On the ECI and CEI of (3, 6)-Fullerenes

Tingzeng Wu¹, Huazhong Lü²

¹School of Mathematics and Statistics, Qinghai Nationalities University, Xining, China.
²School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China.

DOI: 10.4236/am.2020.116034                

Abstract

The eccentricity of a vertex in a graph is the maximum distance from the vertex to any other vertex. Two structure topological indices: eccentric connectivity index and connective eccentricity index involving eccentricity have a wide range of applications in structure-activity relationships and pharmaceutical drug design etc. In this paper, we investigate the eccentric connectivity index and the connective eccentricity index of a (3, 6)-fullerene. We find a relation between the radius and the number of spokes of a (3, 6)-fullerene. Based on the relation, we give the computing formulas of the eccentric connectivity index and the connective eccentricity index of a (3, 6)-fullerene, respectively.

Keywords

Eccentricity, Eccentric Connectivity Index, Connective Eccentricity Index, (3, 6)-Fullerene

1. Introduction

In this paper, we consider finite undirected simple connected graphs and follow the notation and terminology of [1].

Let $G=\left(V\mathrm{,}E\right)$ be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. Let $d\left(v\right)$ denote the degree of a vertex v. For vertices $u\mathrm{,}v\in V\left(G\right)$, the distance $d\left(u\mathrm{,}v\right)$ is defined as the length of the shortest path between u and v in G. The eccentricity $\epsilon \left(v\right)$ of a vertex v is the maximum distance from v to any other vertex.

In organic chemistry, topological indices have a wide range of applications, such as isomer discrimination, structure-property relationships, structure-activity (SAR) relationships and pharmaceutical drug design etc. Recently, two topological indices involving eccentricity have attracted much attention. One is connective eccentricity index, the other is eccentric connectivity index. The connective eccentricity index (CEI briefly), denoted by ${\xi }^{ce}\left(G\right)$, is defined as follows:

${\xi }^{ce}\left(G\right)={\displaystyle \underset{v\in V\left(G\right)}{\sum }}\frac{d\left(v\right)}{\epsilon \left(v\right)}\mathrm{.}$ (1)

Gupta et al. [2] first used CEI to explore the antihypertensive activity of derivatives of N-benzylimidazole. For more background and some known results about CEI, we refer the reader to [3] - [10] and the references therein.

The eccentric connectivity index (ECI for short), denoted by ${\xi }^c\left(G\right)$, is defined as follows:

${\xi }^c\left(G\right)={\displaystyle \underset{v\in V\left(G\right)}{\sum }}d\left(v\right)\epsilon \left(v\right)\mathrm{.}$ (2)

The ECI was first introduced by Sharma et al. [11], which has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature [12] - [18].

In the study of ECI and CEI, a natural problem is how to compute the ECI and CEI for a molecular graph. In this paper, our aim is to investigate the calculation formulas of ECI and CEI of a (3, 6)-fullerene.

An outline of the rest of the paper is to follows. In Section 2, we will present some properties of (3, 6)-fullerenes. In Section 3, we will give the computing formulas of ECI and CEI of a (3, 6)-fullerene.

2. Some Preliminaries

As a member of the fullerene family, (3, 6)-fullerenes has been extensively studied, see [19] [20] [21], among others. A (3, 6)-fullerene is a cubic plane graph whose faces have sizes 3 and 6. Let G be a (3, 6)-fullerene graph with n vertices. By Euler’s formula, G has exactly four faces of size 3 and $\frac{n}{2}-2$ faces of size 6. And the connectivity of G is 2 or 3.

The structure of a (3, 6)-fullerene with connectivity 3 is well known, namely, it is determined by only 3 parameters r, s and t, where $r\ge 1$ is the radius (number of rings), s is the size (number of spokes in each layer and $s\ge 4$ is even), and t is the twist (torsion, $0<t\le s$, $t\equiv r\left(\mathrm{mod}2\right)$ ). So we denote it by $F\left(r\mathrm{,}s\mathrm{,}t\right)$. For example, $F\left(\mathrm{2,4,2}\right)$ and $F\left(\mathrm{2,4,0}\right)$ are depicted in Figure 1, C is a cap of $F\left(\mathrm{2,4,2}\right)$ and $F\left(\mathrm{2,4,0}\right)$.

Figure 1. A (3, 6)-fullerene $F\left(r\mathrm{,}s\mathrm{,}t\right)$ with $r=2$, $s=4$, $t=2$ (or 0) and a cap C of them.

Yang and Zhang [22] characterized the structure of a (3, 6)-fullerene with connectivity 2.

Lemma 1. [22] A (3, 6)-fullerene G has the connectivity 2 if and only if $G\cong T_l$ for some integer $l\ge 2$, where $T_l$ is the tube consisting of l cyclic chains each of two hexagons, capped on each end by a cap of two adjacent triangles, see Figure 2.

3. Main Results

Since a (3, 6)-fullerene is a 3-regular graph, if the eccentricity of every vertex of the (3, 6)-fullerene is known, then the ECI and CEI of the (3, 6)-fullerene can be computed. Thus, the following we will discuss the eccentricity of all vertices of $F\left(r\mathrm{,}s\mathrm{,}t\right)$.

Checking $F\left(r\mathrm{,}s\mathrm{,}t\right)$, it can be known that $F\left(r\mathrm{,}s\mathrm{,}t\right)$ consists of $r-1$ concentric layers of hexagons (i.e. each layer is a cyclic chain of s hexagons) and two caps with torsion t on ends. Thus, the radius, the number of spokes and the twist of $F\left(r\mathrm{,}s\mathrm{,}t\right)$ necessarily affects the eccentricity of every end of $F\left(r\mathrm{,}s\mathrm{,}t\right)$. As an example, we label the eccentricity of every vertex of $F\left(\mathrm{2,4,0}\right)$ and $F\left(\mathrm{2,4,2}\right)$, see Figure 1. Through a lot of illustrations, we find a relation between the radius r and the number of spokes s, and give the following result.

Theorem 1. Let $F\left(r\mathrm{,}s\mathrm{,}t\right)$ be a (3, 6)-fullerene. If $r\ge 2s-1$, then

${\xi }^{ce}\left(F\left(r\mathrm{,}s\mathrm{,}t\right)\right)=6s{\displaystyle \underset{j=0}{\overset{r-1}{\sum }}}\frac{1}{r+j}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\xi }^c\left(F\left(r\mathrm{,}s\mathrm{,}t\right)\right)=9s{r}^2-3sr\mathrm{.}$

Proof. Let $r\ge 2s-1$ in a (3, 6)-fullerene $F\left(r\mathrm{,}s\mathrm{,}t\right)$. Checking the structure of $F\left(r\mathrm{,}s\mathrm{,}t\right)$, we can obtain the following laws:

1) By the definition of eccentricity, we find that the eccentricity of every vertex of $F\left(r\mathrm{,}s\mathrm{,}t\right)$ do not change when the twist t changes. We give an example, see Figure 3.

2) Let $u\mathrm{,}v$ be two vertices of $F\left(r\mathrm{,}s\mathrm{,}t\right)$. The distance $d\left(u\mathrm{,}v\right)$ attains the maximum value only when one of u and v belongs to a vertex of a cap of

$F\left(r\mathrm{,}s\mathrm{,}t\right)$. If r is odd, then the eccentricity of every vertex of $\frac{r+1}{2}$ -layer equal to r, and the eccentricity of every vertex of $\frac{r+1}{2}$ -layer attain the minimum value

in all vertices of $F\left(r\mathrm{,}s\mathrm{,}t\right)$. If r is even, then the eccentricities of the vertex pairs equal to r, and the eccentricities of the vertex pairs attain the minimum value in all vertices of $F\left(r\mathrm{,}s\mathrm{,}t\right)$, where the vertex pairs are adjacent, and one belongs to

Figure 2. A (3, 6)-fullerene $T_l$.

Figure 3. The eccentricity of every vertex of a (3, 6)-fullerene $F\left(r\mathrm{,}s\mathrm{,}t\right)$.

$\frac{r}{2}$ -layer, the other belongs to $\frac{r}{2}$ -layer. Thus, the eccentricity sequence of $F\left(r\mathrm{,}s\mathrm{,}t\right)$ is $\stackrel{2s}{\stackrel{︷}{r\mathrm{,}\cdots \mathrm{,}r}}\mathrm{,}\stackrel{2s}{\stackrel{︷}{r+\mathrm{1,}\cdots \mathrm{,}r+1}}\mathrm{,}\cdots \mathrm{,}\stackrel{2s}{\stackrel{︷}{2r-\mathrm{1,}\cdots \mathrm{,2}r-1}}$.

Combining (1), (2) and arguments above, we have

${\xi }^{ce}\left(G\right)=3\times 2s\times \left(\frac{1}{r}+\frac{1}{r+1}+\cdots +\frac{1}{2r-1}\right)$ (3)

$=6s\left(\frac{1}{r}+\frac{1}{r+1}+\cdots +\frac{1}{2r-1}\right)$ (4)

$=6s{\displaystyle \underset{j=0}{\overset{r-1}{\sum }}}\frac{1}{r+j}$ (5)

and

${\xi }^c\left(G\right)=3\times 2s\times \left[r+\left(r+1\right)+\cdots +\left(2r-1\right)\right]$ (6)

$=6s\left(r^2+\frac{r^2-r}{2}\right)$ (7)

$=9s{r}^2-3sr.$ (8)

The proof is completed.

Theorem 2. Let $T_l\left(l\ge 1\right)$ be a (3, 6)-fullerene. Then

${\xi }^{ce}\left(T_l\right)=(\begin{array}{cc}12& \text{if}l=1,\\ 8& \text{if}l=2,\\ 12{\displaystyle \underset{j=0}{\overset{l-1}{\sum }}}\frac{1}{l+j}& \text{if}l\ge 3.\end{array}\mathrm{<mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>}{\xi }^c\left(T_l\right)=(\begin{array}{cc}12& \text{if}l=1,\\ 72& \text{if}l=2,\\ 18l^2-6l& \text{if}l\ge 3.\end{array}$ (9)

Proof. Checking $T_l$, it is easy to see that the eccentricity of every vertex of $T_l$ is 1. By (1) and (2), we have ${\xi }^{ce}\left(T_1\right)=12$ and ${\xi }^c\left(T_1\right)=12$.

Similarly, checking $T_l$, if $l=2$, then the eccentricity of every vertex of $T_l$ equals to 3. By (1) and (2), we have ${\xi }^{ce}\left(T_1\right)=8$ and ${\xi }^c\left(T_1\right)=72$.

Let $l\ge 3$ in $T_l$. By the structure of $T_l$, it is easy to know that the eccentricity sequence of $T_l$ is $\left(\stackrel{4}{\stackrel{︷}{l,l,l,l}},\stackrel{4}{\stackrel{︷}{l+1,l+1,l+1,l+1}},\cdots ,\stackrel{4}{\stackrel{︷}{2l-1,2l-1,2l-1,2l-1}}\right)$. By (1) and (2), we have

${\xi }^{ce}\left(G\right)=3\times 4\times \left(\frac{1}{l}+\frac{1}{l+1}+\cdots +\frac{1}{2l-1}\right)$ (10)

$=12\left(\frac{1}{l}+\frac{1}{l+1}+\cdots +\frac{1}{2l-1}\right)$ (11)

$=12{\displaystyle \underset{j=0}{\overset{l-1}{\sum }}}\frac{1}{l+j},$ (12)

and

${\xi }^c\left(G\right)=3\times 4\times \left[l+\left(l+1\right)+\cdots +\left(2l-1\right)\right]$ (13)

$=12\left(l^2+\frac{l^2-l}{2}\right)$ (14)

$=18l^2-6l.$ (15)

For notation consistency, $T_l$ can be denoted by $F\left(r\mathrm{,}s\right)$ with $r=l$ and $s=2$, where r is the radius and s is the number of spokes of a (3, 6)-fullerene.

By Theorems 1 and 2, we can obtain the following result.

Theorem 3. Let G be a (3, 6)-fullerene with the radius r and the number of spokes s. If $r\ge 2s-1$. Then

${\xi }^{ce}\left(G\right)=6s{\displaystyle \underset{j=0}{\overset{r-1}{\sum }}}\frac{1}{r+j}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}\text{and}\mathrm{<mi>\; </mi>\; <mi>\; </mi>}{\xi }^c\left(G\right)=9s{r}^2-3sr.$ (16)

4. Discussions

In this paper, we investigate the ECI and CEI of a (3, 6)-fullerene. We obtain an important relation between radius r and the number of spokes s of a (3, 6)-fullerene. That is, if $r\ge 2s-1$, then the twist of a (3, 6)-fullerene does not change the eccentricity of every vertex of the (3, 6)-fullerene. Based on the relation, we give the computing formulas of ECI and CEI of a (3, 6)-fullerene, respectively.

Let us conclude this paper with a question:

Question. How to compute the ECI and CEI of a (3, 6)-fullerene when $r<2s-1$ ?

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 11761056), the Natural Science Foundation of Qinghai Province (No. 2016-ZJ-947Q), the Scientific Research Innovation Team in Qinghai Nationalities University.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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