Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C₈₀, C₁₈₀, and C₂₄₀  ^†

Abstract

We develop the combinatorics of edge symmetry and edge colorings under the action of the edge group for icosahedral giant fullerenes from C₈₀ to C₂₄₀. We use computational symmetry techniques that employ Sheehan’s modification of Pόlya’s theorem and the Möbius inversion method together with generalized character cycle indices. These techniques are applied to generate edge group symmetry comprised of induced edge permutations and thus colorings of giant fullerenes under the edge symmetry action for all irreducible representations. We primarily consider high-symmetry icosahedral fullerenes such as C₈₀ with a chamfered dodecahedron structure, icosahedral C₁₈₀, and C₂₄₀ with a chamfered truncated icosahedron geometry. These symmetry-based combinatorial techniques enumerate both achiral and chiral edge colorings of such giant fullerenes with or without constraints. Our computed results show that there are several equivalence classes of edge colorings for giant fullerenes, most of which are chiral. The techniques can be applied to superaromaticity, sextet polynomials, the rapid computation of conjugated circuits and resonance energies, chirality measures, etc., through the enumeration of equivalence classes of edge colorings.

1. Introduction

Symmetry, combinatorial enumerations, chirality, symmetry-based reactivity, spectroscopy, and the synthesis of fullerenes have been a subject of study over the years [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. Whereas the vertex groups or the automorphism groups of graphs of fullerenes, point group symmetries of molecules, solids, and fullerenes have been quite well studied [1,2,3,4,20,24,26,29,30,31,32,33,34,35,36,37,38,39], this is not the case with the edge symmetry or edge groups, especially concerning giant fullerenes. Although the two symmetry groups are isomorphic to each other, the induced permutations on the edges of the associated molecular graphs by vertex permutations are not always obvious or easily obtained. Such an edge group comprising of edge permutations that preserve the connectivity of the fullerene graph is required for many applications, including edge colorings under the group action. Then, special cases of equivalence classes of edge colorings correspond to equivalence classes of conjugated circuits or resonance structures [43,44,45,46,47,48,49,50,51,52,53] and thus can provide an efficient platform for the rapid computation of resonance energies, conjugated circuits, and other properties that provide insight into aromaticity and superaromaticity [43,44,45,46,47,48,49,50,51,52,53]. Fullerenes are a special class of compounds containing only carbons with 12 pentagons and a varied number of hexagons. Fullerenes are especially interesting candidates for the edge colorings, as they contain a network of alternating arrangements of single and double bonds, which results in a large number of resonance structures or conjugated circuits [43,44,45,46,47,48,49,50,51,52,53], especially for giant fullerenes. Although one can employ high-level ab initio quantum chemical computations on giant fullerenes with supercomputers, techniques such as conjugated circuit theory and the combinatorial enumeration of resonance structures provide a viable and efficient alternating platform for shedding light into the structures and reactivity [43,44,45,46,47,48,49,50,51,52,53] of polycyclic aromatics and thus potentially giant fullerenes. Such combinatorial enumerations have been of interest [54,55,56,57,58,59], and especially Kekulė addition patterns related to edge colorings of smaller fullerenes [60].

Combinatorial enumeration of Kekulė structures, conjugated circuits, and resonance structures of both acyclic and cyclic polyacenes have been the topic of several studies [43,44,45,46,47,48,49,50,51,52,53], as such techniques aid in the computation of topological resonance energies, stabilities, and reactivities of polycyclic, acyclic aromatics, and fullerenes. We note that the various Kekulė structures are special cases of edge colorings of the associated graphs, and thus the enumeration of equivalence classes of edge colorings aid in the simplification of such computations, as all members of a given equivalence class would make the same contribution to the topological resonance energies or the stabilities of fullerenes or polycyclic aromatic compounds.

Experimental studies on fullerenes and giant fullerenes have been focused on the synthesis, structures, chirality, reactivity, and spectroscopy of fullerenes and their derivatives [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Giant fullerene cages and onions of fullerenes are interesting because they can be employed to encapsulate metallic oxides as well as small clusters inside fullerenes such as C₁₈₀ and C₂₄₀ [5,6,7,20,21,22,23,28], and such cages find a variety of applications including the environmental remediation of high-level nuclear wastes [61,62,63]. The chirality of substituted fullerenes such as chlorinated fullerenes and their spectroscopy has also received attention [4,21,24,26,30,33]. The edge connectivity of fullerenes can shed light not only on the halogenation reactions but also on chemisorption. The existence of inequivalent edges in fullerenes would provide an opportunity to study reactivity contrast upon the addition of molecules such as H₂, N₂, and O₂, etc., as inequivalent edges can result in different reactivity patterns. Thus, edge colorings under the edge group action of giant fullerenes can provide insights into reactivity patterns especially in chemisorptive reactions and reactions that involve the attachment of molecules to the edges of fullerenes. The icosahedral giant series of fullerenes with the formula C_60k for k = m² of which C₆₀, C₂₄₀, C₅₄₀, C₉₆₀, etc., are members are especially interesting for their high symmetry and chirality of derivatives of these members. The enumeration of the isomers of polysubstituted giant icosahedral fullerenes was considered by the present author in a recent study [4]. Subsequently, stimulated by the pentagonal face dynamics of nanocones and chirality concepts, Balsubramanian et al. [64] recently outlined a combinatorial scheme for the face colorings of icosahedral giant fullerenes. The present study complements the previous two related works [4,64] in that this is the first study on the edge colorings of giant fullerenes, which is a topic of considerable interest in the context of reactivity, stability, enumeration of Kekulė structures, conjugated structures, achirality, and chirality arising from the edge colorings and edge groups etc., of giant fullerenes. The edge colorings considered here are also relevant to heterofullerenes such as C₄₈N₁₂ [37,65].

Stimulated by the intriguing symmetry, chirality, edge substitution, and reactivity of giant fullerenes, in the present study, we undertake edge symmetry and edge colorings under the group action for a series of giant fullerenes, namely C₈₀, C₁₈₀, and C₂₄₀. We show that the Möbius inversion technique combined with our generalization of Sheehan’s modification⁵⁴ of Pόlya’s theorem to all characters yields powerful combinatorial generating functions for the edge colorings of giant fullerenes for all of the irreducible representations of the symmetry group of fullerenes. We focus on the edge colorings of high symmetry icosahedral giant fullerenes that have been of both experimental and theoretical interest.

2. Möbius Inversion Technique with Generalization of Sheehan’s Theorem to All Characters as Tools for the Combinatorics of Edge Groups and Edge Colorings of Giant Fullerenes

The key advantage of the Möbius inversion technique in the context of cycle index polynomials is that when a group acts on two different sets and if the cycle types for the permutations for the action on the smaller sets are known, then the technique can be used to obtain the cycle types of a larger set of objects. The application of the technique results in the inversion of a polynomial generating function from one set to the other through the divisors, which we used extensively in the context of hypercube enumerations for coloring the various hyperplanes for all of the irreducible representations of the hyperoctahedral group [55]. That is, the coefficient of x^q in the inverted polynomial generating function Q_p(x) shown below obtained from a known polynomial F_d(x) generates the permutational cycle types for a larger set from a smaller set—for example, the edges of a 7D hypercube from the permutational matrix types for the hexeracts of the 7D hypercube [55].

$$Q_p={\displaystyle \sum }_{d/p}\mu \left(p/d\right)F_d\left(x\right),$$

(1)

where the sum is over all divisors d of p, and $\mu \left(p/d\right)$ is the Möbius function that yields the cycle types of larger sets, where F_d(x) is the known polynomial in x as constructed from the permutations of objects of a smaller set with the Group G commonly acting on both sets. The Möbius function, which in the above equation is denoted as $\mu \left(p/d\right)$, takes values 1, −1, −1, 0, −1, 1, −1, 0, 0, 1… for arguments 1 to 10. The Möbius function for any number is 0, +1, or −1 depending on whether it is a perfect square and the number of prime factors in the argument.

Figure 1 shows a regular icosahedron, a vertex–edge combo coloring of the icosahedron with 5 different colors, the C₂₀ dodecahedron, and the C₆₀ buckminsterfullerene—all of which exhibit the icosahedral symmetry. Fullerenes are vertex regular graphs with 3 vertex degrees that contain 12 pentagons. Therefore, for any fullerene, the number of edges is given by 3n/2, where n is the number of vertices. For example, there are 360 edges in the case of the C₂₄₀ icosahedral fullerene. Each of the proper and improper rotational operations of the icosahedron induces a permutation on the set of 360 edges of C₂₄₀, a permutation on the 240 vertices, and a permutation on 122 faces of the giant C₂₄₀ fullerene with the icosahedral point group symmetry. Euler’s formula connects the number of vertices, edges, and edges of the fullerene; thus, we can obtain the number of any one of them when the other two are known. The group action on the edges of the fullerene can be obtained through a number of techniques, including the generation of the edge automorphism group from the vertex automorphism group of the graph. However, we note that the permutational cycle types of group action on the edges can be obtained computationally through the Möbius sum. It can also been seen that certain operations such as the C₅ rotational axis for an icosahedral fullerene cage pass through the center of a pentagon (see Figure 2e) and not through any of the vertices with the exception that for the 12-vertex icosahedron (see Figure 1a,b), the C₅ axis passes through two opposite vertices of the icosahedron. Hence, this is directly related to the Möbius concept of whether the fold-ness of the rotational axis divides the number of objects in the set or not. For example, for the C₅ axis, the number 5 does not divide 12 in the case of the vertices of an icosahedron, thus leaving a remainder of 2 when 5 is divided by 12. This necessitates the C₅ axis to pass through two vertices that are across each other for the icosahedron. For giant fullerenes such as C₂₄₀, the C₅ axis passes through the center of a pentagon, and as 5 is a divisor of 240, it is clear that the vertex permutation will be described by 5⁴⁸. As seen from

Table 1. Character table of I_h, permutation cycles and their lengths for the 60 pent–hex edges and remaining hex–hex edges of C₆₀ (I_h), Icosahedron, C₈₀ (I_h), C₁₈₀ (I_h), and C₂₄₀ (I_h) upon action on edges.^a

CC	E	15C2	12C5	12C52	20C3	I	12S10	12S103	20S6	15σd
C60 (30 hex–hex)	130	12214	56	56	310	215	103	103	65	14213
Icosahedron
(30 edges)
C60 (pent–hex edge)	160	230	512	512	320	230	106	106	610	14 228
C80 (hex–hex);	160	230	512	512	320	230	106	106	610	14 228
chamfered dodecahedron
C80 (pent–hex; edges)	160	230	512	512	320	230	106	106	610	14 228
C180 (hex–hex)	1210	122104	542	542	370	2105	1021	1021	635	182101
C180 (pent–hex) = C60 (pent–hex);Vertices: chamfered dodecahedron	160	230	512	512	320	230	106	106	610	14 228
C240 (hex–hex)	1300	2150	562	562	3100	2150	1030	1030	650	1122144
C240 (pent–hex)	160	230	512	512	320	230	106	106	610	14 228
C60 (pent–hex); chamfered truncated icosahedron
Ag	1	1	1	1	1	1	1	1	1	1
T1g	3	−1	ω	ω *	0	3	ω	ω	0	−1
T2g	3	−1	ω *	ω	0	3	ω	ω *	0	−1
Gg	4	0	−1	−1	1	4	−1	−1	1	0
Hg	5	1	0	0	−1	5	0	0	−1	1
Au	1	1	1	1	1	−1	−1	−1	−1	−1
T1u	3	−1	ω	ω *	0	−3	−ω *	−ω	0	1
T2u	3	−1	ω *	ω	0	−3	−ω	−ω *	0	1
Gu	4	0	−1	−1	−1	−4	1	1	1	0
Hu	5	1	0	0	−1	−5	0	0	1	−1

ω = (1 + $\surd 5$)/2 (golden ratio). ω * = (1 − $\surd 5$)/2. ^a Permutation cycles of 30 pent–hex edges of all fullerenes are shown only once, as they are the same for all icosahedral fullerenes with 12 isolated pentagons.

, the conjugacy class representatives of the icosahedral fullerene group comprises the set {E, C₆, C₅, C₃, C₂, C_i, S₁₀, S₁₂, S₆, σ_d}. For all of the icosahedral fullerenes considered here, all the edges are not equivalent or alternatively, the edges are divided into more than one equivalence class for such giant fullerenes. In such a scenario, it is more suitable to consider a generalization of Pόlya’s theorem outlined by Sheehan [54], which we further generalize to all characters and the edge action of the icosahedral group.

Consider a set D consisting of the edges of a fullerene, and because for all the fullerenes considered here, there exists more than one equivalence classes of edges, we further partition the edge set D of fullerenes into two sets Y₁, Y₂, … Y_m where Y_i is an equivalence class partition of the edges of the fullerene. That is, ${{\displaystyle \cup }}_{i=1}^m{Y}_i=D,Y_i{\displaystyle \cap }=\varnothing ,fori\ne j$, where m = number of the edges of the fullerene cage.

The action of a group G, in this case the icosahedral group of 120 operations, on the set D, induces a permutation in various Y_i sets such that every cycle is contained within the set Y_i. Consequently, an edge coloring of the fullerene can be considered as a map from D to R, where R is the set of colors, but because D is partitioned into various equivalence classes, one may choose an independent set of colors for each set Y_i. Sheehan [54] has considered such a generalization of Pόlya’s theorem for the enumeration under group action for the case where the set D is partitioned into multiple sets. We note that Sheehan’s generalization of Pόlya’s theorem reduces even the Redfield–Read superposition theorem to a special case. Sheehan’s technique [54] is further generalized here to all characters of the irreducible representations of the icosahedral group for the edge colorings of fullerenes. These techniques also yield the number of chiral and all achiral colorings of the fullerene cages for a different distribution of colors.

The generalized character cycle index (GCCI) polynomial for any character of the automorphism group G of the fullerene cage under consideration is defined as

$$P_G^{\chi }=\frac{1}{\left|G\right|}{\displaystyle \sum _{g\epsilon G^{\prime }}\chi (g){\displaystyle \prod _i{\displaystyle \prod _j}}}{s}_{ij}^{c_{ij}(g)}$$

(2)

where the sum is over all the permutation representations of g ∈ G, and c_ij(g) is the number of j-cycles of g ∈ G upon its action on the set Y_i. That is, the second index j in c_ij(g) stands for the length of the cycle (that is a j-cycle) generated in the Y_i set upon the action of g ∈ G. We also note that unlike the ordinary cycle index of Pólya’s enumeration theorem, the generalization considered here generates a GCCI for each irreducible representation of the I_h group for the icosahedral fullerene. Consequently, we employ an even more powerful generating function than the ordinary Pólya or the Sheehan cycle index [54] to enumerate the edge colorings of giant fullerenes for each irreducible representation (IR).

Define [n] as an ordered partition, or the composition of n into p parts such that n₁ ≥ 0, n₂ ≥ 0, …, n_p ≥ 0, ${{\displaystyle \sum }}_{i=1}^p{n}_i=n$. We can obtain a multinomial generating function (GF) in λs with n₁ colors of the type λ₁, n₂ colors of the type λ₂ … n_p colors of the type λ_p, which are defined as follows:

$${\left({\lambda }_1+{\lambda }_2+\cdots ..+{\lambda }_p\right)}^n={{\displaystyle \sum }}_{\left[n\right]}^p\left(\begin{array}{c}n\\ n_1{n}_2..n_p\end{array}\right){\lambda }_1{}^{n_1}{\lambda }_2{}^{n_2}\dots \dots ..{\lambda }_{p-1}{}^{n_{p-1}}{\lambda }_p{}^{n_p}$$

(3)

where $\left(\begin{array}{c}n\\ n_1{n}_2..n_p\end{array}\right)$ are multinomials given by

$$\left(\begin{array}{c}n\\ n_1{n}_2..n_p\end{array}\right)=\frac{n!}{n_1!n_2!\dots \dots n_{p-1}!n_p!}.$$

(4)

Let R be a set of colors partitioned into sets R₁, R₂ … R_m with the same number of partitions as the Y sets such that $R=R_i{{\displaystyle \cup }}^{}{R}_{j,}{R}_i{{\displaystyle \cap }}^{}{R}_j=\varnothing$. Furthermore, let w_ij be the weight of each color r_j in the set R_i. The generating function for each irreducible representation for the edge colorings of giant fullerene cages considered here is given by

$$G{F}^{\chi }\left({\lambda }_1,{\lambda }_2\dots ..{\lambda }_p\right)=P_G^{\chi }\left\{s_{ik}\to \left(w_{i1}^k+w_{i2}^k+\dots .+w_{i,p_i-1}^k+w_{i,p_i}^k\right)\right\}.$$

(5)

where p_i = |R_i|. The GFs obtained above for each IR and various color distributions provides the number of edge coloring with varying color distribution for the different equivalence classes of the edges, independent of each other and those that transform according to the irreducible representation whose character is given by χ. Thus, all of the techniques demonstrated including the Möbius function method, GCCIs, and generating functions were all programmed into FORTRAN ’95 codes using the quadruple precision arithmetic as the number of edge colorings explode combinatorially. We note that in order to make the codes efficient, especially for larger fullerenes, we have taken considerable efforts to optimize the codes by designing efficient algorithms for the computations of multinomial and binomial numbers using recursion as opposed to an explicit evaluation of factorials and identifying the equivalence of multinomials, thereby eliminating duplicative computations. Furthermore, although we have obtained the results for all of the irreducible representations in the ensuing section, we show the results only for the chiral and totally symmetric irreducible representations, as these are the most interesting cases for the present study. However, we point out that the results obtained for other irreducible representations are quite important in several other applications such as the enumeration of nuclear spin functions, nuclear spin statistics, Electron Spin Resonance (ESR) spin function enumerations for the assignment of the observed hyperfine structures, and so on [57,58,59]. Thus, there is a wealth of information latent in these generating functions for the various irreducible representations for giant fullerenes.

3. Results and Discussions

Table 1 displays the various conjugacy classes of the icosahedral group, the permutation orbit structures for the edges for C₈₀ (I_h), C₁₈₀ (I_h), and C₂₄₀ (I_h), and the character values of the I_h group under each conjugacy class. The three structures considered here are shown in Figure 2 together with other structures of icosahedral symmetry in Figure 1. Among the fullerenes considered here, we illustrate the procedure of constructing the GCCIs with C₈₀ (I_h) (the chamfered dodecahedron shown in Figure 2), the related edges of the icosahedron in Figure 1a,b, and the corresponding generating functions. There are two types of edges in C₈₀ as seen from Figure 2 analogous to buckminsterfullerene C₆₀ (Figure 1d): the edges that are shared by a pentagon and hexagon are distinct from the edges shared by two hexagons. Thus, for the case of C₈₀, there are two equivalent class edges: namely, Y₁ and Y₂. Let the first set correspond to the edges shared by a pentagon and hexagon. As there are exactly 12 pentagons in any fullerene, it is evident that the cardinality of the set Y₁ is 60. This leaves 60 edges that are common to two hexagons in the case of C₈₀; or, the cardinality of the set Y₂ is also 60 for the case of C₈₀. However, as can be seen from Table 1, the cardinalities of sets Y₁ and Y₂ are 60 and 30, respectively for the buckminsterfullerene case. From Table 1, we combine the cycle types that are shown for the pent–hex edges together with the cycle types for C₈₀ (hex–hex) edges to obtain the GCCIs for the A_g and A_u irreducible representations of the I_h group shown below:

$$\begin{array}{cc}\hfill P_G^{Ag}=\frac{1}{120}\{s_{11}^{60}{s}_{21}^{60}& +15s_{12}^{30}{s}_{22}^{30}{}^{}\hfill \\ & +24s_{11}^2{s}_{15}^{12}{s}_{25}^{12}+20s_{13}^{20}{s}_{23}^{20}{}^{}+s_{12}^{30}{s}_{22}^{30} +24s_{1,10}^6{s}_{2,10}^6+20s_{16}^6{s}_{26}^6\hfill \\ & +15s_{11}^4{s}_{12}^{28}{s}_{21}^4{s}_{22}^{28}{}^{}{}^{}\}\hfill \end{array}$$

(6)

$$\begin{gathered} P_G^{Au}=\frac{1}{120}\left\{s_{11}^{60}{s}_{21}^{60}{}^{}+15s_{12}^{30}{s}_{22}^{30}{}^{}+24s_{11}^2{s}_{15}^{12}{s}_{25}^{12}+20s_{13}^{20}{s}_{23}^{20}{}^{}-s_{12}^{30}{s}_{22}^{30}-24s_{1,10}^6{s}_{2,10}^6 \\ -20s_{16}^6{s}_{26}^6-15s_{11}^4{s}_{12}^{28}{s}_{21}^4{s}_{22}^{28}{}^{}\right\}. \end{gathered}$$

(7)

We note that it is coincidental that the cardinalities of both equivalence classes for the edge sets of C₈₀ became the same and equal to 60, which is not the case in general. For the C₆₀ buckminster fullerene, the cardinalities of the two sets are 60 and 30, respectively, and thus, we obtain different terms for s_1j and s_2j in each of the terms of the GCCIs for C₆₀ for the A_g and A_u IRs, as shown below:

$$\begin{array}{cc}\hfill P_G^{Ag}\left(C_{60}\right)=\frac{1}{120}\{s_{11}^{60}{s}_{21}^{30}& +15s_{12}^{30}{s}_{21}^2{s}_{22}^{14}{}^{}\hfill \\ & +24s_{15}^{12}{s}_{25}^6+20s_{13}^{20}{s}_{23}^{10}{}^{}+s_{12}^{30}{s}_{22}^{15}+24s_{1,10}^6{s}_{2,10}^3+20s_{16}^{10}{s}_{26}^5\hfill \\ & +15s_{11}^4{s}_{12}^{28}{s}_{21}^4{s}_{22}^{13}{}^{}{}^{}\}\hfill \end{array}$$

(8)

$$\begin{gathered} P_G^{Au}\left(C_{60}\right)=\frac{1}{120}\left\{s_{11}^{60}{s}_{21}^{30}{}^{}+15s_{12}^{30}{s}_{21}^2{s}_{22}^{14}{}^{}+24s_{15}^{12}{s}_{25}^6+20s_{13}^{20}{s}_{23}^{10}{}^{}-s_{12}^{30}{s}_{22}^{15}-24s_{1,10}^6{s}_{2,10}^3-20s_{16}^{10}{s}_{26}^5- \\ 15s_{11}^4{s}_{12}^{28}{s}_{21}^4{s}_{22}^{13}{}^{}{}^{}\right\}. \end{gathered}$$

(9)

Generating functions for the two cases with 4 different colors to color the edges common to a pentagon and hexagon (denoted as pent–hex edges) and 3 different colors (say red, green, and blue) to color the edges common to two hexagons (denoted as hex–hex) for the A_g of C₆₀ are shown first in Equation (10).

$$\begin{array}{cc}\hfill G{F}_G^{Au}=\frac{1}{120}\{(1& +a+b+c{)}^{60}{\left(1+d+e\right)}^{30}\hfill \\ & +15{\left(1+a^2+b^2+c^2\right)}^{30}{\left(1+d+e\right)}^2{\left(1+d^2+e^2\right)}^{14}\hfill \\ & +24{\left(1+a^5+b^5+c^5\right)}^{12}{\left(1+d^5+e^5\right)}^6\hfill \\ & +20{\left(1+a^3+b^3+c^3\right)}^{20}{\left(1+d^3+e^3\right)}^{10}{}^{}+{\left(1+a^2+b^2+c^2\right)}^{30}{\left(1+d^2+e^2\right)}^{15}\hfill \\ & +24{\left(1+a^{10}+b^{10}+c^{10}\right)}^6{\left(1+d^{10}+e^{10}\right)}^3\hfill \\ & +20{\left(1+a^6+b^6+c^6\right)}^{10}{\left(1+d^6+6\right)}^5\hfill \\ & +15{\left(1+a+b+c\right)}^4{\left(1+a^2+b^2+c^2\right)}^{28}{\left(1+d+e\right)}^4{\left(1+d^2+e^2\right)}^{13}{}^{}\}.\hfill \end{array}$$

(10)

The corresponding GF for the A_u IR is obtained in an analogous manner, and it shown as Equation (11):

$$\begin{array}{cc}\hfill G{F}_G^{Au}=\frac{1}{120}\{(1+& a+b+c{)}^{60}{\left(1+d+e\right)}^{30}\hfill \\ & +15{\left(1+a^2+b^2+c^2\right)}^{30}{\left(1+d+e\right)}^2{\left(1+d^2+e^2\right)}^{14}\hfill \\ & +24{\left(1+a^5+b^5+c^5\right)}^{12}{\left(1+d^5+e^5\right)}^6\hfill \\ & +20{\left(1+a^3+b^3+c^3\right)}^{20}{\left(1+d^3+e^3\right)}^{10}{}^{}\hfill \\ & -{\left(1+a^2+b^2+c^2\right)}^{30}{\left(1+d^2+e^2\right)}^{15}\hfill \\ & -24{\left(1+a^{10}+b^{10}+c^{10}\right)}^6{\left(1+d^{10}+e^{10}\right)}^3-20{\left(1+a^6+b^6+c^6\right)}^{10}{\left(1+d^6+6\right)}^5\hfill \\ & -15{\left(1+a+b+c\right)}^4{\left(1+a^2+b^2+c^2\right)}^{28}{\left(1+d+e\right)}^4{\left(1+d^2+e^2\right)}^{13}{}^{}\}.\hfill \end{array}$$

(11)

The above multinomial generating functions can be expanded and simplified for each IR of the I_h group, and these functions offer a greater flexibility for the edge colorings compared to the ordinary Pόlya’s expansion, because the GF has terms partitioned into coloring hex–hex edges and pent–hex edges differently. Thus, the GFs obtained above have the ability to enumerate edge colorings with different specifications:

(a)

if all hex–hex edges of C₆₀ are colored with one color (white) and pent–hex edges are colored with multiple colors;

(b)

if all hex–hex edges are colored with multiple colors—say green, red, blue, burgundy, and cyan (see Figure 1b)—and all hex–pent edges are colored with white; this case also corresponds to coloring the 30 edges of the icosahedron as shown in Figure 1a, b.

(c)

if both pent–hex and hex–hex edges are colored with multiple colors from a single set R of colors without making any distinction between the two sets of edges.

(d)

if all hex–hex edges are colored with colors from a single set, while all pent–hex edges are colored with colors chosen from a different set of colors, thus making a distinction between the pent–hex and hex–hex edges.

For each of the cases listed above, one can obtain a set of GFs for each IR of the I_h group, thus yielding a powerful set of enumerative combinatorial generating functions. Hence, we consider each such case.

For case (a), the edge colorings of C₆₀ are obtained by setting d and e to 0 in Equation (10), which yields Equation (12):

$$\begin{array}{cc}\hfill G{F}_G^{Ag}=\frac{1}{120}\{(1& +a+b+c{)}^{60}+15{\left(1+a^2+b^2+c^2\right)}^{30}\hfill \\ & +24{\left(1+a^5+b^5+c^5\right)}^{12}+20{\left(1+a^3+b^3+c^3\right)}^{20}{}^{}+{\left(1+a^2+b^2+c^2\right)}^{30}\hfill \\ & +24{\left(1+a^{10}+b^{10}+c^{10}\right)}^6+20{\left(1+a^6+b^6+c^6\right)}^{10}\hfill \\ & +15{\left(1+a+b+c\right)}^4{\left(1+a^2+b^2+c^2\right)}^{28}{}^{}{}^{}\}\hfill \end{array}$$

(12)

when the above expressions are expanded and simplified for each IR, the results can be expressed in terms of partitions of integer 60 in 4 parts. For example, the partition [15 15 15 15] of 60 corresponds to the term 1¹⁵a¹⁵b¹⁵c¹⁵ = a¹⁵b¹⁵c¹⁵, and hence, the coefficient a¹⁵b¹⁵c¹⁵ in the multinomial expansion gives the number of ways 60 pent–hex edges of C₆₀ can be colored with 15 white, 15 green, 15 blue, and 15 red colors keeping all hex–hex edges white. The GF for the A_u IR generates the number of chiral pairs for the corresponding chiral partition [15 15 15 15] for the pent–hex colorings. Thus, the sum of the numbers for A_g and A_u enumerated for each color partition yields the total number of all possible pent–hex colorings that encompass both achiral colorings and chiral pairs. In general, for each IR of the I_h group the results enumerated represent the number of colorings that transform according to the IR for the given color partition. These general results for any IR find applications to nuclear spin statistics beyond electrons, for example, spin 3/2 fermions and quarks [57,58,59].

The GF is obtained for the case (b) by setting the edge weights a, b, and c to 0 in Equation (10), which results in Equation (13):

$$\begin{array}{cc}\hfill G{F}_G^{Ag}=\frac{1}{120}\{(1& +d+e{)}^{30}+15{\left(1+d+e\right)}^2{\left(1+d^2+e^2\right)}^{14}\hfill \\ & +24{\left(1+d^5+e^5\right)}^6+20{\left(1+d^3+e^3\right)}^{10}{}^{}+{\left(1+d^2+e^2\right)}^{15}\hfill \\ & +24{\left(1+d^{10}+e^{10}\right)}^3+20{\left(1+d^6+e^6\right)}^5\hfill \\ & +15{\left(1+d+e\right)}^4{\left(1+d^2+e^2\right)}^{13}{}^{}\}.\hfill \end{array}$$

(13)

Expression (13) thus obtained for the A_g IR yields the number of equivalence classes of coloring hex–hex edges with 3 different colors while keeping all the pent–hex colors white. The above expression when generalized to include two more colors with weights f and g—that is, every term (1 + d^k + e^k) in Equation (13) is replaced with (1 + d^k + e^k + f^k + g^k)—we obtain the results shown in

Table 2. Edge colorings of an icosahedron with up to 5 different colors (green, blue, red, cyan, burgundy)—one such coloring is in Figure 1 ^a.

Color Partition	Ag	Au
30 0 0 0 0	1	0
29 1 0 0 0	1	0
28 2 0 0 0	8	3
27 3 0 0 0	46	32
26 4 0 0 0	262	221
25 5 0 0 0	1257	1166
24 6 0 0 0	5113	4912
23 7 0 0 0	17,238	16,874
22 8 0 0 0	49,270	48,620
21 9 0 0 0	119,997	118,996
20 10 0 0 0	251,512	249,995
19 11 0 0 0	456,729	454,727
18 12 0 0 0	722,750	720,125
17 13 0 0 0	1,000,251	997,248
16 14 0 0 0	1,214,376	1,210,944
15 15 0 0 0	1,295,266	1,291,834
28 1 1 0 0	9	6
27 2 1 0 0	113	97
26 3 1 0 0	937	897
25 4 1 0 0	6019	5902
24 5 1 0 0	29,835	29,588
23 6 1 0 0	119,106	118,586
22 7 1 0 0	390,754	389,818
21 8 1 0 0	1,074,073	1,072,500
20 9 1 0 0	2,505,217	2,502,786
19 10 1 0 0	5,009,719	5,006,287
18 11 1 0 0	8,652,111	8,647,535
17 12 1 0 0	12,977,523	12,971,946
16 13 1 0 0	16,969,947	16,963,512
15 14 1 0 0	19,393,980	19,387,116
26 2 2 0 0	1438	1355
25 3 2 0 0	12,025	11,817
24 4 2 0 0	74,698	74,087
23 5 2 0 0	357,162	355,914
22 6 2 0 0	1,367,704	1,365,026
21 7 2 0 0	4,295,434	4,290,858
20 8 2 0 0	11,272,690	11,264,825
19 9 2 0 0	25,045,735	25,034,295
18 10 2 0 0	47,583,250	47,566,805
17 11 2 0 0	77,858,703	77,838,111
16 12 2 0 0	110,297,148	110,271,837
15 13 2 0 0	135,747,564	135,720,108
14 14 2 0 0	145,443,696	145,414,524
26 1 1 1 1	5484	5478
22 2 2 2 2	122,923,788	122,907,252
18 3 3 3 3	266,399,185,320	266,399,082,360
14 4 4 4 4	76,423,258,557,360	76,423,248,055,440
10 5 5 5 5	2,937,587,511,706,920	2,937,587,492,247,480
6 6 6 6 6	11,423,951,476,076,040	11,423,951,359,159,200

^a Numbers under A_u are chiral pairs of colorings and the sum of A_g and A_u is the total number of colorings and A_g–A_u is the number of achiral colorings for each color partition listed in the first column for coloring 30 edges of an icosahedron.

for the colorings of the edges of the icosahedron (Figure 1a,b) with up to 5 different colors under the action of the I_h group. Only some of the color partitions are shown in Table 2, as the number of terms in the multinomial GF for coloring 30 edges with 5 different colors is $\left(\begin{array}{c}30+5-1\\ 30\end{array}\right)=\left(\begin{array}{c}34\\ 30\end{array}\right)=\mathrm{46,376}$.

If all edges are colored with colors chosen from a single set of colors (case (c)), we obtain the GF for each IR by setting w_ij = w_j for all i, which leads to the following GF for A_g for the case with 4 colorings of 120 edges of C₈₀ (I_h) without differentiating the pent–hex and hex–hex edges. Thus, the expression obtained is shown in Equation (14):

$$\begin{array}{cc}\hfill G{F}_G^{Ag}=\frac{1}{120}\{(1& +a+b+c{)}^{120}+15 {\left(1+a^2+b^2+c^2\right)}^{60}\hfill \\ & +24 {\left(1+a^5+b^5+c^5\right)}^{24}+20 {\left(1+a^3+b^3+c^3\right)}^{40}+{\left(1+a^2+b^2+c^2\right)}^{60} \hfill \\ & +24 {\left(1+a^{10}+b^{10}+c^{10}\right)}^{12 }+20 {\left(1+a^6+b^6+c^6\right)}^{20}\hfill \\ & +15{\left(1+a+b+c\right)}^8{\left(1+a^2+b^2+c^2\right)}^{56}{}^{}{ }^{}\}.\hfill \end{array}$$

(14)

The above expression (14) can be expanded; except for the first few terms, there is a combinatorial explosion. Note that the number of terms in the above expansion or the number of ordered partitions of 120 into 4 parts is $\left(\begin{array}{c}120+4-1\\ 120\end{array}\right)=\left(\begin{array}{c}123\\ 120\end{array}\right)=\mathrm{302,621}$. Consequently, an exhaustive construction of the combinatorial tables for all such edge colorings will take too much space, although they have all been generated by our computer code. Hence, we restrict to some of the binomial coloring terms shown in

Table 3. Edge colorings of C₈₀.

[ ]	Ag	Au
120 0	1	0
119 1	2	0
118 2	78	56
117 3	2410	2284
116 4	69,088	68,264
115 5	1,590,094	1,586.216
114 6	30,453,590	30,434,316
113 7	495,768,634	495,690,848
112 8	7,002,404,212	7,002,082,880
111 9	87,138,846,064	87,137,701,732
110 10	967,237,561,188	967,233,448,280
109 11	9,672,354,806,324	9,672,341,633,472
108 12	87,857,190,426,762	87,857,148,114,480
107 13	729,890,339,015,124	729,890,215,403,328
106 14	5,578,447,347,506,340	5,578,446,986,386,416
105 15	39,421,026,791,684,152	39,421,025,819,079,244
104 16	258,700,486,756,281,228	258,700,484,140,064,064
103 17	1,582,638,265,236,654,990	1,582,638,258,686,625,504
102 18	9,056,207,842,369,897,698	9,056,207,825,999,272,476
101 19	48,617,536,803,295,178,082	48,617,536,764,944,543,760
100 20	245,518,560,814,069,700,244	245,518,560,724,389,406,200
99 21	1,169,136,003,716,937,778,470	1,169,136,003,519,309,366,060
98 22	5,261,112,016,541,763,342,090	5,261,112,016,106,995,874,280
97 23	22,416,912,069,826,131,025,350	22,416,912,068,920,923,328,320
96 24	90,601,686,281,500,477,155,795	90,601,686,279,618,732,869,760
95 25	347,910,475,318,534,665,936,000	347,910,475,314,819,617,649,960
94 26	1,271,211,352,122,935,979,909,384	1,271,211,352,115,611,772,064,864
93 27	4,425,698,781,456,914,849,108,248	4,425,698,781,443,161,812,224,368
92 28	14,699,642,381,259,834,553,464,448	14,699,642,381,234,041,957,888,288
91 29	4,663,334,824,397,2017,281,896,744	4,663,334,824,3925,833,868,982,784
90 30	141,454,489,673,359,691,507,520,872	141,454,489,673,277,089,121,719,224
89 31	410,674,324,858,072,502,600,358,552	410,674,324,857,931,154,308,999,872
88 32	1,142,187,966,011,457,101,562,409,164	1,142,187,966,011,215,471,310,707,584
87 33	3,045,834,576,030,378,172,825,647,474	3,045,834,576,029,982,283,356,027,984
86 34	7,793,753,179,842,304,178,443,269,582	7,793,753,179,841,656,145,234,151,672
85 35	19,150,364,956,183,541,440,106,097,990	19,150,364,956,182,523,204,755,942,120
84 36	45,216,139,479,877,519,005,495,375,832	45,216,139,479,875,920,443,015,611,752
83 37	102,652,857,197,558,901,889,038,488,854	102,652,857,197,556,489,675,761,794,864
82 38	224,215,451,247,299,146,518,483,944,134	224,215,451,247,295,509,415,708,269,304
81 39	471,427,359,032,781,098,433,790,040,842	471,427,359,032,775,821,266,598,044,912
80 40	954,640,402,041,380,730,670,203,931,050	954,640,402,041,373,079,676,086,262,720
79 41	1,862,712,979,592,934,866,777,089,399,284	1,862,712,979,592,924,181,779,327,085,024
78 42	3,503,674,413,996,233,035,110,445,055,388	3,503,674,413,996,218,123,774,600,549,088
77 43	6,355,502,425,388,510,434,415,173,583,148	6,355,502,425,388,490,372,453,396,946,368
76 44	11,122,129,244,429,890,961,707,126,809,024	11,122,129,244,429,863,988,882,908,985,904
75 45	18,784,040,501,703,807,575,164,139,407,276	18,784,040,501,703,772,587,371,933,573,536
74 46	30,626,152,991,908,378,993,288,608,599,100	30,626,152,991,908,333,639,766,231,416,080
73 47	48,219,900,455,345,095,423,215,234,731,220	48,219,900,455,345,038,666,516,858,355,040
72 48	73,334,431,942,503,996,127,122,897,840,930	73,334,431,942,503,925,147,462,481,638,080
71 49	107,756,716,323,679,325,348,758,295,151,870	107,756,716,323,679,239,606,867,504,704,160
70 50	153,014,537,179,624,639,361,191,172,071,458	153,014,537,179,624,535,854,656,958,090,680
69 51	210,019,952,991,641,642,300,675,366,021,178	210,019,952,991,641,521,554,044,495,863,704
68 52	278,680,322,238,909,101,292,212,428,312,224	278,680,322,238,908,960,524,975,921,227,960
67 53	357,552,111,551,807,881,353,106,541,651,418	357,552,111,551,807,722,713,006,235,751,856
66 54	443,629,471,740,206,076,931,127,682,726,010	443,629,471,740,205,898,263,250,604,257,152
65 55	532,355,366,088,247,269,202,081,059,624,998	532,355,366,088,247,074,629,686,854,396,112
64 56	617,912,478,495,287,014,817,849,899,446,772	617,912,478,495,286,803,059,724,238148864
63 57	693,796,467,082,427,503,413,465,875,637,192	693,796,467,082,427,280,527,729,050,007,160
62 58	753,606,507,348,154,022,809,487,360,189,168	753,606,507,348,153,788,359,679,050,072,384
61 59	791,925,482,298,060,140,610,210,171,032,240	791,925,482,298,059,902,076,562,608,006,656
60 60	805,124,240,336,361,157,749,996,742,071,252	805,124,240,336,360,915,214,413,591,091,584

,

Table 4. Edge colorings of C₁₈₀.

[ ]	Ag	Au
270 0	1	0
269 1	4	1
268 2	345	294
267 3	27,304	26,862
266 4	1,807,917	1,803,450
265 5	96,020,482	95,988,421
264 6	4,240,271,777	4,240,025,284
263 7	159,913,119,758	159,911,591,426
262 8	5,257,122,017,406	5,257,112,226,096
261 9	153,040,502,687,936	153,040,448,776,803
260 10	3,994,356,526,363,671	3,994,356,224,947,782
259 11	9,441,205,959,837,3881	9,441,205,809,631,1423
258 12	2,037,726,939,692,823,911	2,037,726,932,160,044,746
257 13	40,441,042,267,224,568,708	40,441,042,232,772,631,201
256 14	742,381,989,938,234,707,452	742,381,989,780,438,407,424
255 15	12,669,985,960,338,205,685,410	12,669,985,959,668,812,183,074
254 16	201,927,901,238,727,703,128,192	201,927,901,235,891,706,182,400
253 17	3,017,040,406,724,166,624,480,064	3,017,040,406,712,915,359,645,072
252 18	42,406,179,050,007,222,298,565,790	42,406,179,049,962,711,561,688,884
251 19	562,439,848,452,470,787,556,343,504	562,439,848,452,304,546,365,120,176
250 20	7,058,620,098,077,732,214,695,554,434	7,058,620,098,077,113,629,220,104,300
249 21	84,031,191,643,779,442,794,688,130,060	84,031,191,643,777,256,084,971,286,460
248 22	951,080,305,422,767,422,366,226,704,800	951,080,305,422,759,724,718,170,368,000
247 23	10,255,126,771,515,023,854,648,652,352,800	10,255,126,771,514,997,983,993,766,994,400
246 24	105,542,346,356,842,025,862,721,432,364,005	105,542,346,356,841,939,302,159,520,713,540
245 25	1,038,536,688,151,325,201,078,073,328,559,350	1,038,536,688,151,324,923,452,263,653,73,0150
244 26	9,786,211,099,887,486,562,268,711,079,585,360	9,786,211,099,887,485,675,909,620,716,272,160
243 27	88,438,352,161,946,171,819,410,469,089,895,264	88,438,352,161,946,169,097,774,234,297,660,864
242 28	767,518,556,262,604,268,829,301,580,115,605,632	767,518,556,262,604,260,510,851,755,374,513,280
241 29	6,404,810,021,225,870,079,832,897,841,141,394,432	6,404,810,021,225,870,055,311,800,035,487,058,944
240 30	51,451,973,837,181,156,242,454,272,224,708,534,272	51,451,973,837,181,156,170,504,106,998,886,072,320

and

Table 5. Edge Colorings of C₂₄₀.

[ ]	Ag	Au
360 0	1	0
359 1	5	1
358 2	599	523
357 3	64,695	63,867
356 4	5,742,088	5,732,528
355 5	408,394,963	408,310,967
354 6	24,161,376,447	24,160,630,719
353 7	1,221,853,155,773	1,221,847,572,889
352 8	53,914,173,386,252	53,914,131,840,160
351 9	2,108,642,343,606,824	2,108,642,069,551,032
350 10	74,013,342,528,379,652	74,013,340,742,388,172
349 11	2,354,969,960,511,184,484	2,354,969,949,897,381,076
348 12	68,490,376,233,711,549,198	68,490,376,171,529,817,312
347 13	1,833,434,686,063,143,560,788	1,833,434,685,724,940,441,732
346 14	45,442,988,287,129,208,605,684	45,442,988,285,316,727,072,676
345 15	1,048,218,263,136,994,898,032,224	1,048,218,263,127,866,791,758,384
344 16	22,602,206,298,818,610,054,452,936	22,602,206,298,773,304,345,806,944
343 17	457,362,292,163,918,680,987,994,882	457,362,292,163,705,502,108,517,978
342 18	8,715,292,567,344,302,910,553,154,422	8,715,292,567,343,313,881,950,682,458
341 19	156,875,266,212,189,736,812,649,026,446	156,875,266,212,185,357,989,454,884,534
340 20	2,674,723,288,917,808,199,876,682,916,172	2,674,723,288,91,778,907,356,818,542,3648
339 21	43,305,043,725,335,812,080,811,714,363,506	43,305,043,725,335,731,946,246,037,299,994
338 22	667,291,355,585,855,940,114,394,440,368,422	667,291,355,585,855,608,684,210,761,554,798
337 23	9,806,281,660,348,663,563,010,529,539,308,321	9,806,281,660,348,662,242,732,670,525,711,097
336 24	137,696,538,314,062,477,732,769,369,588,040,480	137,696,538,314,062,472,537,951,467,943,685,984
335 25	1,850,641,474,940,999,672,906,554,426,005,029,888	1,850,641,474,940,999,653,153,302,290,271,031,040
334 26	23,844,803,619,432,111,082,681,856,449,184,387,072	23,844,803,619,432,111,008,453,439,294,860,623,872

.

However, for any given color partition, the respective coefficient can be directly obtained from the GFs without having to expand the entire multinomial function for the edge colorings. Thus, we have constructed the tables of our enumerations for the A_g and A_u IRs of the I_h group for C₈₀, C₁₈₀, and C₂₄₀ only for binomial colorings. Owing to a large number of edges for the giant fullerenes, both the number of terms in the expansion and the coefficients of various terms result in a combinatorial explosion for the edge colorings. Table 3, Table 4 and Table 5 show our computed results for the three giant fullerenes for only binomial colorings (black and white colors) for coloring the edges for the A_g and A_u IRs. Although as shown above, we can obtain a more itemized distribution for the edge coloring such as pent–hex, hex1–hex1, hex1–hex2… colorings depending on the number of equivalence classes of hex–hex edges, for the sake of simplicity, we show in Table 3, Table 4 and Table 5 only the overall binomial color distributions without making further distinction into the types of edges. Table 3 shows the binomial colorings of 120 edges of C₈₀, where in Table 3, k represents the number of black colors while 120-k is the number of white colors. We have shown the results for both A_u and A_g IRs where the numbers for A_u yield the numbers of chiral pairs for each edge-coloring partition shown in Table 3. As can be seen from Table 3, one needs a minimum of two black colors with the remaining 118 edges colored in white in order to produce a chiral edge coloring. As the number of black colors increases, the number of chiral colorings also increases, reaching a maximum of 805,124,240,336,360,915,214,413,591,091,584 chiral pairs. The corresponding number for A_g is 805,124,240,336,361,157,749,996,742,071,252, suggesting that as we approach the peak of the binomial distribution, almost every edge coloring is chiral.

Table 4 shows our computed results as for the binomial edge colorings of C₁₈₀. A striking contrast between C₁₈₀ and C₈₀ as well as C₆₀ is that the number of A_g colorings for k = 1 is 4 for C₁₈₀, which means there are 4 kinds of edges for C₁₈₀, unlike the smaller icosahedral fullerenes. Moreover, there is exactly one chiral pair of edge coloring for the case k = 1 as inferred from Table 4 for the A_u IR. There 5 edge colorings for C₁₈₀ with 269 white colors and 1 black color, among which there exists a chiral pair: a result that was not known until now. For the case of k = 2 or two black colors, it is seen that among 639 edge colorings for C₁₈₀, there are 294 chiral pairs, and the remaining 51 colorings are totally achiral. Analogous to C₈₀, the number of edge colorings grows combinatorially, resulting in an explosion already for k = 30, and hence, numerical results are not shown beyond k = 30. We see from Table 4 that almost all edge colorings of C₁₈₀ are chiral. Although we do not consider the case of C₁₄₀ icosahedral fullerene, the parent cage itself is chiral, as it belongs to the I group. Consequently, every edge coloring of C₁₄₀ is chiral, which is a direct consequence of the chirality of the parent cage. The results in Table 5 for the edge colorings of C₂₄₀, the largest of the fullerenes considered here, reveal that there are 6 edge colorings for the one black color and the remaining white colors, among which there is exactly a chiral pair, and the remaining 4 colorings are achiral.

The numbers for k = 2 (two black colors) suggest that there are 1122 edge colorings with 523 chiral pairs and 76 achiral colorings. As seen from Table 5, for k = 26 (26 black colors), there are 23,844,803,619,432,111,008,453,439,294,860,623,872 chiral pairs of edge colorings and 74,228,417,154,323,763,200 achiral colorings for C₂₄₀.

The total numbers of all edge colorings, the number of chiral pairs, and the number of achiral colorings can be readily derived from the numbers enumerated in the tables for C₈₀ through C₂₄₀. Let N_k(A_g) and N_k(A_u) be the numbers enumerated in the tables for the various values of k. Thus, we obtain:

N_k (Total; Binomial Edge colors) = N_k(A_g) + N_k(A_u)

(15)

C_k (Chiral; Binomial Edge colors) = 2N_k(A_u)

(16)

AC_k (Achiral; Binomial Edge colors) = N_k(A_g) − N_k(A_u).

(17)

We also note that as k approaches 3n/4, where n is the number of vertices of the fullerene, we obtain Equation (18):

$$\underset{k\to \frac{3n}{4}}{\lim }\frac{2C_k}{N_k\left(A_g\right)+N_k\left(A_u\right)}=1.$$

(18)

As can be seen from Table 2, Table 3, Table 4 and Table 5, the numbers of edge colorings rise astronomically even for 2 colorings, and hence, we obtain analytical expressions for any k. Such expressions can be derived from the original GCCI for the A_g and A_u IRs for the edge colorings of the giant C₂₄₀ fullerene cage as shown below:

$$\begin{gathered} {\mathrm{N}}_{\mathrm{k}}(A_g;{\mathrm{C}}_{240};\text{ k black edge colors})=\frac{1}{120}\left\{\left(\begin{array}{c}360\\ k\end{array}\right)+15\eta \left(\frac{k}{2}\right)\left[\left(\begin{array}{c}180\\ \frac{k}{2}\end{array}\right)\right]+24\eta \left(\frac{k}{5}\right)\left(\begin{array}{c}72\\ \frac{k}{5}\end{array}\right)+ \\ +20\eta \left(\frac{k}{3}\right)\left(\begin{array}{c}120\\ \frac{k}{3}\end{array}\right)+\eta \left(\frac{k}{2}\right)\left[\left(\begin{array}{c}180\\ \frac{k}{2}\end{array}\right)\right]+24\eta \left(\frac{k}{10}\right)\left(\begin{array}{c}36\\ \frac{k}{10}\end{array}\right)+20\eta \left(\frac{k}{6}\right)\left(\begin{array}{c}60\\ \frac{k}{6}\end{array}\right)+ \\ 15{{\displaystyle \sum }}_{\left[\lambda \right]}\left(\begin{array}{c}16\\ \left[\lambda \right]\end{array}\right)\left(\begin{array}{c}172\\ k-\left[\lambda \right]\end{array}\right)\right\}, \end{gathered}$$

(19)

where $\eta \left(\frac{k}{d}\right)=1ifddivideskand0otherwise$, where [ ] is an ordered partition of k into 2 parts such that both parts must be even.

$$\begin{gathered} {\mathrm{N}}_{\mathrm{k}}(A_g;{\mathrm{C}}_{240};\text{ k black edge colors})=\frac{1}{120}\left\{\left(\begin{array}{c}360\\ k\end{array}\right)+15\eta \left(\frac{k}{2}\right)\left[\left(\begin{array}{c}180\\ \frac{k}{2}\end{array}\right)\right]+24\eta \left(\frac{k}{5}\right)\left(\begin{array}{c}72\\ \frac{k}{5}\end{array}\right)+ \\ +20\eta \left(\frac{k}{3}\right)\left(\begin{array}{c}120\\ \frac{k}{3}\end{array}\right)-\eta \left(\frac{k}{2}\right)\left[\left(\begin{array}{c}180\\ \frac{k}{2}\end{array}\right)\right]-24\eta \left(\frac{k}{10}\right)\left(\begin{array}{c}36\\ \frac{k}{10}\end{array}\right)-20\eta \left(\frac{k}{6}\right)\left(\begin{array}{c}60\\ \frac{k}{6}\end{array}\right)- \\ 15{{\displaystyle \sum }}_{\left[\lambda \right]}\left(\begin{array}{c}16\\ \left[\lambda \right]\end{array}\right)\left(\begin{array}{c}172\\ k-\left[\lambda \right]\end{array}\right)\right\}, \end{gathered}$$

(20)

where $\eta \left(\frac{k}{d}\right)=1ifddivideskand0otherwise.$

We can also obtain the number of inequivalent ways of labeling all the edges of an icosahedral fullerene with n vertices as follows:

$${\mathrm{N}}_{\mathrm{Lab}}({\mathrm{I}}_{\mathrm{h}};{\mathrm{C}}_{\mathrm{n}};\text{ fullerene wit n vertices})=\frac{\left(\left(3n\right)/2\right)!}{120}.$$

(21)

We can divide the above expression further into pent–hex and hex–hex edges with expressions (22) and (23), respectively:

$${\mathrm{N}}_{\mathrm{Lab}}({\mathrm{C}}_{\mathrm{n}}{\mathrm{I}}_{\mathrm{h}};\mathrm{pent}-\mathrm{hex})=\frac{60!}{120}$$

(22)

$${\mathrm{N}}_{\mathrm{Lab}}({\mathrm{C}}_{\mathrm{n}}{\mathrm{I}}_{\mathrm{h}};\mathrm{hex}-\mathrm{hex})=\frac{\left\{\frac{3n-120}{2}\right\}!}{120}.$$

(23)

4. Conclusions

In the present study, we have outlined powerful combinatorial techniques that synthesized Möbius inversion with the generalized version of Sheehan’s modification of Pόlya’s theorem to all characters of the irreducible representations of the icosahedral group and applied the techniques to enumerate the edge colorings of giant fullerene cages. We developed a computer code that was used to construct the results specifically for the binomial chiral and achiral edge colorings of giant fullerenes. We also showed how the techniques can be divided into combinatorial parts with varied color partitions for the pent–hex and hex–hex edges contained in these giant fullerenes. The existence of chiral colorings even for one black edge coloring was demonstrated for the C₁₈₀ and C₂₄₀ fullerenes. It was shown that the ratios of chiral colorings and total number of edge colorings asymptotically reach unity as the number of black colors becomes almost equal to the number of white colors. The present enumeration of edge colorings, especially with restrictions to nearest neighbor exclusion, can be of immense use in the combinatorics of equivalence classes of Kekulė structures and thus in the computations of stabilities of giant fullerenes using the conjugated circuit method or the topological resonance theory method. The present techniques considered in this study could have applications to NMR, multiple quantum NMR, and ESR where one seeks the number of different types of dipolar couplings or edge colorings. The present enumeration techniques can also be extended in the future to nanomaterials such as carbon nanotubes, carbon nanocones, nanotori, and other nanomaterials of ongoing experimental interest.