | $9n+14$ | 3 |

${\beta}_{1}$ | 2 | 7 | 12 | $9n+15$ | $3\times {2}^{n+1}$ |

${\beta}_{2}$ | 3 | 6 | 8 | $9n+16$ | $3\times {2}^{n+1}$ |

${\beta}_{3}$ | 2 | 5 | 6 | $9n+17$ | $3\times {2}^{n+2}$ |

${\beta}_{4}$ | 2 | 5 | 6 | $9n+18$ | $3\times {2}^{n+2}$ |

${\beta}_{5}$ | 3 | 6 | 8 | $9n+19$ | $3\times {2}^{n+1}$ |

${\beta}_{6}$ | 2 | 5 | 6 | $9n+20$ | $3\times {2}^{n+1}$ |

${\beta}_{7}$ | 2 | 5 | 6 | $9n+21$ | $3\times {2}^{n+1}$ |

${\beta}_{8}$ | 3 | 6 | 8 | $9n+22$ | $3\times {2}^{n+1}$ |

${\beta}_{9}$ | 2 | 7 | 12 | $9n+23$ | $3\times {2}^{n+2}$ |

${\beta}_{10}$ | 4 | 7 | 6 | $9n+24$ | $3\times {2}^{n+2}$ |

${\beta}_{11}$ | 1 | 4 | 4 | $9n+25$ | $3\times {2}^{n+3}$ |

${\beta}_{12}$ | 3 | 9 | 16 | $9n+25$ | $3\times {2}^{n+1}$ |

${\beta}_{13}$ | 1 | 3 | 3 | $9n+26$ | $3\times {2}^{n+1}$ |

Representative | Degree | ${\mathit{S}}_{\mathit{u}}$ | $\mathit{M}\left(\mathit{u}\right)$ | Eccentricity | Frequency |
---|---|---|---|---|---|

${a}_{i}$ | 2 | 7 | 12 | $9n+9i+6=\gamma +6$ | $3\times {2}^{i}$ |

${b}_{i}$ | 3 | 6 | 8 | $\gamma +7$ | $3\times {2}^{i}$ |

${c}_{i}$ | 2 | 5 | 6 | $\gamma +8$ | $3\times {2}^{i+1}$ |

${d}_{i}$ | 2 | 5 | 6 | $\gamma +9$ | $3\times {2}^{i+1}$ |

${e}_{i}$ | 3 | 6 | 8 | $\gamma +10$ | $3\times {2}^{i}$ |

${f}_{i}$ | 2 | 5 | 6 | $\gamma +11$ | $3\times {2}^{i}$ |

${g}_{i}$ | 2 | 5 | 6 | $\gamma +12$ | $3\times {2}^{i}$ |

${h}_{i}$ | 3 | 7 | 8 | $\gamma +13$ | $3\times {2}^{i}$ |

${j}_{i}$ | 1 | 3 | 3 | $\gamma +14$ | $3\times {2}^{i}$ |

${k}_{i}$ | 4 | 8 | 12 | $\gamma +14$ | $3\times {2}^{i}$ |

${l}_{i}$ | 1 | 4 | 4 | $\gamma +15$ | $3\times {2}^{i}$ |

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