On the ECI and CEI of Boron-Nitrogen Fullerenes

The eccentric connectivity index and connective eccentricity index are important topological indices for chemistry. In this paper, we investigate the eccentric connectivity index and connective eccentricity index of boron-nitrogen fullerenes, respectively. And we give computing formulas of eccentric connectivity index and connective eccentricity index of all boron-nitrogen fullerenes with regular structure.


Introduction
All graphs considered in this paper are simple connected graphs.Let G be a graph with vertex set ( ) V G and edge set ( ) notes the degree of v.For vertices ( ) , the distance ( ) , d u v is defined as the length of a shortest path between u and v in G.The eccentricity ( ) v ε of a vertex v is the maximum distance from v to any other vertex.
The chemical information derived through the topological index has been found useful in chemical documentation, isomer discrimination, structure-property correlations, etc.For quite some time there has been rising interest in the field of computational chemistry in topological indices.The interest in topological indices is mainly related to their use in nonempirical quantitative structure-property relationships and quantitative structure-activity relationships.Among various indices, the eccentric connectivity index and connective eccentricity index involving eccentricity have attracted much attention.
The ECI was successfully used for mathematical models of biological activities of diverse nature, see [2]- [7] and the references cited therein.
A novel graph invariant for predicting biological and physical properties-connective eccentricity index (CEI briefly) was introduced by Gupta et al. [8], which was defined as: Some recent results on the CEI of graphs can be found in [9]- [14].
Boron-nitrogen fullerene is a member of the fullerene family, it has been extensively studied [15] [16] [17].A boron-nitrogen fullerene is also called (4,6)-fullerene graph which is a plane cubic graph whose faces have sizes 4 and 6.
Let G be a (4,6)-fullerene graph with n vertices.By Euler's formula, G has exactly six faces of size 4 and 4 2 n − faces of size 6.
A (4,6)-fullerene graph is said to be of dispersive structure if has neither three squares arranged in a line nor a square-cap.According to the quadrilateral positional relationship, Wei and Zhang [18] given a classification of all (4,6)-fullerenes as follows.
In this paper, we aim to investigate the ECI and CEI of a boron-nitrogen fullerene.In next section, we give computing formulas of ECI and CEI of a boron-nitrogen fullerene.

Computing Formula of ECI and CEI of a (4,6)-Fullerene
In this section, according the classification of all boron-nitrogen fullerenes in Lemma 1, we will give computing formula of CEI of a (4,6)-fullerene.
Theorem 2 Let G be a (4,6)-fullerene graph.Then 1) ( ) ( )  Proof.Let G be a cube.Checking the structure of G, we obtain that the eccentricity of every vertex in G is 3.By ( 1) and ( 2), we have Similarly, let G be a hexagonal prism.Checking the structure of G, we obtain that the eccentricity of every vertex in G is 4. By (1) and (2), we obtain □ Theorem 3 Let G be a tubular graph, and let the number of hexagon-layer in G be Proof.Let v be a vertex of G.By the structure of G (see Figure 1), we know that the eccentricity of v equal to the distance between v and u (or u').Thus, the eccentricity sequence of G is ( ).By ( 1) and ( 2), we obtain T. Z. Wu, Y. Wu DOI: 10.4236/am.2018.98061894 Applied Mathematics  ; Proof.Checking G, we see that G is symmetry.By the symmetry of G, it is easy to obtain that the eccentricity sequence of G is (   The proof of the theorem is now complete.□

Conclusion
In this note, we give calculation formulas of ECI and CEI of all (4,6)-fullerenes

Theorem 4
Let G be a(4,6)-fullerene graph having lantern structure, and let the number of hexagon-layer in G be 1 k ≥ , see Figure1.Then T. Z. Wu, Y. Wu DOI: 10.4236/am.2018.98061895 Applied Mathematics with regular structures, respectively.In the future, we will discuss the calculation of ECI and CEI of a (4,6)-fullerene graph of dispersive structure, and try to give an algorithm.