In this paper, we consider finite undirected simple connected graphs and follow the notation and terminology of .
Let be a graph with vertex set and edge set . Let denote the degree of a vertex v. For vertices , the distance is defined as the length of the shortest path between u and v in G. The eccentricity 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 , is defined as follows:
Gupta et al.  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  -  and the references therein.
The eccentric connectivity index (ECI for short), denoted by , is defined as follows:
The ECI was first introduced by Sharma et al. , which has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature  - .
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   , 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 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 is the radius (number of rings), s is the size (number of spokes in each layer and is even), and t is the twist (torsion, , ). So we denote it by . For example, and are depicted in Figure 1, C is a cap of and .
Figure 1. A (3, 6)-fullerene with , , (or 0) and a cap C of them.
Yang and Zhang  characterized the structure of a (3, 6)-fullerene with connectivity 2.
Lemma 1.  A (3, 6)-fullerene G has the connectivity 2 if and only if for some integer , where 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 .
Checking , it can be known that consists of 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 necessarily affects the eccentricity of every end of . As an example, we label the eccentricity of every vertex of and , 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 be a (3, 6)-fullerene. If , then
Proof. Let in a (3, 6)-fullerene . Checking the structure of , we can obtain the following laws:
1) By the definition of eccentricity, we find that the eccentricity of every vertex of do not change when the twist t changes. We give an example, see Figure 3.
2) Let be two vertices of . The distance attains the maximum value only when one of u and v belongs to a vertex of a cap of
. If r is odd, then the eccentricity of every vertex of -layer equal to r, and the eccentricity of every vertex of -layer attain the minimum value
in all vertices of . 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 , where the vertex pairs are adjacent, and one belongs to
Figure 2. A (3, 6)-fullerene .
Figure 3. The eccentricity of every vertex of a (3, 6)-fullerene .
-layer, the other belongs to -layer. Thus, the eccentricity sequence of is .
Combining (1), (2) and arguments above, we have
The proof is completed.
Theorem 2. Let be a (3, 6)-fullerene. Then
Proof. Checking , it is easy to see that the eccentricity of every vertex of is 1. By (1) and (2), we have and .
Similarly, checking , if , then the eccentricity of every vertex of equals to 3. By (1) and (2), we have and .
Let in . By the structure of , it is easy to know that the eccentricity sequence of is . By (1) and (2), we have
For notation consistency, can be denoted by with and , 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 . Then
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 , 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 ?
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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