Atlas of ρ, ρE, and TM–EC for Fullerenes Isomers

Introduction

Mathematical chemistry is concerned with the application of mathematical and computational techniques in chemistry, but it should not be confused with computational chemistry. Major areas of research in mathematical chemistry include chemical graph theory. This branch of mathematical chemistry is dealing with the topological description of molecules as well as the mathematical modeling of chemical phenomena. This part of mathematical chemistry is used to model physical properties of molecules. To do this, some structural numbers named topological indices based on the graphical structure of molecules are defined. The main problem of molecular graph theory is ability of these models and their errors. In our study, we will consider simple molecular graphs without loops and multiple edges.

In chemical graph theory, two different sets of objects are usually considered: atoms and chemical bonds. Define V and E as the set of all atoms and chemical bonds. Then the pair G = (V,E) has a topological structure named “molecular graph”. A chemical bond e connects two atoms u and v and it is convenient to write e = uv. We also use the terms vertex and edge for atom and chemical bond, respectively. The degree of a vertex v, deg_G(v), is defined as the number of edges in the form of vx, where x is an arbitrary vertex of G. The graph G is said to be r−regular, if the degree of all vertices are equal to r. An r−regular molecular graph with r = 3 is called a cubic graph.

If G and H are graphs, V(H) ⊆ V(G) and E(H) ⊆ E(G) then H is called a subgraph of G written as H ≤ G. A path P in a molecular graph G is a subgraph with the property that V(P) = {v₁, v₂, …, v_n} and E(P) = {v₁v₂, v₂v₃, …, v_n−1v_n}. The number n − 1 is said represents the length of P denoted by l(P). A path of minimal length connecting vertices x and y is called a shortest path between them. A molecular graph is called connected if we cannot partition its vertex set into subsets A and B in such a way that no vertex of A is adjacent to a vertex in B. A molecular graph is said to be disconnected if it is not connected. In general, a molecular graph G is called k−connected if there does not exist a set of k − 1 vertices in G whose removal disconnects the graph. The molecular graph G is called planar if we can embed G in the plane. It is well−known that G is planar if and only if it can be drawn on the sphere in such a way that its edges intersect only at their end points.

A fullerene is a carbon sphere−like molecule such that its molecular graph is planar, cubic, 3−connected and all faces are pentagon or hexagon. By Euler’s Theorem, it is easy to prove that the number of pentagons and hexagons in an n−vertex fullerene is 12 and n/2 – 10, respectively. The study of these molecular graphs started after the discovery of buckminsterfullerene by Kroto et al. (Kroto et al., 1985).