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

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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*

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

*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 $\stackrel{2s}{\stackrel{\ufe37}{r\mathrm{,}\cdots \mathrm{,}r}}\mathrm{,}\stackrel{2s}{\stackrel{\ufe37}{r+\mathrm{1,}\cdots \mathrm{,}r+1}}\mathrm{,}\cdots \mathrm{,}\stackrel{2s}{\stackrel{\ufe37}{2r-\mathrm{1,}\cdots \mathrm{,2}r-1}}$.

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*

${\xi}^{ce}\left({T}_{l}\right)=(\begin{array}{cc}12& \text{if}\text{\hspace{0.17em}}l=1,\\ 8& \text{if}\text{\hspace{0.17em}}l=2,\\ 12{\displaystyle \underset{j=0}{\overset{l-1}{\sum}}}\frac{1}{l+j}& \text{if}\text{\hspace{0.17em}}l\ge 3.\end{array}\mathrm{}\text{and}\mathrm{}{\xi}^{c}\left({T}_{l}\right)=(\begin{array}{cc}12& \text{if}\text{\hspace{0.17em}}l=1,\\ 72& \text{if}\text{\hspace{0.17em}}l=2,\\ 18{l}^{2}-6l& \text{if}\text{\hspace{0.17em}}l\ge 3.\end{array}$ (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 $\left(\stackrel{4}{\stackrel{\ufe37}{l,l,l,l}},\stackrel{4}{\stackrel{\ufe37}{l+1,l+1,l+1,l+1}},\cdots ,\stackrel{4}{\stackrel{\ufe37}{2l-1,2l-1,2l-1,2l-1}}\right)$. 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)

$=18{l}^{2}-6l.$ (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*

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

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