1. Introduction
A fullerene is a spherically shaped molecule consisting of carbon atoms in which every carbon ring forms a pentagon or a hexagon. Every atom of a fullerene has bonds with exactly three neighboring atoms. The molecule may be a hollow sphere, ellipsoid, tube, or many other shapes and sizes. Fullerenes are the subject of intense research in chemistry, and they have found promising technological applications, especially in nanotechnology and materials science [
1,
2].
Molecular graphs of fullerenes are called
fullerene graphs. A fullerene graph is a 3-connected planar graph in which every vertex has degree 3, and every face has size 5 or 6. By Euler’s polyhedral formula, the number of pentagonal faces is always 12. It is known that fullerene graphs having
n vertices exist for all even
and for
. The number of all non-isomorphic fullerene graphs can be found in [
3,
4,
5]. The set of fullerene graphs with
n vertices will be denoted as
. The number of faces of graphs in
is
and, therefore, the number of hexagonal faces is
. Despite the fact that the number of pentagonal faces is very small compared to the number of hexagonal faces, their location is crucial to the shape and properties of fullerene molecules. Fullerene graphs without adjacent pentagons, i.e., each pentagon is surrounded only by hexagons, satisfy the isolated pentagon rule (IPR), and are called
IPR fullerene graphs. They are considered as molecular graphs of thermodynamic stable fullerene compounds. The number of all non-isomorphic IPR fullerene graphs was reported, for example, in [
5,
6]. Mathematical studies of fullerenes include applications of topological and graph theory methods, information theory approaches, design of combinatorial and computational algorithms, etc. (see selected articles [
3,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27]). A comprehensive bibliography on mathematical methods and its applications can be found in [
1,
2,
4,
12,
28,
29]. The set of IPR fullerene graphs with
n vertices will be denoted as
.
The vertex set of a graph
G is denoted by
. The number of vertices of
G is called its
order. The distance
between vertices
is the number of edges in a shortest path connecting
u and
v in
G. The maximal distance between vertices of a graph
G is called the
diameter of
G. Vertices are
diametrical if the distance between them is equal to the diameter of a graph. By
transmission of
, we mean the sum of distances from vertex
v to all the other vertices of
G,
. Transmissions of vertices are used for the design of many distance-based topological indices [
30]. Usually, a topological index is a graph invariant that maps a family of graphs to a set of numbers such that values of the invariant coincide for isomorphic graphs. The
Wiener index is a topological index defined as a half of the sum of vertex transmissions:
It was introduced as a structural descriptor for tree-like organic molecules by Harold Wiener [
31]. The definition of the index in terms of distances between vertices of a graph was given by Haruo Hosoya [
32]. The Wiener index that has found important applications in chemistry (see books and reviews [
33,
34,
35,
36,
37,
38,
39,
40,
41]). Various aspects of the theory and practice of the Wiener index of fullerene graphs are discussed in many works [
7,
8,
11,
12,
13,
15,
16,
17,
18,
19,
20,
22,
24,
42].
The number of different vertex transmissions in a graph
G is known as the
Wiener complexity [
43] (or the
Wiener dimension [
7]),
. This graph invariant can be regarded as a measure of transmission variety. A graph is called
transmission irregular if all vertices of the graph have pairwise different transmissions, i.e., it has the largest possible Wiener complexity. It is obvious that a transmission irregular graph has the identity automorphism group. Various properties of transmission irregular graphs were studied in [
43,
44,
45]. It was shown that almost all graphs are not transmission irregular. Several infinite families of transmission irregular graphs were constructed for trees, 2-connected graphs, and 3-connected cubic graphs in [
44,
46,
47,
48,
49].
In this paper, we present some results of studies of the Wiener complexity and the Wiener index of fullerene graphs. In particular, we are interested in two questions: does a transmission irregular fullerene graph exist and can a graph with the maximal Wiener complexity has the maximal Wiener index?
2. Wiener Complexity of Fullerene Graphs
The Wiener complexity of fullerene graphs was examined for fullerene and IPR-fullerene graphs with
and
vertices, respectively. A typical distribution of the numbers of fullerene graphs with fixed number of vertices with respect to values of
is shown in
Figure 1. The number of graphs of
(100 faces) is 177,175,687. Denote by
the maximal Wiener complexity among all fullerene graphs with
n vertices, i.e.,
, where
or
. Let
be a difference between order and the Wiener complexity,
. If a transmission irregular graph exists, then
. It is obvious that a transmission irregular graph has the identity automorphism group. The behavior of
when the number of vertices
n increases is shown in
Figure 2. The bottom and top lines correspond to all fullerene graphs and to IPR fullerene graphs, respectively. Explicit values of
can be found in
Table 1 and
Table 2.
Proposition 1. There do not exist transmission irregular fullerene graphs with vertices and IPR fullerene graphs with vertices.
Since the almost all fullerene graphs have no symmetries, we believe that transmission irregular graphs exist for a large number of vertices (when the interval of possible values of transmissions will be sufficiently large with respect to the number of vertices).
Problem 1. Does there exist a transmission irregular fullerene graph (IPR fullerene graph)? If yes, then what is the smallest order of such graphs?
3. 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? Denote by the maximal Wiener index among all fullerene graphs with n vertices, i.e., , where or .
Numerical data for the Wiener indices of fullerene graphs in
Table 1 and
Table 2 show that Wiener indices of graphs with maximal
are not maximal. Here, three columns
,
W, and
D contain the maximal Wiener complexity, the Wiener index, and the diameter of graphs with
, respectively. Three columns
,
, and
D contain the maximal Wiener index of graphs with
n vertices, the Wiener complexity, and the diameter of graphs with
. Based on data of the tables, one can make the following observations for the corresponding graphs:
Values of the maximal Wiener complexity of all fullerene graphs do not decrease except for one case: and . For IPR fullerene graphs, we have four exceptions (see pairs ; ; , and ).
Wiener indices of fullerene and IPR fullerene graphs with maximal () are not maximal except graphs of order with ().
Several fullerene or IPR fullerene graphs of fixed n may have the maximal complexity .
Almost all fullerene graphs with fixed have distinct Wiener indices except for two graphs of order 46 with and and two IPR fullerene graphs of order 204 with and 185,306.
Given n, only one fullerene graph has the maximal Wiener index while there are pairs of IPR fullerene graphs with the same maximal W (see graphs of order 132, 186, 204, 222, and 258).
The diameter D of graphs with fixed is less than or equal to the diameter of graphs with a maximal Wiener index for all fullerene graphs (with seven exceptions for IPR fullerene graphs of order 90, 92, 96, 100, 102, 116, and 124).
Fullerene graphs with the maximal Wiener index have the maximal diameter among all fullerenes. The values of the Wiener complexity of these graphs can vary greatly. This can be partially explained by the appearance of symmetries in graphs with .
It is of interest how the pentagons are distributed among hexagons for fullerene graphs of
with the maximal Wiener complexity
,
. Does there exist any regularity in the distribution of pentagons?
Table 3 gives some information on the occurrence of pentagonal parts of a particular size in these graphs (an isolated pentagon forms a part). Here,
N is the number of graphs in which pentagons form
isolated connected parts. The considered fullerene graphs contain at most eight isolated parts.
Table 4 shows how many fullerene graphs with the maximal Wiener complexity have isolated pentagons. Here,
N is the number of graphs having
isolated pentagons. These fullerene graphs contain at most five isolated pentagons.
Does there exist an IPR fullerene graph with maximal Wiener complexity
(lines of
Figure 2 will have an intersection for
)? We believe that the answer to this question will be positive for sufficiently large
n.
4. Fullerene Graphs with the Maximal Wiener Index
The Wiener index of fullerene graphs was studied in [
7,
8,
12,
13,
15,
16,
17,
18,
19,
20,
42]. A class of fullerene graphs of tubular shapes is called
nanotubical fullerene graphs. They have cylindrical shape with the two ends capped by subgraphs containing six pentagons and possibly some hexagons called
caps (see an illustration in
Figure 3).
Consider fullerene graphs with the maximal Wiener indices (see
Table 1). Five graphs of
–
and
contain one pentagonal part and the other 102 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 diagrams of the parts are depicted in
Figure 4. Cap of types a–c have symmetries. The number of fullerene graphs having a given part is shown near diagrams.
Characteristics of the corresponding nanotubes are reported in [
9]. We assume that a type of a cap is determined by the type of its pentagonal part. Types of caps of fullerene graphs are presented in column
t of
Table 1. Constructive approaches for enumeration of various caps were proposed in [
25,
26]. Consider every kind of cap type.
1.
Type a. Caps of type
a define (5,0)-nanotubical fullerene graphs. The structure of graphs of this infinite family
is clear from an example in
Figure 5a. Analytical formulas for diameter and the Wiener index of such fullerene graphs were obtained in [
7,
18]. To indicate the order of graph
G, we will use notation
. We rewrite the formulas in terms of
n.
Proposition 2 ([
7,
18]).
Let be a nanotubical fullerene graph with caps of type a. It has vertices, . Then, , , and , , , and for , Based on numerical data of
Table 1, the similar results have been obtained for fullerene graphs of order
with caps of the other three types.
2.
Type b. Caps of type
b define (3,3)-nanotubical fullerene graphs. The structure of graphs of the corresponding family
is clear from examples of
Figure 5b. Vertices marked by
v should be identified in every graph.
Table 1 contains 28 such graphs.
Proposition 3. Let be a nanotubical fullerene graph with caps of type b. It has vertices, . Then, , , and for , Two caps of type b have adjacent pentagonal rings only for . If fullerene graphs with caps of types a and b have the same number of vertices (), then the graph with caps of type a has the maximal Wiener index.
3.
Type c. Caps of type
c define (4,2)-nanotubical fullerene graphs. Fullerene graphs with caps of type
c will be splitted into two disjoint families,
. The corresponding graphs are marked in column
t of
Table 1 by
(14 graphs) and
(13 graphs). The numbers of vertices of graphs of
and
are given in
Table 5. The orders of graphs of
do not coincide with the orders of graphs from the set
. By analysis of 3D-models of fullerene graphs, it is possible to determine the mutual position of their pentagonal parts. As an example, fragments of graphs with
(case
) and
(case
) vertices are depicted in
Figure 6.
Proposition 4. - (a)
Let be a nanotubical fullerene graph of family . Then, for , - (b)
Let be a nanotubical fullerene graph of family . Then, for ,
The Wiener complexity and the diameter of are shown in Table 5. 4.
Type d. Caps of type
d define (5,1)-nanotubical fullerene graphs. Such graphs will be also splitted into two disjoint families,
. The both families have 13 members (see graphs with marks
and
in column
t of
Table 1). The numbers of vertices of graphs of
and
are shown in
Table 5. The orders of graphs of
do not coincide with the orders of graphs from the set
. An example of the mutual position of pentagonal parts of graphs from these families with
and
vertices are shown in
Figure 6.
Proposition 5. - (a)
Let be a nanotubical fullerene graph of family . Then, and for , - (b)
Let be a nanotubical fullerene graph of family . Then, and for ,
The Wiener complexity and the diameter of are shown in Table 5. The above considerations of fullerene graphs with vertices lead to the following conjectures for all fullerene graphs.
Conjecture 1. If a fullerene graph of an arbitrary sufficiently large order has the maximal Wiener index, then it is a nanotubical fullerene graph with caps of types a–d and its Wiener index is given by Propositions 2–5.
Conjecture 2. The Wiener complexity and the diameter of fullerene graphs of an arbitrary sufficiently large order having the maximal Wiener index are given in Propositions 2–5.
5. IPR Fullerene Graphs with the Maximal Wiener Index
Numerical data for the Wiener indices of IPR fullerene graphs of order
are presented in
Table 2. The structure of the table is the same as for fullerene graphs. Graphs with maximal Wiener index (column
) are nanotubical fullerene graphs with two identical caps. All caps are defined by four fragments shown in
Figure 7. Caps of type a–c have no symmetries. The number of fullerene graphs having a given cap is shown near diagrams. Note that it is difficult to separate caps from each other when IPR fullerene graphs have a shape close to spherical one. Therefore, caps of types
can be recognized only for fullerene graphs of a sufficiently large number of vertices when graphs became nanotubical. A type of cap is indicated in column
t of
Table 2. Consider graphs with these caps.
1.
Type a. Caps of type
a define (5,5)-nanotubical fullerene graphs. Caps of this type can be positioned relative to each other in two ways. The structure of the corresponding graphs of this infinite families
is described by examples in
Figure 8. Analytical formulas for the Wiener index of these graphs are presented in [
17]. We rewrite the formulas in terms of
n.
Proposition 6 (a) Let be a nanotubical IPR fullerene graph of family . It has vertices, and k is odd. Then, , , and (b) Let be a nanotubical IPR fullerene graph of family . It has vertices, and k is even. Then, , , and 2.
Type b. Caps of type
b define (9,0)-nanotubical fullerene graphs. The caps can be positioned relative to each other in four ways
as shown in
Figure 9. Since graphs with fragments of cases
and
almost always have the same Wiener index, we split graphs into two infinite families. Graphs having structures
and
form family
and
, respectively. Each family contains two graphs with the same maximal Wiener index for fixed
n.
Table 6 shows the number of vertices of the families. The difference between
n of the neighboring graphs is 36. The numbers in table’s sells are the number of hexagonal rings (
k) in the fragments connecting two caps (see
Figure 9). The graph of order 168 with caps of type
has almost maximal Wiener index,
116,097, while the maximal Wiener index is
116,100. Remember that the graph of order 150 with the maximal Wiener index,
87,335, has caps of type
a. The graph of order 150 with caps of type
has the second maximal Wiener index,
86,379, and the graph of type
has the third maximal Wiener index,
86,373. The graphs of order 240 with caps of types
and
have the second maximal Wiener index,
302,034 (the graph with the caps of type
a gives maximum
W).
Proposition 7. - (a)
Let be a nanotubical IPR fullerene graph of family . It has vertices, . Then, and - (b)
Let be a nanotubical IPR fullerene graph of family . It has vertices, . Then, and
3. Types c and d. Caps of types c and d generate many cases of their mutual arrangement. By analyzing the structure of fullerene graphs, we have preliminarily identified several families. The number of vertices of eight families of graphs with the caps of type c can be written as , , where . For nine families of graphs with the caps of type d, we have , , where .
Conjecture 3. If an IPR fullerene graph of an arbitrary sufficiently large order has the maximal Wiener index, then it is a nanotubical fullerene graph with caps of types a–d and its Wiener index is given by Propositions 6 and 7 for graphs of types a and b.