On the Wiener complexity and the Wiener index of fullerene graphs

TL;DR

This paper investigates the Wiener complexity and Wiener index of fullerene graphs, providing calculations for graphs up to 216 vertices, analyzing their structures, and deriving formulas for specific graph families.

Contribution

It offers the first extensive calculations of Wiener complexity and index for fullerene graphs up to 216 vertices, and derives formulas for certain graph families.

Findings

Calculated Wiener complexity and index for fullerene graphs up to 216 vertices.

Identified structures with maximal Wiener complexity and Wiener index.

Derived formulas for Wiener index of specific fullerene graph families.

Abstract

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.

Tables

Table 1. Table 1: Distribution of fullerene graphs of F 196 subscript 𝐹 196 F_{196} with respect to the Wiener complexity C W subscript 𝐶 𝑊 C_{W} ( N 𝑁 N is the number of graphs).

$C_W$	$N$	$C_W$	$N$	$C_W$	$N$	$C_W$	$N$	$C_W$	$N$
13	1	45	162	73	4635	101	289459	129	3966428	157	1787105
14	2	46	176	74	5426	102	333904	130	4148282	158	1534799
15	3	47	178	75	6367	103	380828	131	4323678	159	1296521
16	1	48	134	76	7143	104	437958	132	4482070	160	1080839
19	1	49	101	77	8442	105	498054	133	4630277	161	885355
21	1	50	86	78	9680	106	564113	134	4766726	162	713786
23	5	51	78	79	10834	107	637114	135	4889247	163	564259
24	3	52	90	80	12451	108	718248	136	4999476	164	438608
25	1	53	114	81	13990	109	804261	137	5082927	165	331636
26	4	54	100	82	16160	110	899361	138	5147054	166	246578
27	7	55	112	83	18120	111	1000968	139	5186406	167	178749
28	1	56	132	84	20406	112	1110963	140	5195292	168	126604
29	4	57	173	85	23226	113	1231994	141	5177020	169	87271
30	4	58	247	86	26153	114	1357031	142	5131736	170	58523
31	4	59	268	87	29857	115	1490391	143	5052595	171	38065
32	8	60	325	88	34232	116	1632573	144	4943731	172	23910
33	10	61	429	89	39558	117	1783764	145	4809637	173	14592
34	8	62	551	90	46466	118	1941577	146	4643527	174	8433
35	13	63	619	91	55187	119	2108524	147	4450101	175	4630
36	12	64	846	92	65082	120	2278437	148	4236185	176	2549
37	33	65	1039	93	77669	121	2459465	149	4005961	177	1318
38	27	66	1268	94	92772	122	2636351	150	3750830	178	653
39	48	67	1587	95	110504	123	2825077	151	3479586	179	306
40	60	68	1777	96	130842	124	3016435	152	3198568	180	130
41	78	69	2267	97	155105	125	3210085	153	2912936	181	71
42	121	70	2704	98	181583	126	3401907	154	2624386	182	26
43	132	71	3279	99	214088	127	3594118	155	2339326	183	8
44	153	72	4016	100	249142	128	3784693	156	2059994	184	5

Table 2. Table 2: Maximal Wiener complexity C n subscript 𝐶 𝑛 C_{n} and g n = n − C n subscript 𝑔 𝑛 𝑛 subscript 𝐶 𝑛 g_{n}=n-C_{n} of fullerene graphs of F n subscript 𝐹 𝑛 F_{n} ( N 𝑁 N is the number of graphs with C n subscript 𝐶 𝑛 C_{n} ).

$n$	$C_n$	$g_n$	$N$	$n$	$C_n$	$g_n$	$N$	$n$	$C_n$	$g_n$	$N$	$n$	$C_n$	$g_n$	$N$
20	1	19	1	72	56	16	6	122	109	13	5	172	160	12	1
24	2	22	1	74	61	13	1	124	111	13	5	174	164	10	1
26	2	24	1	76	63	13	1	126	115	11	1	176	165	11	1
28	5	23	1	78	64	14	2	128	117	11	1	178	167	11	2
30	7	23	1	80	66	14	2	130	118	12	3	180	168	12	1
32	9	23	1	82	71	11	1	132	121	11	1	182	171	11	1
34	10	24	2	84	70	14	2	134	123	11	3	184	172	12	4
36	14	22	1	86	73	13	3	136	124	12	1	186	177	9	1
38	18	20	1	88	73	15	7	138	127	11	2	188	177	11	1
40	19	21	1	90	79	11	1	140	131	9	1	190	180	10	1
42	22	20	1	92	80	12	1	142	131	11	1	192	181	11	1
44	25	19	1	94	82	12	1	144	132	12	2	194	183	11	2
46	25	21	4	96	84	12	2	146	134	12	4	196	184	12	5
48	30	18	1	98	86	12	1	148	136	12	1	198	187	11	2
50	35	15	1	100	89	11	1	150	138	12	4	200	189	11	2
52	36	16	1	102	89	13	4	152	141	11	1	202	193	9	1
54	37	17	1	104	91	13	3	154	144	10	1	204	192	12	1
56	40	16	1	106	93	13	3	156	144	12	3	206	195	11	3
58	43	15	2	108	96	12	1	158	147	11	1	208	198	10	1
60	44	16	3	110	98	12	2	160	148	12	2	210	199	11	1
62	46	16	3	112	100	12	1	162	151	11	2	212	199	13	8
64	49	15	5	114	102	12	1	164	153	11	1	214	202	12	3
66	50	16	2	116	102	14	4	166	155	11	2	216	204	12	4
68	56	12	1	118	106	12	3	168	157	11	2	-
70	56	14	1	120	110	10	2	170	159	11	2	-

Table 3. Table 3: Maximal Wiener complexity and Wiener indices of fullerene graphs.

$n$	$C_n$	$W$	$D$	$W_m$	$C_W$	$D$	$t$		$n$	$C_n$	$W$	$D$	$W_m$	$C_W$	$D$	$t$
20	1	500	5	500	1	5			84	70	19939	13	21754	21	15	$c1$
24	2	804	5	804	2	5					20076	13
26	2	987	6	987	2	6	$b$		86	73	21404	13	23467	8	16	$b$
28	5	1198	6	1198	5	6					21521	13
30	7	1431	6	1435	3	6	$a$				21593	13
32	9	1688	6	1696	3	7	$b$		88	73	22359	13	24714	21	16	$c2$
34	10	1973	7	1978	10	7					22421	13
		1978	7								22604	13
36	14	2288	7	2298	8	7	$c1$				22616	13
38	18	2627	7	2651	4	8	$b$				22619	13
40	19	3001	7	3035	4	8	$a$				22750	14
42	22	3397	8	3415	19	8	$d1$				22939	14
44	25	3830	8	3888	4	9	$b$		90	79	23923	14	27155	9	17	$a$
46	25	4285	8	4322	19	9	$d2$		92	80	25731	15	28256	8	17	$b$
		4289	8						94	82	26793	14	28910	44	17	$d2$
		4289	8						96	84	28274	14	31418	24	17	$c1$
		4291	8								28317	15
48	30	4795	9	4858	12	9	$c1$		98	86	30068	15	33651	9	18	$b$
50	35	5310	9	5455	5	9	$a$		100	89	31196	15	36580	10	19	$a$
52	36	5876	9	5994	13	10	$c2$		102	89	32984	15	36206	47	18	$d1$
54	37	6475	9	6558	22	10	$d1$				33070	15
56	40	7114	10	7352	5	11	$b$				33226	15
58	43	7782	10	7910	25	11	$d2$				33505	15
		7822	10						104	91	34402	15	39688	9	19	$b$
60	44	8437	10	8880	6	11	$a$				34529	15
		8466	10								36801	17
		8490	10						106	93	36648	16	40278	47	19	$d2$
62	46	9202	10	9651	6	12	$b$				36664	16
		9220	11								37594	17
		9250	11						108	96	38033	15	43578	27	19	$c1$
64	49	9988	11	10410	15	12	$c2$		110	98	40154	16	48005	11	21	$a$
		9993	11								41419	17
		10003	11						112	100	41940	17	48234	27	20	$c2$
		10013	11						114	102	43885	16	49318	52	20	$d1$
		10016	11						116	102	45437	16	53832	10	21	$b$
66	50	10814	11	11126	30	12	$d1$				46632	17
		10842	11								46798	17
68	56	11714	11	12376	6	13	$b$				47927	18
70	56	12589	11	13505	7	13	$a$		118	106	47059	15	54310	50	21	$d2$
72	56	13407	11	14298	18	13	$c1$				47489	16
		13448	11								47697	16
		13453	11						120	110	49143	16	61630	12	23	$a$
		13567	12								49606	17
		13578	12						122	109	51344	16	62011	11	22	$b$
		13766	12								51456	16
74	61	14521	12	15563	7	14	$b$				51933	17
76	63	15867	13	16554	18	14	$c2$				52974	17
78	64	16834	13	17398	37	14	$d1$				53070	17
		16877	13						124	111	54105	17	64170	30	22	$c2$
80	66	17727	13	19530	8	15	$a$				55050	18
		17832	13								55789	18
82	71	19075	13	19918	38	15	$d2$				57358	19
											57473	19

Table 4. Table 3: Maximal Wiener complexity and Wiener indices of fullerene graphs ( continue ).

$n$	$C_m$	$W$	$D$	$W_m$	$C$	$D$	$t$		$n$	$C_m$	$W$	$D$	$W_m$	$C$	$D$	$t$
126	115	57238	18	65286	57	22	$d1$		178	167	141743	23	174510	65	31	$d2$
128	117	60434	18	70976	11	23	$b$				150696	25
130	118	63736	19	77655	13	25	$a$		180	168	139697	21	200780	18	35	$a$
		63922	19						182	171	144410	21	192971	16	32	$b$
		65396	20						184	172	146581	22	197130	45	32	$c2$
132	121	62917	17	76538	33	23	$c1$				147054	21
134	123	64935	17	80763	12	24	$b$				153615	23
		65225	17								154923	23
		68161	19						186	177	167300	26	198046	82	32	$d1$
136	124	69838	19	83274	33	24	$c2$		188	177	154868	21	211776	16	33	$b$
138	127	72311	19	84398	62	24	$d1$		190	180	169849	24	235405	19	37	$a$
		73771	19						192	181	163370	22	222778	48	33	$c1$
140	131	73644	19	96280	14	27	$a$		194	183	187947	27	231763	17	34	$b$
142	131	79852	20	91518	56	25	$d2$				191290	27
144	132	77934	18	97914	36	25	$c1$		196	184	174774	23	236394	48	34	$c2$
		78924	18								178192	24
146	134	86095	20	102947	13	26	$b$				178529	25
		86287	21								179284	25
		87298	21								184011	24
		87442	21						198	187	177296	23	237198	87	34	$d1$
148	136	86432	20	105834	36	26	$c2$				189530	25
150	138	87886	19	117705	15	29	$a$		200	189	180683	23	273830	20	39	$a$
		88860	19								192365	25
		92732	21						202	193	210388	28	251318	71	35	$d2$
		93898	21						204	192	189450	22	265274	51	35	$c1$
152	141	92988	20	115416	13	27	$b$		206	195	221909	28	275427	18	36	$b$
154	144	97359	21	115270	59	27	$d2$				221995	28
156	144	95579	19	122938	39	27	$c1$				222097	28
		96997	19						208	198	201644	23	280554	51	36	$c2$
		98864	21						210	199	238572	29	316255	21	41	$a$
158	147	100055	19	128851	14	28	$b$		212	199	207617	23	299176	18	37	$b$
160	148	103952	20	142130	16	31	$a$				207975	23
		108170	22								209707	23
162	151	104909	19	133206	72	28	$d1$				211942	24
		116278	23								215779	24
164	153	110088	20	143288	14	29	$b$				228285	26
166	155	117531	21	142838	62	29	$d2$				228507	26
		119485	23								228922	26
168	157	114316	18	151898	42	29	$c1$		214	202	226652	25	297030	74	37	$d2$
		126632	23								247352	29
170	159	123193	22	169755	17	33	$a$				250978	29
		130548	24						216	204	220131	23	312858	54	37	$c1$
172	160	129708	22	162474	42	30	$c2$				226928	25
174	164	131354	23	163478	77	30	$d1$				240920	27
176	165	130105	20	175312	15	31	$b$				270770	33

Table 5. Table 4: The number of graphs with N p subscript 𝑁 𝑝 N_{p} isolated pentagonal parts.

$N_p$	1	2	3	4	5	6	7	8
$N$	9	8	27	61	42	40	7	1

Table 6. Table 5: The number graphs of with N 5 subscript 𝑁 5 N_{5} isolated pentagons.

$N_5$	0	1	2	3	4	5
$N$	23	56	44	44	24	4

Table 7. Table 6: Parameters of fullerene graphs with n ≤ 216 𝑛 216 n\leq 216 vertices and caps of types c 𝑐 c and d 𝑑 d . Here k ≥ 0 𝑘 0 k\geq 0 for all expressions.

family $T_{c1}$			family $T_{c2}$
$n$	$C_W$	$D$	$n$	$C_W$	$D$
$60k+36$	$15k+9^{\ast }$	$10k+7$	$60k+76$	$15k+18$	$10k+14$
$60k+48$	$15k+12$	$10k+9$	$60k+88$	$15k+21$	$10k+16$
$60k+72$	$15k+18$	$10k+13$	$60k+52$	$15k+{12}^{\ast }$	$10k+10$
$60k+84$	$15k+21$	$10k+15$	$60k+64$	$15k+15$	$10k+12$
family $T_{d1}$			family $T_{d2}$
$n$	$C_W$	$D$	$n$	$C_W$	$D$
$60k+66$	$25k+{32}^{\ast }$	$10k+2$	$60k+106$	$15k+47$	$10k+19$
$60k+78$	$25k+37$	$10k+4$	$60k+58$	$15k+{35}^{\ast }$	$10k+11$
$60k+102$	$25k+47$	$10k+8$	$60k+82$	$15k+{41}^{\ast }$	$10k+15$
$60k+54$	$25k+{27}^{\ast }$	$10k$	$60k+94$	$15k+44$	$10k+17$

Equations

$$W(G)=\sum _{\{u,v\}\subseteq V(G)}d(u,v)=\frac{1}{2}\sum _{v\in V(G)}tr(v).$$

$$W(G_n)$$

Full text

1 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 $n\geq 24$ and for $n=20$. The number of all non-isomorphic 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 $u,v\in V(G)$ 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 $v\in V(G)$ is defined as the sum of distances from $v$ to all the other vertices of $G$, $tr(v)=\sum_{u\in V(G)}d(v,u)$. 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, 29, 30, 35, 38, 39]),

[TABLE]

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].

The number of different vertex transmissions in a graph $G$ is known as the Wiener complexity [2] (or the Wiener dimension [1]), $C_{W}(G)$. This graph invariant is a measure of transmission variety. A graph is called transmission irregular if it has the largest possible Wiener complexity over all graphs of a given order, i. e., vertices of the graph have pairwise different transmissions. Various properties of transmission irregular graphs were studies in [2, 3, 34]. It was shown that almost all graphs are not transmission irregular. Infinite families of transmission irregular graphs were constructed for trees, 2-connected graphs and 3-connected cubic graphs in [3, 13, 14, 15, 16].

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 there exist of a transmission irregular fullerene graph and can a graph with the maximal Wiener complexity has the maximal Wiener index?

2 Distribution of graphs with respect to their Wiener complexity

Distributions of fullerene graphs with respect to their Wiener complexity have been obtained for $n\leq 216$ vertices ($f\leq 110$ faces). As an illustration, we present data for graphs with 196 vertices (100 faces). The number of graphs of $F_{196}$ is 177 175 687. Distribution of graphs of this family with respect to $C_{W}$ is presented in Table 1. A graphical representation of these data is shown in Fig. 1.

3 Wiener complexity of fullerene graphs

Denote by $C_{n}$ the maximal Wiener complexity among all fullerene graphs with $n$ vertices, i. e., $C_{n}=\max\{C_{W}(G)\,|\,G\in F_{n}\}$. Fullerene graphs with maximal Wiener complexity have been examined for $n\leq 216$ 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. Since the minimal $g_{n}$ is equal to $9$, we can formulate the following statement.

Proposition 1**.**

There do not exist transmission irregular fullerene graphs with $n\leq 216$ vertices.

Since the almost all fullerene graphs have no symmetries, we believe that transmission irregular graphs exist for large 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?

4 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 $n\leq 216$ 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.

• •

Wiener indices of fullerene graphs with fixed $C_{n}$ ($|\,F_{n}|>1$) are not maximal except graphs of order $n=28$ with $W=1198$ ($|\,F_{28}|=2$).

• •

Almost all fullerene graphs with fixed $C_{n}$ have distinct Wiener indices. The only exception are graphs of order 46 with $W=4289$ (the sequences of their vertex transmissions are distinct).

• •

The diameter of graphs with fixed $C_{n}$ are not maximal for $n\geq 52$.

• •

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)?

5 Graphs with the maximal Wiener index

Wiener index of fullerene graphs are studied in [1, 5, 6, 19, 21, 23, 22, 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 $G_{n}$.

Proposition 2**.**

[1]* Let $G_{n}$ be a nanotubical fullerene graph with caps of type $a$. It has $n=10k$ vertices, $k\geq 2$. Then $C_{W}(G_{n})=k$, $D(G_{n})=2k-1$, and $W(G_{20})=500$, $W(G_{30})=1435$, $W(G_{40})=3035$, and for $n\geq 50$,*

[TABLE]

Based on numerical data of Table 3, the similar results have been obtained for fullerene graphs of order $n\leq 216$ with caps of the other three types.

2. Type $b$. The structure of graphs of the corresponding family $T_{b}$ is clear from examples of Fig. 5b. Vertices marked by $v$ should be identified in every graph. Table 3 contains 26 such graphs.

Proposition 3**.**

Let $G_{n}$ be a nanotubical fullerene graph with caps of type $b$. It has $n=6k-4$ vertices, $k\geq 5$. Then $C_{W}(G_{n})=\lceil k/2\rceil$, $D(G_{n})=k+1$, and for $n\geq 26$,

[TABLE]

Two caps of type $b$ have adjacent pentagonal rings only for $k=5$ . If fullerenes with caps of types $a$ and $b$ have the same number of faces ($n=10k$), then the graph with caps of type $a$ has the maximal Wiener index.

3. Type $c$. Fullerene graphs with caps of type $c$ will be splitted into disjoint families, $T_{c}=T_{c1}\cup T_{c2}$. 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 $T_{a}\cup T_{b}$.

Proposition 4**.**

a) Let $G_{n}$ be a nanotubical fullerene graph of family $T_{c1}$. Then for $n\geq 36$,

[TABLE]

*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 $n\geq 52$,

[TABLE]

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, $T_{d}=T_{d1}\cup T_{d2}$. 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 set $T_{a}\cup T_{b}\cup T_{c}$.

Proposition 5**.**

a) Let $G_{n}$ be a nanotubical fullerene graph of family $T_{d1}$. Then $W(G_{42})=3415$ and for $n\geq 54$,

[TABLE]

*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 with mark ): $C_{W}(G_{66})=30$ instead of $32$ and $C_{W}(G_{54})=22$ instead of $27$.

b) Let $G_{n}$ be a nanotubical fullerene graph of family $T_{d2}$. Then $W(G_{46})=4322$ and for $n\geq 58$,

[TABLE]

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 $n\leq 216$ vertices lead to the following conjectures for all fullerene graphs.

Conjecture 1**.**

If a fullerene graph of an arbitrary 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 order having the maximal Wiener index are given in Propositions 2–5.

Acknowledgements

This work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).