On the Wiener complexity and the Wiener index of fullerene graphs

Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex $v$ of a graph is the sum of distances from $v$ to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order $n \le 216$ are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wiener index of several families of graphs are obtained.


Introduction
A fullerene is a spherically shaped molecule consists of carbon atoms in which every carbon ring is either a pentagon or a hexagon and every atom has bonds with exactly three other atoms. The molecule may be a hollow sphere, ellipsoid, tube, or many other shapes and sizes. Fullerenes have been the subject of intense research, both for their chemistry and for their technological applications, especially in nanotechnology and materials science [7,11].
Molecular graphs of fullerenes are called fullerene graphs. A fullerene graph is a 3-connected 3-regular planar graph with only pentagonal and hexagonal faces. By Euler's polyhedral formula, the number of pentagonal faces is always 12. It is known that fullerene graphs with n vertices exist for all even n ≥ 24 and for n = 20. The number of all nonisomorphic fullerene graphs can be found in [8,20,26]. The set of fullerene graphs with n vertices will be denoted as F n . The number of faces of graphs in F n is f = n/2 + 2 and, therefore, the number of hexagonal faces is n/2 − 10. Despite the fact that the number of pentagonal faces is negligible compared to the number of hexagonal faces, their location is crucial to the shape and properties of fullerene molecules. Fullerenes where no two pentagons are adjacent, i. e., each pentagon is surrounded by five hexagons, satisfy the isolated pentagon rule and called IPR fullerene. The number of all non-isomorphic IPR fullerenes was reported, for example, in [25,26]. They are considered as thermodynamic stable fullerene compounds. Description of mathematical properties of fullerene graphs can be found in [4,7,11,19,20,36].
The vertex set of a graph G is denoted by V (G). The number of vertices of G is called its order. By distance d(u, v) between vertices u, v ∈ V (G) we mean the standard distance of a simple graph G, i. e., the number of edges on a shortest path connecting these vertices in G. The maximal distances between vertices of a graph G is called the diameter D(G) of G. Vertices are diametrical if the distance between them is equal to the diameter of a graph. The transmission of vertex v ∈ V (G) is defined as the sum of distances from v to all the other vertices of G, tr(v) = u∈V (G) d(v, u). Transmissions of vertices are used for design of many distance-based topological indices [37]. Usually, a topological index is a graph invariant that maps a set of graphs to a set of numbers such that invariant values coincide for isomorphic graphs. A half of the sum of vertex transmissions gives the Wiener index that has found important applications in chemistry (see selected books and reviews [12, 17, 18, 28-30, 35, 38, 39]), The Wiener index was introduced as structural descriptor for acyclic organic molecules by Harold Wiener [40]. The definition of the Wiener index in terms of distances between vertices of a graph was first given by Haruo Hosoya [31]. representation of these data is shown in Fig. 1.

Wiener complexity of fullerene graphs
Denote by C n the maximal Wiener complexity among all fullerene graphs with n vertices, i. e., C n = max{C W (G) | G ∈ F n }. Fullerene graphs with maximal Wiener complexity have been examined for n ≤ 216 vertices. Let g n be a difference between order and the Wiener complexity, g n = n − C n . Then a transmission irregular graph has g n = 0. It is obvious that a transmission irregular graph has the identity automorphism group.
The behavior of g n when the number of vertices n increases is shown in Fig. 2. The bottom and top lines correspond to all fullerene graphs and to IPR fullerene graphs, respectively. Explicit values of C n and quantity g n of the graphs are presented in Table 2.

Graphs with the maximal Wiener complexity
In this section, we study the following problem: can the Wiener index of a fullerene graph with the maximal Wiener complexity be maximal? Numerical data for the Wiener indices of fullerene graphs of order n ≤ 216 are presented in Table 3. Here three columns C n , W , and D are the maximal Wiener complexity, the Wiener index and the diameter of graphs with C n , respectively. Three columns W m , C W , and D contain the maximal Wiener index, the Wiener complexity and the diameter of graphs with W m .
Based on data of Tables 2 and 3, one can make the following observations.
• Several fullerene graphs of fixed n may have the maximal Wiener complexity C n while the only one fullerene graph has the maximal Wiener index.   • The diameter of graphs with fixed C n are not maximal for n ≥ 52.
• Fullerene graphs with the maximal Wiener index have the maximal diameter. The values of the Wiener complexity C W can vary greatly. This can be partially explained by the appearance of symmetries in graphs.
It is of interest how the pentagons are distributed among hexagons for fullerene graphs with the maximal Wiener complexity (see Tables 2 and 3). Does there exist any regularity in the distribution of pentagons? Table 4 gives some information on the occurrence of pentagonal parts of a particular size. Here N is the number of graphs in which pentagons form N p isolated connected parts.   Table 5 shows how many fullerene graphs with the maximal Wiener complexity have isolated pentagons (an isolated pentagon forms a part). Here N is the number of graphs having N 5 isolated pentagons. Does there exist an IPR fullerene graph with maximal Wiener complexity C n (lines of Fig. 2 will have intersection)?

Graphs with the maximal Wiener index
Wiener index of fullerene graphs are studied in [1, 5, 6, 19, 21-24, 27, 32, 33]. There is a class of fullerene graphs of tubular shapes, called nanotubical fullerene graphs. They are cylindrical in shape, with the two ends capped by a subgraph containing six pentagons and possibly some hexagons called caps (see an illustration in Fig. 3).  Consider fullerene graphs with the maximal Wiener indices (see Table 3). Five graphs of F 20 -F 28 and F 34 contain one pentagonal part and other 93 graphs possess two pentagonal parts. Two pentagonal parts of every fullerene graph are the same and contain diametrical vertices. Therefore such graphs are nanotubical fullerene graphs with caps containing identical pentagonal parts. All types of such parts are depicted in Fig. 4. The number of fullerene graphs having a given part is shown near diagrams. A type of a cap is determined by the type of its pentagonal part. Types of caps of fullerene graphs are presented by the corresponding notation in column t of Table 3. Constructive approaches for enumeration of various caps were proposed in [9,10]. Consider every kind of cap types.
1. Type a. Caps of type a define so-called (5,0)-nanotubical fullerene graphs. The structure of graphs of this infinite family T a is clear from an example in Fig. 5a. Diameter and the Wiener index of such fullerene graphs were studied in [1]. To indicate the order of graph G, we will use notation G n . Based on numerical data of Table 3, the similar results have been obtained for fullerene graphs of order n ≤ 216 with caps of the other three types.  Fig. 5b. Vertices marked by v should be identified in every graph. Table 3 contains 26 such graphs. 3. Type c. Fullerene graphs with caps of type c will be splitted into disjoint families, The corresponding graphs are marked in column t of Table 3 by c1 (13 graphs) and c2 (12 graphs). The numbers of vertices of graphs are given in Table 6. The orders of graphs of T c do not coincide with the orders of graphs from the set T a ∪ T b .
Proposition 4. a) Let G n be a nanotubical fullerene graph of family T c1 . Then for n ≥ 36, The Wiener complexity and the diameter of G n are shown in Table 6. One value should be corrected for k = 0 (see a cell of Table 6 with mark *): C W (G 36 ) = 8 instead of 9.
b) Let G n be a nanotubical fullerene graph of family T c2 . Then for n ≥ 52, The Wiener complexity and the diameter of G n are shown in Table 6. One value should be corrected for k = 0: C W (G 52 ) = 13 instead of 12.
4. Type d. Fullerene graphs with caps of type d will be also splitted into two disjoint families, T d = T d1 ∪ T d2 . The both families have 12 members (see graphs with marks d1 and d2 in column t of Table 3). The numbers of vertices of graphs of T d are shown in Table 6. The orders of graphs of T d do not coincide with the orders of graphs from the The Wiener complexity and the diameter of G n are shown in Table 6. Two values should be corrected for k = 0 (see cells of Table 6   The Wiener complexity and the diameter of G n are shown in Table 6. Two values should be corrected for k = 0: C W (G 58 ) = 25 instead of 35 and C W (G 82 ) = 38 instead of 41.
The above considerations of fullerene graphs with n ≤ 216 vertices lead to the following conjectures for all fullerene graphs.