#
Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C_{80}, C_{180}, and C_{240} ^{ †}

^{†}

## Abstract

**:**

_{80}to C

_{240}. 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 C

_{80}with a chamfered dodecahedron structure, icosahedral C

_{180}, and C

_{240}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

_{180}and C

_{240}[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 H

_{2}, N

_{2}, and O

_{2}, 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 = m

^{2}of which C

_{60}, C

_{240}, C

_{540}, C

_{960}, 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 C

_{48}N

_{12}[37,65].

_{80}, C

_{180}, and C

_{240}. We show that the Möbius inversion technique combined with our generalization of Sheehan’s modification

^{54}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

^{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].

_{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.

_{20}dodecahedron, and the C

_{60}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 C

_{240}icosahedral fullerene. Each of the proper and improper rotational operations of the icosahedron induces a permutation on the set of 360 edges of C

_{240}, a permutation on the 240 vertices, and a permutation on 122 faces of the giant C

_{240}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 C

_{5}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 C

_{5}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 C

_{5}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 C

_{5}axis to pass through two vertices that are across each other for the icosahedron. For giant fullerenes such as C

_{240}, the C

_{5}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 5

^{48}. As seen from Table 1, the conjugacy class representatives of the icosahedral fullerene group comprises the set {E, C

_{6}, C

_{5}, C

_{3}, C

_{2}, C

_{i}, S

_{10}, S

_{12}, S

_{6}, 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.

_{1}, Y

_{2}, … Y

_{m}where Y

_{i}is an equivalence class partition of the edges of the fullerene. That is, ${{\displaystyle \cup}}_{i=1}^{m}{Y}_{i}=D,\text{}{Y}_{i}{\displaystyle \cap}=\varnothing ,\text{}for\text{}i\ne j$, where m = number of the edges of the fullerene cage.

_{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.

_{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).

_{1}≥ 0, n

_{2}≥ 0, …, n

_{p}≥ 0, ${{\displaystyle \sum}}_{i=1}^{p}{n}_{i}=n$. We can obtain a multinomial generating function (GF) in λs with n

_{1}colors of the type λ

_{1}, n

_{2}colors of the type λ

_{2}… n

_{p}colors of the type λ

_{p}, which are defined as follows:

_{1}, R

_{2}… R

_{m}with the same number of partitions as the Y sets such that $R={R}_{i}{{\displaystyle \cup}}^{\text{}}{R}_{j,\text{}}{R}_{i}{{\displaystyle \cap}}^{\text{}}{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

_{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

_{80}(I

_{h}), C

_{180}(I

_{h}), and C

_{240}(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 C

_{80}(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 C

_{80}as seen from Figure 2 analogous to buckminsterfullerene C

_{60}(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 C

_{80}, there are two equivalent class edges: namely, Y

_{1}and Y

_{2}. 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 Y

_{1}is 60. This leaves 60 edges that are common to two hexagons in the case of C

_{80}; or, the cardinality of the set Y

_{2}is also 60 for the case of C

_{80}. However, as can be seen from Table 1, the cardinalities of sets Y

_{1}and Y

_{2}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 C

_{80}(hex–hex) edges to obtain the GCCIs for the A

_{g}and A

_{u}irreducible representations of the I

_{h}group shown below:

_{80}became the same and equal to 60, which is not the case in general. For the C

_{60}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 C

_{60}for the A

_{g}and A

_{u}IRs, as shown below:

_{g}of C

_{60}are shown first in Equation (10).

_{u}IR is obtained in an analogous manner, and it shown as Equation (11):

_{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 C
_{60}are colored with one color (white) and pent–hex edges are colored with multiple colors; - (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.

_{h}group, thus yielding a powerful set of enumerative combinatorial generating functions. Hence, we consider each such case.

_{60}are obtained by setting d and e to 0 in Equation (10), which yields Equation (12):

^{15}a

^{15}b

^{15}c

^{15}= a

^{15}b

^{15}c

^{15}, and hence, the coefficient a

^{15}b

^{15}c

^{15}in the multinomial expansion gives the number of ways 60 pent–hex edges of C

_{60}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].

_{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 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 $\left(\begin{array}{c}30+5-1\\ 30\end{array}\right)=\left(\begin{array}{c}34\\ 30\end{array}\right)=\mathrm{46,376}$.

_{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 C

_{80}(I

_{h}) without differentiating the pent–hex and hex–hex edges. Thus, the expression obtained is shown in Equation (14):

_{g}and A

_{u}IRs of the I

_{h}group for C

_{80}, C

_{180}, and C

_{240}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 C

_{80}, 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.

_{180}. A striking contrast between C

_{180}and C

_{80}as well as C

_{60}is that the number of A

_{g}colorings for k = 1 is 4 for C

_{180}, which means there are 4 kinds of edges for C

_{180}, 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 C

_{180}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 C

_{180}, there are 294 chiral pairs, and the remaining 51 colorings are totally achiral. Analogous to C

_{80}, 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 C

_{180}are chiral. Although we do not consider the case of C

_{140}icosahedral fullerene, the parent cage itself is chiral, as it belongs to the I group. Consequently, every edge coloring of C

_{140}is chiral, which is a direct consequence of the chirality of the parent cage. The results in Table 5 for the edge colorings of C

_{240}, 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.

_{240}.

_{80}through C

_{240}. 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:

_{k}(Total; Binomial Edge colors) = N

_{k}(A

_{g}) + N

_{k}(A

_{u})

_{k}(Chiral; Binomial Edge colors) = 2N

_{k}(A

_{u})

_{k}(Achiral; Binomial Edge colors) = N

_{k}(A

_{g}) − N

_{k}(A

_{u}).

_{g}and A

_{u}IRs for the edge colorings of the giant C

_{240}fullerene cage as shown below:

## 4. Conclusions

_{180}and C

_{240}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.

## Funding

## Conflicts of Interest

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**Figure 1.**A regular icosahedron (

**a**), a combo of vertex–edge coloring of the icosahedron with 5 colors (

**b**), where an edge incident on two vertices is defined with the corresponding vertex colors—for example, blue-red edge (

**b**), C

_{20}dodecahedral fullerene (

**c**), and C

_{60}Buckminsterfullerene (

**d**)—all with I

_{h}icosahedral symmetry. The C

_{5}axes pass through each vertex of the icosahedron colored to show the pentagon, and the C

_{3}axes pass through the center of triangular face, while the C

_{2}axes pass through the center of an edge.

**Figure 2.**Strucural representations for C

_{80}, C

_{180}, and C

_{240}fullerene cages considered here for edge colorings. Panel (

**d**) is reproduced from https://robertlovespi.net/ and it is copyright-free.

**Table 1.**Character table of I

_{h}, permutation cycles and their lengths for the 60 pent–hex edges and remaining hex–hex edges of C

_{60}(I

_{h}), Icosahedron, C

_{80}(I

_{h}), C

_{180}(I

_{h}), and C

_{240}(I

_{h}) upon action on edges.

^{a}

CC | E | 15C_{2} | 12C_{5} | 12C_{5}^{2} | 20C_{3} | I | 12S_{10} | 12S_{10}^{3} | 20S_{6} | 15σ_{d} |
---|---|---|---|---|---|---|---|---|---|---|

C_{60} (30 hex–hex) | 1^{30} | 1^{2}2^{14} | 5^{6} | 5^{6} | 3^{10} | 2^{15} | 10^{3} | 10^{3} | 6^{5} | 1^{4}2^{13} |

Icosahedron | ||||||||||

(30 edges) | ||||||||||

C_{60} (pent–hex edge) | 1^{60} | 2^{30} | 5^{12} | 5^{12} | 3^{20} | 2^{30} | 10^{6} | 10^{6} | 6^{10} | 1^{4} 2^{28} |

C_{80} (hex–hex); | 1^{60} | 2^{30} | 5^{12} | 5^{12} | 3^{20} | 2^{30} | 10^{6} | 10^{6} | 6^{10} | 1^{4} 2^{28} |

chamfered dodecahedron | ||||||||||

C_{80} (pent–hex; edges) | 1^{60} | 2^{30} | 5^{12} | 5^{12} | 3^{20} | 2^{30} | 10^{6} | 10^{6} | 6^{10} | 1^{4} 2^{28} |

C_{180} (hex–hex) | 1^{210} | 1^{2}2^{104} | 5^{42} | 5^{42} | 3^{70} | 2^{105} | 10^{21} | 10^{21} | 6^{35} | 1^{8}2^{101} |

C_{180} (pent–hex) = C_{60} (pent–hex);Vertices: chamfered dodecahedron | 1^{60} | 2^{30} | 5^{12} | 5^{12} | 3^{20} | 2^{30} | 10^{6} | 10^{6} | 6^{10} | 1^{4} 2^{28} |

C_{240} (hex–hex) | 1^{300} | 2^{150} | 5^{62} | 5^{62} | 3^{100} | 2^{150} | 10^{30} | 10^{30} | 6^{50} | 1^{12}2^{144} |

C_{240} (pent–hex) | 1^{60} | 2^{30} | 5^{12} | 5^{12} | 3^{20} | 2^{30} | 10^{6} | 10^{6} | 6^{10} | 1^{4} 2^{28} |

C_{60} (pent–hex); chamfered truncated icosahedron | ||||||||||

A_{g} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

T_{1g} | 3 | −1 | ω | ω * | 0 | 3 | ω | ω | 0 | −1 |

T_{2g} | 3 | −1 | ω * | ω | 0 | 3 | ω | ω * | 0 | −1 |

G_{g} | 4 | 0 | −1 | −1 | 1 | 4 | −1 | −1 | 1 | 0 |

H_{g} | 5 | 1 | 0 | 0 | −1 | 5 | 0 | 0 | −1 | 1 |

A_{u} | 1 | 1 | 1 | 1 | 1 | −1 | −1 | −1 | −1 | −1 |

T_{1u} | 3 | −1 | ω | ω * | 0 | −3 | −ω * | −ω | 0 | 1 |

T_{2u} | 3 | −1 | ω * | ω | 0 | −3 | −ω | −ω * | 0 | 1 |

G_{u} | 4 | 0 | −1 | −1 | −1 | −4 | 1 | 1 | 1 | 0 |

H_{u} | 5 | 1 | 0 | 0 | −1 | −5 | 0 | 0 | 1 | −1 |

^{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.

**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 | A_{g} | A_{u} |
---|---|---|

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.

[ ] | A_{g} | A_{u} |
---|---|---|

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 |

[ ] | A_{g} | A_{u} |
---|---|---|

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 |

[ ] | A_{g} | A_{u} |
---|---|---|

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 |

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## Share and Cite

**MDPI and ACS Style**

Balasubramanian, K.
Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C_{80}, C_{180}, and C_{240} . *Symmetry* **2020**, *12*, 1308.
https://doi.org/10.3390/sym12081308

**AMA Style**

Balasubramanian K.
Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C_{80}, C_{180}, and C_{240} . *Symmetry*. 2020; 12(8):1308.
https://doi.org/10.3390/sym12081308

**Chicago/Turabian Style**

Balasubramanian, Krishnan.
2020. "Combinatorics of Edge Symmetry: Chiral and Achiral Edge Colorings of Icosahedral Giant Fullerenes: C_{80}, C_{180}, and C_{240} " *Symmetry* 12, no. 8: 1308.
https://doi.org/10.3390/sym12081308