Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties

: This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modiﬁcation of P ó lya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperﬁne patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperﬁne patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.


Introduction
Two-dimensional nanomaterials, especially those arising from different forms of graphenes, for example, circumscribed-cyclopolyarenes, tessellations of kekulenes, septulenes and octulenes, nanobelts, and various forms of macrocycles that have pores have received significant attention over the years . These nanomaterials not only possess interesting electronic properties but their pores with desirable electronic and geometrical features have opened up a plethora of novel applications . Related one-dimensional nanomaterials such as carbon nanotubes and heteronanotubes and nanowires composed of B/N, Ga/N, C/N, Ga/As have been the topic of a number of studies because these materials exhibit very interesting electronic, geometrical, topological, chiral, polarizability, and optical properties . Heteronuclear one-dimensional nanomaterials have novel optoelectronic properties and thus find many applications in the fabrication of nanodevices, for example, photonics, nanoelectronics, nanochirality and so forth . Two-dimensional nanomaterials and mesoporous materials with cavities, for example, nanobelts comprising of several kekulene/octulene moieties and tubular forms of carbons and nitrogens are becoming increasingly important, as they exhibit optimal cavities for sequestration and transport of both anions such as chloride ions and cations such as toxic heavy metal ions including actinyl ions in high level nuclear wastes and other nuclear environmental geo/biochemical applications [60][61][62][63][64][65]. Specifically, these environmental actinide studies have considered high coordination complexes of actinyls with mesoporous silica [61], minerals such as carbonates [63], phosphates [64] and solvated complexes [62,65]. Our understanding of such actinide complexes and the related laser spectroscopic studies [64] requires a detailed knowledge of their symmetries, coordination spheres, electronic structure, and symmetries of various electronic states and their geometries. For example, as shown in [63] the plutonyl carbonates form high coordination and high symmetry complexes which in turn govern their spectroscopic selection rules and the symmetry aspects of their electronic states. The high coordination and symmetries exhibited by these actinide complexes are in turn influenced by the symmetries of the 5f and 6d orbitals of actinides [62][63][64][65] within the symmetry of molecular or nanosphere environment. The symmetry, combinatorial, graph-theoretical and topological properties of polycyclic aromatics with cavities have been studied over the years . Such polycyclic aromatics, for example, those made by circumscribing cyclopolyarenes are especially intriguing from the standpoint of enumerations of isomers, their electronic and magnetic properties which have given rise to a number of interesting concepts such ring currents, topological delocalization energies, conjugated circuits, aromaticity and superaromaticity . Cycloarenes are derived from a synthesis of angular and linear annulations of benzene rings that result macro-cyclic systems with cavities that are especially suitable for sequestration of both anions and cations. As they exhibit intriguing magnetic as well as electronic properties, several graphtheory-based and combinatorial methods have been developed and employed as robust alternatives over the years to study the spectroscopic properties, electronic structures, and energetics of polycyclic aromatics, nanotubes and fullerenes including giant fullerenes over the decades . Among several computational methods, topologically-based methods such as the conjugated circuit method, ring currents, topological resonance theories, aromatic sextets, spectral and matching polynomials etc., and combinatorial enumeration methods have been of considerable use in rapid and robust enumerations of structures and spectra as well as estimation for their stabilities , The advent of kekulene made of 12 fused benzene rings with a central cavity with D 6h symmetry stimulated significant interest in superaromaticity or superbenzene as a consequence of its planar cyclic conjugation [1][2][3][4]. A novel electronic feature of kekulene is that its π-electrons are delocalized within benzene rings as opposed to the entire framework providing a platform to study aromaticity in a new dimension. Moreover their structures with cavities offer intriguing possibilities for the environmental management, organic chemistry, rational drug design and delivery, as they possess optimal electronic and geometrical features for both trapping of toxic ions to transportation of heavy metal ions, halide anions, drugs and so forth .
Carbon nanotubes and their heteroanalogs such as GaN, GaAs nanowires and related decorations with metals such as gold have opened up a new vista and a new state of matter that has culminated into significant research activities over the years . These nanotubes exhibit intriguing optical properties and chirality, and thus their relative stabilities and reactivities in different forms such as zigzag, armchair, chiral etc., have been the subject matters of intense scrutiny over the years [34,35,39,57]. Heteronanotubes are especially interesting because of their structures that contain intertwining helical patterns . Moreover chirality of heteronanotubes is caused by intriguing chiral interface states that give rise to very interesting conductivity patterns [35,39]. In particular heteronanotubes made of Ga/N, Ga/As, C/N and B/N as well as transition metals are promising materials as building blocks of metal organic frameworks. These nanomaterials find a variety of applications due to their unusual hyperpolarizability and conductive properties-all of which arise because of interesting combinatorics. These materials exhibit interesting aesthetics and thus the entire collection of nanowires, fullerenes, graphenes, giant fullerenes, nanotubes, and their heteroanalogs have been studied over the years . Both structures and dynamics arising from rearrangements of pentagons on nanocones through Stone-Wales rotations of faces, and thus face colorings of fullerenes have been of interest [130]. Combinatorial studies can provide enlightening topological information on the isomerization Symmetry 2022, 14, 34 3 of 39 paths pertinent to the dynamics of pentagons on nanocones. The structures, enumeration of isomers arising from substituents of heteroatoms, chirality, spectroscopy, and topological indices of all these nanomaterials have been studied in recent times [120][121][122][123][124][125][126][127][128][129][130][131][132][133][134]. Furthermore other related fields of biological interest such as computational toxicology, drug discovery and design have been benefited by the techniques of combinatorial and topological origin such as quantitative molecular similarity analysis, quantitative structure activity relations, and targeted therapeutic approaches for computer aided drug discovery [135][136][137][138][139][140][141][142][143]. Quantitative descriptors are derived from their structures which in turn can yield quantitative measures from their molecular topologies, quantum chemically derived electronic parameters, shape, and other biological descriptors [135][136][137][138][139][140][141][142][143]. Combinatorial techniques not only facilitate the constructions of large data sets or libraries of a number of chemical compounds including nanomaterials but also aid in rapid computations of chemical properties of combinatorial libraries of molecules [106][107][108][109][110][111][112][120][121][122][123][124][125][126][127][128][129][130][131][132][133][134][143][144][145][146][147]. The related metal organic frameworks have opened up an entire field of reticular chemistry and MOFs and related COFs have enormous potentials in numerous applications including water harvesting [148,149]. Furthermore symmetry-based combinatorial enumeration of electronic states and geometries of heteronuclear clusters such as Ga m As n [150,151] are critical to planning and interpretations of computations of heteronuclear clusters, heteronuclear nanomaterials and nanowires comprised of Ga/N, Ga/As, B/N, C/N, and so forth. Relativistic electronic structure computations of molecules and clusters [152] are benefited by the inherent double group symmetries when spin-orbit coupling is included. The symmetry interplay between Jahn-Teller and spin-orbit effects is critical to our understanding of laser spectroscopic studies of clusters containing heavy atoms such as gold [153] and so forth. Detailed understanding of clusters and heteronano materials and heteronanowires is enhanced by the enumeration of structures, spectroscopic patterns, nuclear spin statistics and the symmetry properties of the electronic potential energy surfaces of such heteronuclear clusters [150,151] and molecules.
Topological entropy of a nanomaterial is an important information-theoretic concept which appears to characterize the information content or order/disorder content of such nanomaterials. Consequently, graph entropy has been a subject matter of some recent works on tessellations of kekulenes and metal organic frameworks [144,154]. The relative stabilities of various phases of graphenes and nanotubes, for example, the chiral form, zigzag, armchair, and so forth, have been the topic of several studies [35,39], it is believed that graph entropies can provide additional insights into these materials. Phase transformation of armchair to zigzag graphene edge structures, since the synthesis of single-walled carbon nanotubes, has been of interest, especially as such studies could provide insights into the origin of chirality in nanotubes [39]. The relative stabilities of different phases depend on their Gibbs free energies which in turn depend on both energetics and entropies [39][40][41]144]. Stimulated by such vast studies on these nanomaterials, we have undertaken the present review of this interesting subject with focus on their combinatorial properties.
The objective of this review is to outline symmetry-based combinatorial techniques, and in particular, Sheehan's modification of Pólya's theorem that the present author has generalized to all irreducible representations as well as computational techniques based on such enumerations including walks and sequences [134,[155][156][157][158][159]. Group-theoretically based combinatorial structures for all irreducible representations of the point groups of the nanomaterials such as tessellations of cyclopolyarenes, heteronanotubes, etc., are applied to the enumerations of chiral structures, achiral structures and position isomers of such materials. We have also reviewed the used of topologically based techniques such as distance degree sequence vectors for partitioning the vertices of nanomaterials for rapid and robust computations of their thermodynamic and spectroscopic properties. We have applied the techniques to demonstrate the rapid construction of the ESR hyperfine patterns of nanomaterials as well as 13 C NMR and proton NMR spectroscopic patterns including multiple quantum NMR. This review brings together combinatorics, group theory and topological techniques with applications to heteronanotubes, circumscribed-|Y i |= 2m, for a nanotube of even length n, for all i, 1 ≤ i ≤ n/2.
(1) |Y 1 |= m, |Y i |= 2m |for a nanotube of odd length n, for all i = 1, 2 ≤ i ≤ (n + 1)/2. (2) Consequently, a vertex coloring of the set of vertices in D, a map from the sets D to R, with R as the set of colors and D further divided into Y-sets can be represented as: Sheehan's modification [157] of Pólya's theorem [156] facilitates partitioning of colorings for the various equivalence classes, and thus it is more powerful than the ordinary Pólya's combinatorial enumeration. The current author has further generalized Sheehan's theorem to all characters of irreducible representations of the group of action, and thus a direct technique has been developed to enumerate both chiral and achiral colorings. Suppose the generalized character cycle index (GCCI) for the character χ of the irreducible representation Γ is defined by where the sum is over all permutation representations of g ∈ G; c ij (g) is the number of j-cycles of g ∈ G contained in the set Y i upon its action on the members of the objects in the D set which may be for example carbon centers of a nanotube. The index i varies from 1 to n/2 or (n + 1)/2 for even and odd n, respectively. The second index j is the orbit length for the orbit of a permutation contained within the Y i set for the action of g ∈ G.
A number of nanostructures contain cross sections with cyclic rotational symmetries, and hence the rotational subgroup for the cyclic part yields a cycle index provided by the Euler totient function. Therefore such an enumeration scheme can be applied to a variety of nanostructures such as cylindrical nanotubes, circumscribed-kekulenes, septulenes, octulenes, tessellations of kekulenes and nanosheets and nanobelts derived from graphenes.
We can obtain multinomial generators from the GCCIs shown above for each of the irreducible representations in the group G acting on the structure. For this purpose we introduce [n] as an ordered partition of n into p parts given that n 1 ≥ 0, n 2 ≥ 0, . . . , n p ≥ 0, ∑ p i=1 n i = n. Let us assign arbitrary weights λs and n 1 colors of the type λ 1 , n 2 , colors of the type λ 2 . . . n p colors of the type λ p . Then a multinomial generator is constructed by a generalization of Pólya's enumeration to all IRs using the multinomial expansion: n n 1 n 2 . . n p are multinomials defined as n n 1 n 2 . . n p = n! n 1 !n 2 ! . . . . . . n p−1 !n p !
Furthermore, the coloring palette, R can be divided into sets R 1 , R 2 . . . such that R i for an odd m and |R i | = p i Hence in the most general case, the weight w ij is assigned for each color r j in the set R i . In such a set up the multinomial generator for each IR for coloring the vertices of structures that can be divided is obtained as follows: The current author provided a geometrical interpretation for the above expansion. That is, the multinomial function thus obtained for each IR different colors yield the equivalence classes of vertex colorings that transform according to the IR with the character χ. It can thus be seen that the number of such multinomial generators equals the number of IRs of the acting group.

Enumeration of Hetero-Substituted Polyarenes and Polysubstituted Polyarenes
Circum-polyarenes are usually planar circumscribed structures comprising of macrocycles that are reminiscent of donut structures (See Figure 1) with D mh point group symmetries and the combinatorics of these structures were considered by the author in [124]. In this section we consider these cicum-polyarenes of considerable recent interest. A special case of such structures are kekulenes for m = 6 for ( Figure 1, Structure 1). Likewise other circumpolyarenes are obtained with holes, for example, when m = 7 for septulenes ( Figure 1, Structure 2), m = 8 corresponds to octulenes (Figure 1, Structure 3) and so forth. Table 1 displays such a general series comprising of circum-polyarenes together with their general formula for various values of m and n. We define m as the cyclicity and n as the number of circumscribings around the primitive structure (n = 1). As most structures of circumpolyarenes are considered to be planar, the combinatorics of polysubstituted isomers of such circum-polyarenes can be constructed within the rotational subgroup or the D m group. The GCCIs of such structures depend on the point groups as well as the set D (carbon nuclei or protons) and also if the cardinality of the set D is odd or even. Let D be the set of carbon nuclei of polyarenes (structures are displayed in Figures 1 and 2). When carbon vertices are colored in effect, we would be enumerating the hetero-polyarenes for example, aza-polyarenes. For circumscribed m-cyclic polyarenes let n be the order of circumscribing (for example, for Structure 8 in Figure 2, n = 2). In the case of regular kekulene (Structure 1; Figure 1), it can be seen that n = 1 and m = 6 and for circumscribed kekulene (Structure 8; Figure 2), n = 2 and m = 6 while the corresponding values for circumscribed septulenes are n = 2 and m = 7 For the sake of self-completeness and illustrations, we have used the same notations and reproduced the equations from Ref. [124].
Symmetry 2021, 13, x FOR PEER REVIEW 6 of 40 the set of carbon nuclei of polyarenes (structures are displayed in Figures 1 and 2). When carbon vertices are colored in effect, we would be enumerating the hetero-polyarenes for example, aza-polyarenes. For circumscribed m-cyclic polyarenes let n be the order of circumscribing (for example, for Structure 8 in Figure 2, n = 2). In the case of regular kekulene (Structure 1; Figure 1), it can be seen that n = 1 and m = 6 and for circumscribed kekulene (Structure 8; Figure 2), n = 2 and m = 6 while the corresponding values for circumscribed septulenes are n = 2 and m = 7 For the sake of self-completeness and illustrations, we have used the same notations and reproduced the equations from Ref. [124].    All structures in Figure 2 exhibit D6h symmetry and they arise from circumcising the graphene structure with holes of various sizes. In particular, the kekulene series belong to the CpHq series with p = 6(n 2 + 4n + 3) and q = 6 (n + 3). Likewise structures 7,10,14 ( Figure  2) correspond to the series CpHq where p = 6(n 2 + 6n + 5) and q = 6 (n + 5) and structures 9,13 correspond to CpHq where p = 6(n 2 + 8n + 7) and q = 6 (n + 7), and the subsequent  For doubly-circumscribed septulene, etc. Hence different cases of polysubstituted circumscribed Cyclopolyarenes can be encumbered by the general formalism outlined in Section 2. For any n-circumscribed m-cyclopolyarene one obtains the GCCI for even m as: if m is even, (8) where k = m(n 2 + 4n + 3)/2 − (n + 1) and n is even, sum is over all divisors d of m, and ϕ(d) is the Euler totient function defined in the previous section.
The GCCI for odd m is given by and k = [m(n 2 + 4n + 3) − (n + 1)]/2 (10) The above GCCIs can be applied to any circum-polyarenes, for example, kekulene ( Figure 1, Structure 1) which corresponds to case m = 6 and n = 1. In this case, the GCCI for the totally symmetric representation is obtained by (11): The GCCI when applied to 2-circumscribed-kekulene (structure 8 in Figure 2) or n = 2 and m = 6 we arrive at: For septulene (structure 2 in Figure 1) which corresponds to m = 7 and n = 1 the GCCI becomes: The 2-circumscribed septulene corresponds to circumscribing Structure 2 in Figure 1 or m = 7 and n = 2, that is, C 105 H 35 . By substituting m = 7 and n = 2 in the general equation we obtain: For octulene we substitute m = 8 and n = 1in the general expression to obtain: Likewise for a 2-circumscribed octulene, C 120 H 40 , we compute the GCCI by substituting m = 8 and n = 2 in Equation (8) resulting in Equation (16): Next we take up polysubstitution of circum-polyarenes that is, replacing some or all of the protons of circum-polyarenes with other atoms such as F, Cl, Br, I, etc. For this scenario we construct the GCCIs, with the object set D defined as the set of protons of the circum-polyarenes. We obtain the following cases for circum-polyarenes depending on the parities of m and n, where m is the cyclicity and n is the circumscribing order.
Case (4): m odd, n even, D: protons: For kekulene with m = 6 and n = 1 we obtain For a 2-circumscribed kekulene (Str 8, Figure 2), m = 6, n = 2 we obtain Equation (22): The cycle index of septulene, m = 7, n = 1, we obtain, For a 2-circumscribed septulene, m = 7, n = 2 the GCCI is given by For an octulene, m = 8, n = 1 we obtain All structures in Figure 2 exhibit D 6h symmetry and they arise from circumcising the graphene structure with holes of various sizes. In particular, the kekulene series belong to the C p H q series with p = 6(n 2 + 4n + 3) and q = 6(n + 3). Likewise structures 7,10,14 ( Figure 2) correspond to the series C p H q where p = 6(n 2 + 6n + 5) and q = 6(n + 5) and structures 9,13 correspond to C p H q where p = 6(n 2 + 8n + 7) and q = 6(n + 7), and the subsequent members of the series are given by C p H q , p = 6(n 2 + 10n + 9) and q = 6(n + 9). Therefore all coronoids with D 6h symmetry form the series C p H q with p = 6(n 2 + 2n(m + 1)+ 2m + 1) and q = 6(n + 2m + 1) for positive integers m and n. The GCCIs that are derived for the circumscribed kekulenes (D 6h groups) can also be employed for all other circumscribed coronoids that exhibit D 6h point groups. For instance, the GCCI for carbon nuclei of Str 11, Figure 2, with m = 6 and n = 3 is displayed in Equation (27): The corresponding cycle index for the protons of Str 11 (m = 6 and n = 3: Figure 2) with 36 protons is: Structure 13 ( Figure 2) corresponds to C 162 H 54 and thus the GCCI for the carbons is given by (29): The number of polysubstituted isomers is enumerated from the GCCI by Pólya's substitution, that is, substituting for every s k by (w 1 k + w 2 k + w 3 k + w 4 k ) in the GCCI. To illustrate we obtain the expression (30) for kekulenes by such a substitution: The coefficient of w 1 b1 w 2 b2 w 3 b3 w 4 b4 in (30) generates the number of heterosubstituted kekulenes or colorings of carbon vertices by b 1 substituents of first kind, b 2 substituents of second kind, b 3 substituent of third kind, and b 4 substituents of fourth kind. Table 2 shows some of the terms thus obtained for 3 colors (w 1 , w 2 , w 3 ). As seen from Table 2 five monosubstituted compounds are enumerated. There are 109 isomers for disubstitution and so on. In accord with a binomial distribution, the maximum distribution of isomers is reached for substitution of carbons by 16 substituents of the first kind, 16 substituents of the second kind and 16 substituents of the third kind, etc., which can be inferred from Table 2 There exists no chirality as kekulene is a planar macrocycle. Table 3 displays the isomers of C 48 H x F y Cl z which correspond to tri-substituted kekulenes or. In order to enumerate these isomers, we invoke the proton cycle index for kekulene derived from the general equation, that is, Equation (19). In order to enumerate the isomers of C 48 H x F y Cl z we substitute every s k by (w 1 k + w 2 k + w 3 k ) in Equation (19). The coefficients for various terms w 1 x w 2 y w 3 z are shown in Table 3 yield the isomers of C 48 H x F y Cl z . As can be seen from Table 3, There are 3 isomers for (23,1,0) or 3 isomers for C 48 H 23 F (see ,  Table 3). Likewise, the number of isomers for C 48 H 12 F 12 is 226,150 and 788,825,460 isomers are enumerated for C 48 H 8 F 8 Cl 8. The results displayed in Table 3 required such an elegant combinatorial technique and the computer codes that we have developed for up to 10 substituents.
Both kekulene and septulene have the same number of monosubstituted isomers for the substitution carbon centers. There are 124 isomers for disubtituted hetero-septulenes that contain two N atoms. The maximum is reached at 536,056,343,620,384,863,061,500 for (19,19,18) for carbon colorings for septulenes. The combinatorics for the protons implies 3 isomers for monosubstituion of septulene with say Cl. Likewise we obtain 34 isomers for a dichloro-septulenes; the maximum is reached at 45,574,776,390 for (10,9,9). The results for octulenes can also be obtained in an analogous manner and they can be found in ref. [124].  Combinatorial identities for all n-circum-m-polyarenes that contain u substituents for carbon centers are obtained as: Case (1): m even Case (2): m odd The combinatorial identities for proton substituents are given: Case (1): m even, n odd, protons: Case (2): m even, n even, protons: Case (3): m odd, n odd, protons: Case (4): m odd, n even, protons:

Applications to 13 C, Proton NMR and Multiple Quantum NMR of Polyarenes
Combinatorial techniques discussed in the previous sections can be applied to 13 C NMR, proton NMR and multiple-quantum NMR patterns. The number of 13 C NMR signals or the number of equivalence classes of carbons for the cycloarenes is the number of isomers for the monosubstitution; the coefficient of (n − 1, 1) where n is the number of carbons enumerates the NMR signals. There are 5 monosubstituted isomers for carbon replacements of kekulene (see Table 2) suggesting that there are 5 13 C NMR signals for kekulenes. Analogously there are five 13 C NMR signals for septulene and the same for octulene. By using the general combinatorial identities derived for isomer counts, we prove below that for all cycloarenes with D mh groups, there are 5 13 C NMR signals: Consequently, the coefficient of w 1 8m−1 w 2 in (41) can be seen to be 1 2m A similar simplification can be carried out to gather the coefficient for m even and n = 1, and it can be shown to be 5. It can be shown that for any n-circumscribed cyclopolyarene, the number enumerated for monosubstitution is given by Equation (40): As can be seen from the above expression, the number of 13 C NMR signal enumeration for all circumscribed polyarenes is only dependent on n, the order of circumscribing and not on the cyclicity or m.
The combinatorial methods can be applied to enumerate the proton NMR signals for any order n of circumscribing and the result is shown below: (n + 5)/2 if n is odd, (n + 4)/2 if n is even.
For example, as seen from the above expression for n = 3, that is, for the triply circumscribed polyarenes, the number of proton NMR signals is 4.
Two-quantum or n-2 quantum NMR spectra bear direct relation to structurally dependent dipolar couplings compared to ordinary 1-quantum NMR spectra. Likewise enumeration of triangular interactions contain information on 3-quantum and so on. That is, from a graph-theoretical standpoint 2-quantum spectra depend on various equivalence classes of edges in the graph while n-1 quantum NMR depends only the vertex automorphisms. The GCCI polynomials can be applied for both bosons and fermions. For multiple quantum NMR of protons or 13 C (fermions) the two possible spin orientations can be represented by α or β. Thus the combinatorial generators for multiple quantum NMR of circumscribed-cyclopolyarenes for 13 C nuclei are given as follows for even and odd m: where k = m(n 2 + 4n + 3)/2 − (n + 1), sum is over all divisors d of m, and ϕ(d) is the Euler totient function.
As a special case, for 13 C multiple quantum NMR of kekulene and circumscribed kekulene the generating function for the totally symmetric representation is given by Equation (44).

Applications to Nanotubes: Enumerations & Chirality
In this section we consider the applications of the GCCI combinatorial methods for the enumeration of chiral and achiral isomers of nanotubes of any cross section and length. For example, a cylindrical nanotube with a square cross section is shown in Figure 3 and hence the symmetry group is D 4h (m = 4). In the general case of a cylindrical nanotube with a cross-section composed of m vertices, the GCCI for such a cylindrical nanotube length n are obtained with four expressions depending on the parities of m and n: m odd; n odd; σ h plane passes through the central layer; each of m C 2 axes passes through a vertex of the central layer; σ v/ σ d planes pass through n vertices: m odd; n even; σ h plane does not pass through any vertex of the tube; each of m C 2 axes passes through the centers of edges; σ v/ σ d planes pass through n vertices: m even; n even; each of m C 2 axes passes through the centers of the edges; m/2 σ v planes pass through 2n vertices; m/2 σ d planes pass through the centers of the edges: In the above expressions, the sum is over divisors d of m, ϕ(d) is the Euler totient function defined as follows: The above product is computed over all prime numbers p that divide d. The Euler totient function is expressible in terms of the Möbius as shown in Equation (51): where the sum is computed over all prime divisors of d and µ(d) is the Möbius function. The GCCIs can de exemplified by considering a cylindrical nanotube with a cross section of 10-beaded necklace. For such a tube, the character table of the D10h point group with 16 conjugacy classes and 16 IRs needs to be considered. We note that character values g and g −1 which are golden ratio and its inverse, respectively; an accidental degeneracy of The GCCIs can de exemplified by considering a cylindrical nanotube with a cross section of 10-beaded necklace. For such a tube, the character table of the D 10h point group with 16 conjugacy classes and 16 IRs needs to be considered. We note that character values g and g −1 which are golden ratio and its inverse, respectively; an accidental degeneracy of the GCCIs arises, resulting in several GCCIs of two dimensional IRs to become identical. Moreover even though the golden ratio is irrational, the sum g + g −1 and g − g −1 is an integer resulting in integral GCCIs for all IRs of the group D mh . Note that the GCCIs of the A 1g and A 1u IRs are of special interest, as these GCCIs enumerate the achiral, chiral as well as all stereo-position isomers for the colorings of the nanotubes for various vertex colorings. Furthermore, the GCCIs enumerate both heteronanotubes of different kinds and polysubstituted nanotubes including fluorochloro nanotubes and hydrogenated nanotubes, etc. Balasubramanian et al. [129] have applied these techniques to a variety of nanotube enumerations and we shall consider some of their salient findings. Figures 4 and 5 show nanotubes of cross section C 6 of even length m (20) and odd m (21), respectively with a D 6h symmetry while the a tube with C 4 cross section is considered in Figure 3. Application of the formulae derived earlier, for example, for the simplest case of a tube with 3 layers (n = 3) and C 6 (m = 6) and odd length ( Figure 5)  Likewise for a tube of C4 cross section (Figure 3) of length 99, the cycle indices are given by: The cycle indices thus obtained in the above illustration is for the entire set D of all mn vertices of the nanotube. To illustrate the Sheehan's modification, the explicit partitions of equivalence classes of the vertices are considered. For the example under consideration, we use the case of a tube with C6 cross section and length of 3. For this case, we obtain two equivalences classes Y1 and Y2 where the first class is for the central layer and hence contains 6 vertices. The set Y2 for this case consists of the top and bottom equivalent layers and thus 12 vertices. The GCCIs thus obtained for the D6h symmetry nanotubes are show below with explicit partitions of the vertices so that Sheehan's modification can be applied: We shall demonstrate the flexibility of the above GFs in that different scenarios with given restrictions can be considered in coloring palettes as a result of partitioning of vertices of the central layers and vertices of the top/bottom layers. Consequently, we obtain the results shown below: The cycle indices thus obtained in the above illustration is for the entire set D of all mn vertices of the nanotube. To illustrate the Sheehan's modification, the explicit partitions of equivalence classes of the vertices are considered. For the example under consideration, we use the case of a tube with C 6 cross section and length of 3. For this case, we obtain two equivalences classes Y 1 and Y 2 where the first class is for the central layer and hence contains 6 vertices. The set Y 2 for this case consists of the top and bottom equivalent layers and thus 12 vertices. The GCCIs thus obtained for the D 6h symmetry nanotubes are show below with explicit partitions of the vertices so that Sheehan's modification can be applied: As the vertices are partitioned, the Sheehan technique facilitates the assignment of different coloring palettes for the distinct equivalence classes. Consider a hexagonal cylinder with 3 layers wherein the color weights 1, a, b, c are assigned for the central layer, and color weights 1, d, e for the top and bottom layers. In this setup the Sheehan's modification yields the following generating functions for the A 1g and A 1u IRs of the D 6h group: (1 + a + b + c) 6 (1 + d + e) 12 + 2 1 + a 6 + b 6 + c 6 1 + d 6 + e 6 2 +2 1 + a 3 + b 3 + c 3 2 1 + d 3 + e 3 4 + 1 + a 2 + b 2 + c 2 3 1 + d 2 + e 2 6 +3(1 + a + b + c) 2 1 + d 2 + e 2 6 + 3 1 + a 2 + b 2 + c 2 3 1 + d 2 + e 2 6 + 1 + a 2 + b 2 + c 2 3 1 + d 2 + e 2 6 + 2 1 + a 3 + b 3 + c 3 2 1 + d 6 + e 6 2 +2 1 + a 6 + b 6 + c 6 1 + d 6 + e 6 2 + (1 + a + b + c) 6 +3(1 + a + b + c) 6 1 + d 2 + e 2 6 + 3 1 + a 2 + b 2 + c 2 3 1 + d 2 + e 2 6 (58) We shall demonstrate the flexibility of the above GFs in that different scenarios with given restrictions can be considered in coloring palettes as a result of partitioning of vertices of the central layers and vertices of the top/bottom layers. Consequently, we obtain the results shown below: The GF(A 1g ) for the case (a) is computed by setting d and e to 0 in Equation (58): The GF thus obtained for A 1g can likewise be computed for other IRs and the coefficient obtained for each term enumerates the colorings of the vertices of the central layer with the corresponding color palette that transform according to the IR. That is, for instance, the partition [2 2 1 1] is enumerated by the term 1 2 a 2 bc or the number of ways to color the central vertices with 2 white, 2 blue, 1 green and 1 red, such that only the central vertices are colored keeping the top/bottom layers constant.
The case (c) is the most common single-set-coloring scheme, as it involves coloring of all vertices of the nanotube with colors chosen from a single set of colors. Here no distinction is made between the central vertices and the vertices of the top/bottom layers, and thus all vertices are placed in one D set. Here the GF for each IR is obtained by replacing w ij = w j for all i and colors are selected from a single color set R. For four-coloring of the vertices, the GF (A 1g ) is shown below for the tube with C 6 cross section and 3 layers: Tables 4 and 5 display the combinatorial results for the C 6 -tubes shown in Figures 4 and 5, respectively. Tables 4 and 5 were generated from the corresponding GFs obtained from their GCCIs for the binomial colorings of these tubes which are shown below: C 6 tube of length 20: C 6 tube of length 21:    In order to generate a chiral coloring, at least 2 black colors are needed, as seen from Tables 4 and 5 Figures 8 and 9 show achiral and chiral isomers for the case of 21 blue colors (nitrogen) and remaining carbons for the case of C 6 tube with length 21. Figure 9 shows one of chiral isomers enumerated (Table 5) out of 17,889,827,492,074,590,075,716 chiral pairs for the partition [105 21] whereas the corresponding achiral isomer is depicted in Figure 8. Figures 10 and 11 illustrate achiral colorings for equal number of grey and blue colors for the C 6 tubular nanotube of and even and odd lengths containing alternating arrangement of carbon and nitrogen atoms, respectively.  Table 4 for the partition [ 100 20].  Table 4 for the partition [ 100 20].  Table 4 for the partition [ 100 20].    Table 4 Table 5 for the partition [ 105 21].  Table 5 for the partition [ 105 21].  Table 5 for the partition [ 105 21].   Table 4 for the partition [60 60].  Table 4 for the partition [60 60]. Figure 10. Tubular nanotube of cross section C6 of even length with alternating arrangement of carbon and nitrogen atoms: One of achiral isomers enumerated in Table 4 for the partition [60 60]. Figure 11. Tubular nanotube of cross section C 6 of odd length with alternating arrangement of carbon and nitrogen atoms: One of achiral isomers enumerated in Table 5

Applications to Tessellations of Kekulenes, Nanobands, C 60 Polymers, Spectroscopy & Topology
Tessellations of polyarenes such as kekulenes and octulenes are of considerable interest because they are excellent candidates for sequestering both anions such as Cl − (Figure 12 Left). Moreover heterosubstituted tessellations of these structures, for example, crown ether analogs and porphyrin analogs can be candidates for sequestration of toxic heavy metal ions such as Cd 2+ (Figure 12 Right) as well as actinyl ions in high level nuclear wastes such as UO 2 2+ and PuO 2 2+ . Another variation to the heavy metal ion trap made possible by polyphenolic kekulene us shown in Figure 12 (bottom) and thus one could make tessellations of these structures possessing multiple cavities for efficient trapping of heavy metal ions. Consequently, such derivatives of graphenes with cavities have been proposed as molecular belts for the sequestration and transport of both anions and cations [25,32,35]. Furthermore phase transformations among various topological configurations such as the square, armchair and zigzag structures have been considered in previous studies [39][40][41] for both carbon nanotubes and various structures arising from graphene sheets. Furthermore enthalpies of formation and Gibbs free energies of such large systems are extremely challenging to compute from the ab initio techniques for example, Gaussian-3 theories. Our understanding of the phase transformations among various tessellations needs the Gibbs free energies of these systems which depend not only the enthalpies but also the entropies of different phases. As there are a large number of vibrational modes for tessellations of kekulenes, computations of thermodynamic properties by ab initio theory would be a mammoth task. Hence robust topological techniques [144] based on graph theory have been developed for large tessellations of kekulenes for the characterization of the structures and spectra using machine learning/artificial intelligence methods. Machine learning that can integrate topology with ab initio techniques through by partitions into equivalence classes can be extremely valuable. Topological methods dissect such large tessellations into edge partitions of tessellations of cyclopolyarenes, which can then be harnessed for rapid and robust computations of enthalpies of formations through bond partitions. Analytical expressions for graph-theoretically based information theoretic entropies of such large tessellations have been derived [144]. In this process two different structures that exhibit the same graph entropies were discovered, and they are shown in Figure 13. The existence of isentropic tessellations for kekulenes is quite interesting as it appears to be previously unknown. Hence these structures were taken up for further studies [144] including computing measures of contrasts in their electronic and other spectra that we discus in the ensuing paragraphs.  There exists greater variations in entropies among the various possible tessellations for relatively smaller tessellations whereas for larger tessellations, different configurations converge to the entropies of the 2D graphitic sheet with holes. Among the smaller tessellations, the zigzag tessellation exhibits the largest entropy whereas the armchair tessellation exhibits the lowest entropy. Liu et al. [39] have investigated the energy changes from the armchair to zigzag graphene structures when studying nanotube chirality selection and chemical control. Thess et al. [37] and Okada [41] have both independently shown that the armchair structure is more stable than the zigzag structure because of the possibility of a triple bond at the edges for the armchair structure. The binding energy/atom and the state density at the edge seems to compete resulting in an enhanced stability of about 15% for the armchair relative to the zigzag structure.
The stabilities of various phases are determined by the Gibbs free energies and thus bond enthalpies and entropies compete in the determination of their free energies. Consequently, the relative entropies of different phases can provide important insights into the relative stabilities of various tessellations or phases. Hence the topological entropies imply that the zigzag structure is more stable than the armchair but an opposite energetic trend is seen for the two structures. The two tessellations of kekulenes shown in Figure 13 contain the same number of vertices (360) and C−C edges (468), and identical topological edge partitions ( Figure 13). Table 6 displays the various topological indices, graph spectra and energetics for the two isentropic kekulene structures. Note that the vertex degree based topological indices such as the Padmakar-Ivan, Zagreb-1, Zagreb-2, ABC, and Randić indices are identical for the two isentropic kekulene tessellations (Table 6). On the other hand, the distance-dependent topological distances such as the Wiener, hyper Wiener, Mostar, Szeged, Gutman, Harary and Balaban, indices differ for the two isentropic kekulene tessellations. The graph spectra and their spectral degeneracies were computed for the two tessellations (see Table 6). The two tessellations in Figure 13 yield different spectral patterns, as AHK(2) belongs to the nonabelian D6h group and thus contains twodimensional irreducible representations whereas the RK(3,3) structure with a D2h abelian There exists greater variations in entropies among the various possible tessellations for relatively smaller tessellations whereas for larger tessellations, different configurations converge to the entropies of the 2D graphitic sheet with holes. Among the smaller tessellations, the zigzag tessellation exhibits the largest entropy whereas the armchair tessellation exhibits the lowest entropy. Liu et al. [39] have investigated the energy changes from the armchair to zigzag graphene structures when studying nanotube chirality selection and chemical control. Thess et al. [37] and Okada [41] have both independently shown that the armchair structure is more stable than the zigzag structure because of the possibility of a triple bond at the edges for the armchair structure. The binding energy/atom and the state density at the edge seems to compete resulting in an enhanced stability of about 15% for the armchair relative to the zigzag structure.
The stabilities of various phases are determined by the Gibbs free energies and thus bond enthalpies and entropies compete in the determination of their free energies. Consequently, the relative entropies of different phases can provide important insights into the relative stabilities of various tessellations or phases. Hence the topological entropies imply that the zigzag structure is more stable than the armchair but an opposite energetic trend is seen for the two structures. The two tessellations of kekulenes shown in Figure 13 contain the same number of vertices (360) and C−C edges (468), and identical topological edge partitions ( Figure 13). Table 6 displays the various topological indices, graph spectra and energetics for the two isentropic kekulene structures. Note that the vertex degree based topological indices such as the Padmakar-Ivan, Zagreb-1, Zagreb-2, ABC, and Randić indices are identical for the two isentropic kekulene tessellations (Table 6). On the other hand, the distance-dependent topological distances such as the Wiener, hyper Wiener, Mostar, Szeged, Gutman, Harary and Balaban, indices differ for the two isentropic kekulene tessellations. The graph spectra and their spectral degeneracies were computed for the two tessellations (see Table 6). The two tessellations in Figure 13 yield different spectral patterns, as AHK(2) belongs to the nonabelian D 6h group and thus contains two-dimensional irreducible representations whereas the RK(3,3) structure with a D2h abelian symmetry contains only uni-dimensional irreducible representations. Thus the spectral difference index and the root mean square of the spectral difference for the two isentropic structures were computed to measure their contrasts. These spectral difference measures are defined as follows:    [75], for kekulene is 0.8744β is reproduced by the spectra computed here; the HOMO-LUMO gap of kekulene at 631G* = 3.58 eV, 6-311g(d) = 3.55 eV and 6-311g(2d) = 3.54 eV. The above-defined spectral indices of these two structures reveal that the isentropic structures are energetically quite close but they are certainly not degenerate (Table 6). Moreover the HOMO-LUMO energy gaps are different for the two structures, in that the armchair structure's HOMO-LUMO gap is greater, although by only~1.5% compared to the square structure; the armchair structure is kinetically more stable than the square tessellation. The total π-electronic energies imply that the rectangular tessellation is roughly a kcal/mol more stable than the armchair structure. The square tessellation exhibits a less symmetric D2h group compared to the armchair (D 6h ) and thus the greater stability of the square tessellation is analogous to the energy stabilization induced by symmetry-breaking in E⊗e Jahn-Teller distortion [144]. This is a consequence of doubly-degenerate HOMO and LUMO of the armchair tessellation both of which transform as the two-dimensional E representations of the D 6h group. In contrast, the HOMO and LUMO of the rectangular structures are one dimensional. Consequently, the metamorphosis from the armchair to the rectangular tessellation leads to symmetry and hence stabilizing the rectangular structure.
Both 13 C and proton NMR spectral patterns can be generated for the two isentropic tessellations by combinatorial generation of their vertex partitions. The same technique can be employed to generate the ESR hyperfine structures and multiple quantum NMR spectra of these structures. In order to accomplish this, we invoke the distance degree sequence vectors (DDSV) of graphs introduced by Bloom et al. [158]. In general for any vertex v in a graph G can be assigned an integer sequence that corresponds to the number of vertices at distances 0, 1, 2, . . . , ev, where ev is defined as the eccentricity of v in G. Hence we can assign a p-tuple vector (Di0, Di1, Di2, . . . , Dij, . . . Dip) for each vertex vi in the graph where Dij is defined as the number of vertices at distance j from vi. This can be carried out by the use of the graph distance matrix generator. The DDSV is can be computed by the computer code developed by the author [159]. As the two tessellations shown in Figure 13 contain 360 vertices there are 360 such DDSV tuples of variable lengths. It can be seen that two equivalent vertices under the graph automorphism would have the same DDSV although the converse is not true for all graphs. For tessellations of kekulene structures, the DDSV offers a viable alternative to the graph vertex partitioning problem, which is in general a O(n!) problem as there are n! ways to label a graph of n vertices in the most general case. The DDSV vectors of variable lengths are concatenated in order to generate an integral label. As these integral labels rapidly grow, the labels are represented as real numbers in quadruple precision. In this setup the algorithm to generate vertex partitions simplifies to O(n 2 ) where n is the number of vertices.
The vertex partitions for both carbon and hydrogen atoms were computed using the DDSV algorithm for the two kekulene tessellations in Figure 13 with the objective of generating their ESR hyperfine and NMR patterns. Through the application of the DDSV computational technique, the vertices of RK(3,3) are partitioned into 2 4 4 88 partition of 360 vertices whereas the vertices of AHK(2) are partitioned into the 6 4 12 28 partition. Consequently, there are 92 equivalence classes of vertices for RK (3,3) whereas there are 32 equivalence classes for AHK (2). Hence these carbon vertex partitions yield the numbers of 13 C NMR signal signals for the two tessellations shown in Figure 13. In an analogous manner, the number of proton NMR signals for the AHK(2) tessellation is obtained as 14. That is, 144 protons are divided 14 classes with the partition 6 4 12 10 for AHK (2). On the other hand, for the RK(3,3) structure the DDSV technique partitions the protons into 38 classes with the partition 2 4 4 34 for the RK(3,3) (Figure 13), as shown in Table 6 for the two structures.
The machine generation of the ESR hyperfine patterns of the two tessellations can be made possible by combinatorial generating functions analogous to the ones obtained in the previous sections. For example, for the AHK(2) structure, the ESR GF can be constructed from the equivalence class partitions assuming that all 13 C nuclei are coupled equally to the unpaired electron. Such radicals can be generated by a single deprotonation of the kekulene tessellation. Suppose we include only the 13 C nuclear-electron coupling in order to devise techniques for the machine learning of the ESR hyperfine pattern. As there are 4 equivalence classes of 13 C nuclei with each class containing 6 members, and 28 classes with 12 nuclei in each class, the net ESR generating function for the hyperfine structure arising from 13 C-e coupling is given by Equation (69):  (69) In an analogous manner we obtain the combinatorial ESR for the RK(3,3) as: When the above GFs are binomially expanded for the 13 C nuclei-electron couplings we arrive at too many lines for graphical representation of the hyperfine pattern. Furthermore the 13 C−e-hyperfine coupling constant depends on the Euclidian distance between the unpaired electron density and the geometrical position of the nuclei. Hence one can reduce the combinatorial complexity by considering only equivalence class of nuclei that are nearest to the unpaired electron of the radical. Thus if the radical is generated at the inner periphery of the kekulene ring via deprotonating one of the six protons, we obtain a more amenable ESR hyperfine structure. In this event we generate the equivalence class partition for the AHK(2) as 6 2 12 3 whereas for RK (3,3) it is 2 2 4 11 yielding the ESR GFs shown below as Equations (71) and (72) (2) These combinatorial GFs for the two tessellations generate 7 2 13 3 lines for the AHK(2) structure and 3 2 5 11 lines for the RK(3,3) structure, respectively. The computed ESR hyperfine patterns for the two kekulene tessellations are displayed in Figure 14a,b, respectively. One can also obtain the proton ESR hyperfine structures for the two tessellations as well as the multiple quantum NMR patterns in a manner analogous to the techniques demonstrated in the previous sections.
The other set of structures for which Euler totient based combinatorial techniques would apply are nanobands or a necklace-choker composed of hexagons as displayed in Figure 15. The structures can be tailored to various pore sizes so that they serve as sequestering agents for the complexation of metal anions or halide ions. These materials also offer optimal sites for functionalization or substitutions of carbon sites with heteroatoms such as nitrogen atoms in order to synthesize nanobands that could complex with metal ions analogous to porphyrins. equally to the unpaired electron. Such radicals can be generated by a single deprotonation of the kekulene tessellation. Suppose we include only the 13 C nuclear-electron coupling in order to devise techniques for the machine learning of the ESR hyperfine pattern. As there are 4 equivalence classes of 13 C nuclei with each class containing 6 members, and 28 classes with 12 nuclei in each class, the net ESR generating function for the hyperfine structure arising from 13 C-e coupling is given by Equation (69): GF(ESR: 13 C;AHK(2)) = ∏ ( + ) ∏ + In an analogous manner we obtain the combinatorial ESR for the RK(3,3) as: GF(ESR: 13 C;RK(3,3)) = ∏ ( + ) ∏ + When the above GFs are binomially expanded for the 13 C nuclei-electron couplings we arrive at too many lines for graphical representation of the hyperfine pattern. Furthermore the 13 C−e-hyperfine coupling constant depends on the Euclidian distance between the unpaired electron density and the geometrical position of the nuclei. Hence one can reduce the combinatorial complexity by considering only equivalence class of nuclei that are nearest to the unpaired electron of the radical. Thus if the radical is generated at the inner periphery of the kekulene ring via deprotonating one of the six protons, we obtain a more amenable ESR hyperfine structure. In this event we generate the equivalence class partition for the AHK(2) as 6 2 12 3 whereas for RK (3,3) it is 2 2 4 11 yielding the ESR GFs shown below as Equations (71) and (72) These combinatorial GFs for the two tessellations generate 7 2 13 3 lines for the AHK(2) structure and 3 2 5 11 lines for the RK(3,3) structure, respectively. The computed ESR hyperfine patterns for the two kekulene tessellations are displayed in Figure 14a,b, respectively. One can also obtain the proton ESR hyperfine structures for the two tessellations as well as the multiple quantum NMR patterns in a manner analogous to the techniques demonstrated in the previous sections. The other set of structures for which Euler totient based combinatorial techniques would apply are nanobands or a necklace-choker composed of hexagons as displayed in Figure 15. The structures can be tailored to various pore sizes so that they serve as sequestering agents for the complexation of metal anions or halide ions. These materials also offer optimal sites for functionalization or substitutions of carbon sites with heteroatoms such as nitrogen atoms in order to synthesize nanobands that could complex with metal ions analogous to porphyrins. The other set of structures for which Euler totient based combinatorial techniques would apply are nanobands or a necklace-choker composed of hexagons as displayed in Figure 15. The structures can be tailored to various pore sizes so that they serve as sequestering agents for the complexation of metal anions or halide ions. These materials also offer optimal sites for functionalization or substitutions of carbon sites with heteroatoms such as nitrogen atoms in order to synthesize nanobands that could complex with metal ions analogous to porphyrins. In a recent study Sabirov et al. [161] have considered various arrangements arising from (C 60 ) n polymers such as the zigzag and linear configurations of C 60 polymers. These authors have studied the topological indices such as the Wiener indices, roundness and graph entropies of these structures such as the ones in Figure 16. Combinatorial tools developed herein can be applied for the enumeration of heteronuclear fullerene polymers arising from different configurations shown in Figure 16. Analogous to the spectroscopic studies that were outlined for the tessellations of kekulenes, future research should be devoted for the machine generation of the spectra of fullerene polymers ( Figure 16). By extending the linkers between the two fullerenes with small alkane chains, one can obtain longer polymers. The longer linkers would then provide sufficient flexibility to generate necklaces of fullerenes where each bead would be a C 60 cage. For such structures, the techniques outlined here on the basis of Euler's totient functions can become extremely useful and applicable. In a recent study Sabirov et al. [161] have considered various arrangements arising from (C60)n polymers such as the zigzag and linear configurations of C60 polymers. These authors have studied the topological indices such as the Wiener indices, roundness and graph entropies of these structures such as the ones in Figure 16. Combinatorial tools developed herein can be applied for the enumeration of heteronuclear fullerene polymers arising from different configurations shown in Figure 16. Analogous to the spectroscopic studies that were outlined for the tessellations of kekulenes, future research should be devoted for the machine generation of the spectra of fullerene polymers ( Figure 16). By extending the linkers between the two fullerenes with small alkane chains, one can obtain longer polymers. The longer linkers would then provide sufficient flexibility to generate necklaces of fullerenes where each bead would be a C60 cage. For such structures, the techniques outlined here on the basis of Euler's totient functions can become extremely useful and applicable.

Conclusions, Helical Structures, Fullerene Polymers and Other Structural Derivatives & Future Perspectives
This review considered combinatorics and topology of circum-polycycloarenes, heteronanotubes and tessellations of cyclopolyarenes such as kekulenes in different configurations, for example, armchair, zigzag and square, etc. We showed the power of Sheehan's modifica-tion of Pólya's theorem when generalized to all the characters of the point symmetry groups of nanostructures. Topologically-derived edge partition techniques revealed the existence of isentropic kekulene tessellations, that is, the existence of two kekulene tessellations with the same graph entropies. These structures were also shown to be quite close in their energy separations. Consequently, their spectral differences, ESR hyperfine patterns and NMR signal patterns were combinatorially constructed using the DDSV techniques followed by the generating function methods. While these combinatorial techniques were shown to be very powerful in their applications to novel nanomaterials that were considered here, there are several emerging structures comprising of fullerenes.
The emerging field of reticular metal organic frameworks, mesoporous cages, zeolites, sodalite materials, other nanomaterials such nanobelts 2D-nanosheets and other nanomaterials [25,32,35,60,61,[144][145][146][147][148][149][150] could all be benefited by such robust graph-theoretical and combinatorial tools for the rapid computations of their properties and creation of combinatorial libraries of these structures. Consequently, the advent of these novel materials has rekindled our research interest in such interesting applications of combinatorics, group theory, graph theory and topological indices. Such techniques would offer robust and rapid computational tools for the computations of their thermodynamic, optoelectric, spectroscopic, phase transformation and chiral properties. We envisage several nanowires and 2D-sheets of such molecules [25,32,35,[47][48][49][59][60][61][144][145][146][147][148][149][150] to be synthesized in the future, which would also provide a platform for the combinatorics of big data pertinent to these structures. Finally we believe that combinatorial and graph-theoretical techniques would be of considerable value for the enumeration and computation of electronic properties of materials such as gallium arsenide, GaN nanowires, topological characterization of 2D materials, zeolites, their helical structures made from kekulenes, and the emerging novel expanded kekulenes, and so forth [9,17,23,. There are a number of combinatorial techniques and applications of variants of Pólya's theorem and related applications to graphs and chemical enumerations, and readers are referred to references [162][163][164][165][166][167][168][169][170][171].