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8.1 Introduction

Fullerenes are molecules consisting of tri-connected polyhedral cages of various coverings or tessellations. When there is a single type of polygonal faces, the covering is called Platonic; when there are two types of faces, the covering is called Archimedean (Diudea 2010a). A molecule can be represented by a molecular graph. A graph \(\ G(V,E) \) is an ordered pair of two sets, V and E, V = V(G) being a finite nonempty set and E = E(G) a binary relation defined on V (Harary 1969; Diudea et al. 2002). A graph can be visualized by representing the elements of V by points (i.e., vertices) and joining pairs of vertices (i, j) by a bond (i.e., edge) if and only if (i, j) ∈ E(G). The number of vertices in G equals the cardinality v = |V| of this set. A graph is said connected if any two vertices, i and j, are the endpoints of a path; otherwise, it is disconnected. The molecular graphs are in general connected graphs.

Spanning fullerenes can be designed by deleting, from their molecular graph, some vertices/atoms or edges/bonds, thus resulting in open structures with di-connected atoms which can further be joined with atoms of the same or different repeating units in designing periodic nanostructures, as those encountered in nano-dendrimers, in crystals or quasicrystals. Note that spanning of fullerenes can be obtained in laboratory by irradiating the closed cages by electron or ion beams, while many of the molecular constructions to be presented in the following can be seen as potentially real structures. Since fullerenes can be designed from the Platonic polyhedra, tetrahedron T, cube C, octahedron Oct, dodecahedron Do, and icosahedron Ico, by applying some operations on maps, the spanning fullerenes can be designed by such sequences of operations, the “opening” Op operation included.

A map is a discretized (closed) surface. Among the most important map operations we mention are the following: dual Du, medial Med, P k -mapping, k = 3–5, Leapfrog Le, Chamfering Q, and Capra Ca. These operations are implemented in the CVNET software package (Stefu and Diudea 2005). More about map operation the reader can find in Diudea (2010a) and Stefu et al. (2005).

Dendrimers are hyper-branched structures with a well-tailored architecture. Their endgroups can be functionalized, thus modifying their initial properties. Dendrimers have gained a wide range of applications in supramolecular chemistry, particularly in host-guest reactions and self-assembly processes. Promising applications come from polyamidoamine dendrimers as gene transfer vectors and peptide dendrimers as antipeptide antibodies and synthetic vaccines (Diudea 2010a, p. 80).

The number of edges emerging from each branching point is called the progressive degree (Diudea and Katona 1999), p (i.e., the edges increasing the number of points of a newly added generation). It equals the classical vertex degree d minus 1: p = d − 1.

The stepwise growth of a dendrimer follows a mathematical progression. A first problem in studying the topology of dendrimers is that of enumerating its constitutive parts: vertices, edges, or fragments.

The number of vertices in the ith orbit (i.e., that located at distance i from the center) of a regular dendrimer can be expressed as a function vertex degree d:

$$ {v_i}=(2-z)(d+z-1){(d-1)^{(i-1) }};\quad i>0 $$
(8.1)

where z = 1 for a monocentric dendrimer and z = 0 for a dicentric one. By using the progressive degree p, relation (8.1) becomes

$$ {v_i}=(2-z)(p+z){p^{(i-1) }};\quad i>0 $$
(8.2)

For the core, the number of vertices is \( {v_0}=2-z \), while the number of external vertices (i.e., the endpoints) is obtained by

$$ {v_r}=(2-z)(p+z){p^{(r-1) }} $$
(8.3)

where r is the radius of the dendrimer and equals the number of its orbits/generations.

The total number of vertices v(D), in dendrimer, is obtained by summing the populations on all orbits:

$$ v(D)=(2-z)+(2-z)(p+z)\sum\limits_{i=1}^r {{p^{(i-1) }}} $$
(8.4)

By developing the sum in (8.4), one obtains

$$ v(D)=(2-z)+(2-z)(p+z)\left( {\frac{{{p^r}-1}}{p-1 }} \right)=\frac{{2({p^{r+1 }}-1)}}{p-1 }-z{p^r} $$
(8.5)

A useful recurrence enables one to calculate v(D) from the number of vertices of the precedent term of a dendrimer family (i.e., a homologous series of dendrimers, having the same progressive degree, p) \( v({D_{r+1 }})=pv({D_r})+2 \), irrespective of monocentric or dicentric the dendrimer is.

The term nano-dendrimer refers here to hyper-branched structures of which branching nodes represent nanotube junctions (i.e., spanned fullerenes), while the bonds joining them are nanotubes of various length.

Crystals are (MacKay 1981; Hargittai and Hargittai 2010) periodic structures, of which (one-type) unit cells, consisting of one or more atoms or other identical components, repeat a large number of times by three noncoplanar translations. Corresponding atoms in each unit cell have almost identical surrounding, while the fraction of atoms near the surface is small and the effects of the surface can be neglected.

Quasicrystals are quasiperiodic structures (Levine and Steinhardt 1984), showing more than one type of repeating unit, or the same unit repeating quasi-regularly. Quasicrystals can have the topology of multi-tori, particularly of those with icosahedral symmetry. These kinds of periodic structures will be exemplified in the following.

Multi-tori are structures of high genera, consisting of more than one tubular ring (Diudea and Petitjean 2008). They are supposed to result by self-assembly of some repeat units or monomers; their geometry is eventually superimposed on surfaces of negative curvature, like FRD or P-surface, and shows a high porosity. Multi-tori can be designed starting from small cages, for example, the Platonic solids. Modeling of porous structures has been previously reported by Mackay and Terrones (1991), Lenosky et al. (1992) and Terrones and Mackay (1993), etc. Such structures appear in spongy carbon (already synthesized, Benedek et al. 2003), in schwarzites (named in honor of mathematician Schwarz 1865, 1890), or in zeolites (natural aluminosilicates).

8.2 Design of Spanning Fullerenes

There are at least two ways to design a tubular nano-junction: (1) by sequences of map operations and (2) by spanning appropriate fullerenes.

8.2.1 Junctions by Map Operations

Three basic map operations Leapfrog Le, Chamfering (Quadrupling) Q, and Capra Ca (the reader is invited to consult: Diudea 2004, 2005; Diudea et al. 2003, 2006a; Stefu et al. 2005), associated with the opening Op operation, are most often used to transform small polyhedral objects (basically, the Platonic solids) into tubular junctions (Diudea and Nagy 2007). These transforms preserve the symmetry of the parent map. Figure 8.1 presents a realization by Leapfrog Le operation (Nagy et al. 2011).

Fig. 8.1
figure 00081

Design of a tetrapodal junction by Leapfrog Le map operation

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8.2.2 Junctions by Spanning Fullerenes

Spanning a fullerene graph can be done by deleting/braking some edges (and vertices), thus getting a particular patch (eventually identical to a circulene molecule). Figure 8.2 illustrates the case of coronene-preserving opening (left) and sumanene-preserving opening (right) of fullerene C84_20−T d (Diudea 2010b). Figure 8.3 presents three octahedral nano-junctions derived from the hypothetical fullerene C168 that consists of eight (disjoint) sumanene patches (Szefler et al. 2012a), lying in the corners of cube, while on cube faces having octagons, Sum_CZ_192 and Sum_CA_216 are obtained by opening operations Op(1a) and Op(2a), respectively; the third one, Sum_OctS2LeX_168, results by simply deleting the alternating edges of octagon, Op(−a). Observe that the last numbers in the name of structures represent the number of carbon atoms. Other particular monomeric units will be presented below.

Fig. 8.2
figure 00082

Opening C84_20-T d : as coronene, 4[6:66], v = 84, e = 114, f 6 = 28, g = 2 (left – also designed by the sequence of map operations – see Fig. 8.1) and sumanene, 4[6:(5,6)3], v = 84, e = 114, f 5 = 12, f 6 = 16, g = 2 (right) patches, respectively; optimized structures are shown in the bottom row

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Fig. 8.3
figure 00083

Sumanene motif decorating octahedral nano-junctions: Sum_CZ_192 (left), Sum_CA_216 (middle), and Sum_OctS2LeX_168 (right)

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The spanned cages can be used to build dendrimers or other periodic nanostructures by joining with other units (the same or not) or by identifying common substructures (see below).

Table 8.1 lists the energetics of these hypothetical nano-junctions, optimized at the Hartree-Fock level of theory, as hydrogen-ended molecules. When compared to the values for C60, the reference structure in nanoscience, one can see the spanned fullerenes show at least (or higher) the stability of the reference. The extent of strain, as given by POAV theory (Haddon 1987, 1990), is favorable for the opened fullerenes (Table 8.1, last column) in comparison to that of C60, the structures included in this table being real candidates to the status of real molecules.

Table 8.1 Total energy Etot and HOMO-LUMO gap, at Hartree-Fock HF (HF/6-31 G(d,p)) level of theory, for some hypothetical nano-junctions and C60 reference nanostructure
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8.3 Dendrimers

Dendrimers (Fig. 8.4) can be designed from monomers with tetrahedral symmetry, either by joining the two-connected terminal vertices or by identification of appropriate (open) faces (Diudea 2010b). Dendrimers at 2nd generation are illustrated in Fig. 8.5 (Diudea 2010c).

The corresponding dimers (Fig. 8.6, top) are “intercalated” ones. A possible linear evolution of the above monomers is presented in Fig. 8.6, bottom (Diudea 2010b).

Other examples come from polybenzenes (Fig. 8.7), which are nanostructures consisting of benzene ring as the main motif (O’Keeffe et al. 1992) of tessellation.

8.4 Crystal-Like Networks

When monomers have octahedral symmetry, they can embed in the P-type surface in getting crystal-like networks which belong to the space group P n 3m. Figure 8.8 illustrates nets showing the sumanene 6:(5,6)3 motif (Szefler et al. 2012a).

Other examples of crystal-like networks are given in Fig. 8.9 (Szefler and Diudea 2013): their covering consists of motifs of hexagon triples (unit C_3HexZ_104) and heptagon triples (unit C_3HepA_104), respectively, for which the energetic data are given in Table 8.2.

Again the open structures (i.e., nano-junctions) appear less strained than the reference C60 fullerene and, according to the total energy (calculated on the optimized structures at Hartree-Fock HF and DFT levels of theory) and HOMO-LUMO gap, are at least as stable as the reference molecule. The strain of heptagon triple motif is one order of magnitude lower than that of hexagon triple one and two orders of magnitude lower than that of the reference fullerene, also reflected in the values of TE/C. In view of a possible identification among nano-materials, IR and RAMAN spectra have been simulated (Szefler and Diudea 2013). Polybenzenes can also be embedded in the P-type surface (Fig. 8.10).

Fig. 8.4
figure 00084

Dendrimers by monomers with coronene (left) and sumanene (right) motifs, at 1st generation

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Fig. 8.5
figure 00085

Dendrimer with coronene motif at 2nd generation (two different views)

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Fig. 8.6
figure 00086

Linear evolution of the monomers with coronene (left column) and sumanene (right column) motifs

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Fig. 8.7
figure 00087

Polybenzene dendrimer (left) and its “intercalated” dimer (right)

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Fig. 8.8
figure 00088

P-type surface embedding of monomers in Fig. 8.3: Sum_CA_216 (left), Sum_CZ_192 (middle), and Sum_OctS2LeX_168 (right)

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Fig. 8.9
figure 00089

Top row: C_3HexZ_104 (v = 104; e = 144; f 6 = 36; g = 3) and its p-type crystal-like net (v(3,3,3) = 2,808); bottom row: C_3HepA_104 (v = 104; e = 132; f 7 = 24; g = 3) and its p-type crystal-like net (junctions designed by map operation sequences Op 2a (S 2(C)) and Op(S 1(C)), respectively)

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Table 8.2 Energies (total energy per carbon atom TE/C, in Hartree H; HOMO-LUMO gap, HL gap, in electron volts; POAV strain energy per carbon atom S/C, kcal/mol) of the optimized structures at HF and DFT level of theory
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Fig. 8.10
figure 000810

Polybenzenes embedded in the P-type surface: p-BCZ_48 (top) and p-BCA_96 (bottom) and the corresponding units (right column)

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D-type surface can also embed a polybenzene structure (Fig. 8.11). This last net shows the topology of D6-diamond: a face-centered cube f cc-structure, belonging to the space group Pn3m (Szefler and Diudea 2012).

Fig. 8.11
figure 000811

Polybenzene embedded in the D-type surface: f cc-BTA_48 and its f cc-dimer (right)

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Calculations have been performed at HF/6-31 G(d, p) and B3LYP/6-311 + G(d, p) level of theory, respectively, on Gaussian 09 software package (2009).

Stability of the monomers and the corresponding dimmers with respect to the reference C60 fullerene can be compared from data in Table 8.3. Note the three types of dimers originating in BTA_48, function of the identified face/ring and conformation: (BTA_48)2_84_dendrim (R 12, intercalate, Table 8.3, entry 2, and Fig. 8.7, right), (BTA_48)2_88_ f cc (R 8, Table 8.3, entry 3, and Fig. 8.11, right), and (BTA_48)2_90_MT (R 12, eclipsed, Table 8.3, entry 4, and Fig. 8.12, bottom, left) (Szefler et al. 2012b).

Table 8.3 Total energy E tot per atom (kcal/mol) and HOMO-LUMO gap, at Hartree-Fock HF level of theory, strain by POAV theory and HOMA index in benzene-patched units and their dimers and C60 reference structure, as well
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Fig. 8.12
figure 000812

Left column: BTA_48 (top) formed by spanning the cage Le(P4(T))_48 (by deleting the blue bonds – middle) and its multi-torus BMTA2_90 dimer ((BTA_48)2_90_MT in Table 8.3, entry 4). Right column: BTZ_24 (top) originating in S2(T)_28 = C28 (the four blue points located at the center of pentagon triples to be deleted – middle) and its multi-torus BMTZ2_48 dimer

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One can see that the strain in polybenzenes, as calculated by POAV theory, is far less than that in C60, and the overall stability is at least as that of the reference fullerene. Infrared and Raman spectra have been simulated (Szefler et al. 2012b) in view of possible use in laboratory identifying such structures, tessellated with the simplest benzene ring motif. The high values of HOMA index of aromaticity (Krygowski and Ciesielski 1995, 1996), in the last column of Table 8.3, suggest that the benzene ring geometry is not too much modified in comparison to the free benzene molecule (HOMA value = 1), while the hexagonal rings in C60 are much more affected (HOMA = 0.493) by the presence of pentagons (see also Cysewski and Szefler 2010). The HOMA value is even dropped in case of dimer (BCZ_48)2_96 (Table 8.3, entry 6) because the units bound directly at the benzene ring. The loss in pi-electron resonance is partly compensated by the loss in strain energy, visible when compared with the BCZ_48 unit (Table 8.3, entry 5). However, in an infinite network, the strain will drop even more (Szefler et al. 2012b), and the geometry approaches to that of the unit-free molecule.

8.5 Quasicrystal Nanostructures

Multi-tori are designed by using “eclipsed” dimers (Diudea and Nagy 2007), as shown in Fig. 8.12, bottom row. The dimer BMTA2_90 was included in Table 8.3 (as (BTA_48)2_90_MT, entry 4). The unit BTZ_24, due to its simplicity, can form only the dimer BMTZ2_48, leading to multi-tori. The multi-tori bearing the benzene patch will have B as a prefix in their name. Next, because the opening faces show either “zigzag” or “armchair” endings, “Z” or “A” will be added as a suffix to their name. The number of repeating units and/or number of atoms will be added after the letters.

The BMTX2 dimers, because of their “eclipsed” conformation, will form pentagonal hyper-rings BMTXCy5 (Fig. 8.13, bottom) in a self-assembly process. These can further evolve to the multi-torus BMTX17 (X = Z, Fig. 8.3, left), of which reduced graph is just C17, the structure proposed by Diudea (2010d) as the seed for the diamond D5. By analogy to D5, a hyper-dimer BMTX34 can be designed (X = A, Fig. 8.3, right). We mention that its reduced graph is C34, the repeating unit of the triple periodic structure of D5. Recall that D5 is an mtn triple periodic 3-nodal net, named ZSM-39 (a structure found in clathrates of type II), of point symbol net: {55.6}12{56}5 and 2[512] + [512.64] tiling. It is also known as the fcc_C34 structure (Blase et al. 2010; Diudea et al. 2011), because of its face-centered cubic lattice that belongs to the space group Fd3m. Thus, we can expect a 3D network derived from these benzene-patched units, similar to D5.

Fig. 8.13
figure 000813

Left column: BMTZ17_408 and its hyper-pentagon MMTZCy5_120. Right column: BMTA34_1332 and its hyper-pentagon BMTACy5_210

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A spherical multi-torus BMTX20 (Fig. 8.14, left column) can be constructed and is a g = 21 multi-torus, with a well-defined core: Core(BMTA20)_180 = −f 5(Le 2,2 (Do)) while Core (BMTZ20)_120 = −d 5(S2(Ico). In the above, −f 5 means deletion of all pentagonal faces in the Leapfrog (2,2) transform of the dodecahedron Do, and −d 5 is deletion of vertices of degree d = 5, in the transform of Icosahedron = Ico by the septupling S2 operation. Also, −d 5(S2(Ico)) = Op(Le(Ico)). Recall that g is the genus of the surface that embeds a structural graph and accounts for the number of simple tori of which consists that graph (Diudea and Szefler 2012).

Fig. 8.14
figure 000814

Top row: multi-torus BMTA20_1_780 (left) and its core (−f 5(Le 2,2(Do)), right); bottom row: multi-torus BMTZ20_1_480 (left) and its core (−d 5(S2(Ico), right))

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A linear array BMTX20_k, k = 1,2,… with the repeating unit formed by two units superimposing one pentagonal hyper-face (i.e., BMTXCy5), rotated to each other by an angle of pi/5 as in the “dimer” BMTA20_2 (Fig. 8.15, top, left). Next, the structure can evolve with a one-dimensional periodicity, as shown in BMTA20_4 (Fig. 8.15, top, right) or in the hyper-cycle BMTZCy20_5 (Fig. 8.15, bottom, left). Twelve units of BMTX20 can form a spherical array (of icosahedral symmetry), as in case of BMTZSp20_12 (Fig. 8.15, bottom, right), of which core is just BMTZ20 (a 13th unit).

Fig. 8.15
figure 000815

Top row: the repeat unit BMTA20_2_1350 (left) and a rodlike BMTA20_4_2490 (right); bottom row: multi-tori BMTZCy20_5_1800 (left) and BMTZSp20_12_3120 (twofold symmetry)

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Theorem 8.1

In multi-tori built up from open tetrahedral units, the genus of structure equals the number of its units plus one, irrespective of the unit tessellation.

Demonstration comes out from construction and is illustrated on the multi-tori BMTXCy5 (Fig. 8.13, bottom row): there are five units open to be inserted in exactly five simple tori and one more torus that join all the above five units, thus demonstrating the first part of the theorem (Diudea and Szefler 2012).

For the second part, we apply the Euler’s theorem (1758):

$$ v-e+f=\upchi =2(1-g) $$
(8.6)

where \( v=|V(G)| \) is the number of vertices/atoms, \( e=|E(G)| \) is the number of edges/bonds, and f is the number of faces of the graph/molecule. In the above, g is the genus of the (orientable) surface S on which a molecular graph is embedded. The genus is related to the Gaussian curvature of the surface S by means of Euler’s characteristic \( \chi \) of S (Gauss-Bonnet 1853) theorem as for g = 0 (case of sphere) \( \chi >0 \) (positive curvature); for g = 1 (case of torus) \( \chi =0 \), while for g > 1 (surfaces of high genera), \( \chi <0 \), S will show a negative curvature. More about surfaces of negative curvature the reader can find in Diudea and Nagy (2007).

To complete the demonstration, we will use the data in Table 8.4 providing the values of g in several BMTX multi-tori, tessellation differing as X = A or Z.

Table 8.4 Euler formula calculation in multi-tori BMTX
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The number of tetrahedral units BMTX1 in the linear array of BMTX20_k (Table 8.4, entries 3–6) is u = 20 k − 5(k − 1) = 15 k + 5, according to the construction mode. The term −5(k − 1) accounts for the superimposed hyper-rings, BMTXCy5. In case of BMTZCy20_5 (Table 8.4, entry 7), the formula is u = 20 k − 5 k = 15 k, k = 5, the last hyper-ring unit being omitted because of the cyclic structure. Thus, the drop in g is of 5 units for each fivefold hyper-cycle (compare Table 8.4, entries 6 and 7).

In case of the spherical array BMTZSp20_12 (Table 8.4, entry 9), u = 20 k − 2 [5(k − 1)], k = 12. Remark the twice subtraction of the term 5(k − 1), in case of the spherical array, which accounts for the difference in g to the linear array of k = 12 (Table 8.4, entries 8 and 9): 186 − 131 = 55 = 5(12 – 1). This drop in g, in case of the spherical array, seems to parallel the well-known result that sphere is the minimal surface among all known solid objects. The number u is also related to the number of simple faces/rings as follows: \( u={f_8}/6 \) in case BMTA and \( u={f_6}/4 \) in case BMTZ.

On the ground of Theorem 8.1, the spherical array BMTZSp20_12 seems to be the minimum g (lower bound) while BMTZ20_k the maximum g (upper bound) among all the studied structures (Diudea and Szefler 2012). We just demonstrated the following:

Theorem 8.2

The genus in multi-tori shows the lower-bound value in structures of icosahedral symmetry, while the upper-bound value is shown in linear structures provided the same number of (open) tetrahedral units.

Note these icosahedral multi-tori represent quasicrystal nanostructures; quasicrystals have been Nobel prized in 2011.

Carbon atom orbit analysis in BMTZSp20_12 revealed a 6.82 massive class (2,580 atoms, about 83 %), located inside, of the same signature as in polybenzene, and two smaller classes, of signature 6.8 (360 atoms), and 6 (180 atoms), disposed outside of the spherical structure. Compare with % of 6.82 in the linear array BMTZ20_k (about 74 % at k = 12) and in BMTA20_k (about 26 %, at k = 9). Knowing the (calculated by O’Keeffe et al. 1992) stability of polybenzene, consisting of only 6.82 atoms (in the infinite triple periodic net), the orbit analysis provides a “topological” proof of stability of the spherical array BMTZSp20_12 (Diudea and Szefler 2012).

8.6 Omega Polynomial in Polybenzenes

In a connected graph G(V, E), with the vertex set V(G) and edge set E(G), two edges e = uv and f = xy of G are called codistant e co f if they obey the relation (John et al. 2007)

$$ d(v,x)=d(v,y)+1=d(u,x)+1=d(u,y) $$
(8.7)

which is reflexive, that is, e co e holds for any edge e of G, and symmetric, if e co f, then f co e. In general, relation co is not transitive; if “co” is also transitive, thus it is an equivalence relation, then G is called a co-graph, and the set of edges \( C(e):=\{f\in E(G);f\;co\;e\} \) is called an orthogonal cut oc of G, E(G) being the union of disjoint orthogonal cuts: \( E(G)={C_1}\cup {C_2}\cup \ldots \cup {C_k},\ {C_i}\cap {C_j}=\emptyset, i\ne j \). Klavžar (2008) has shown that relation co is a theta Djoković (1973)–Winkler (1984) relation.

We say that edges e and f of a plane graph G are in relation opposite, e op f, if they are opposite edges of an inner face of G. Note that the relation co is defined in the whole graph while op is defined only in faces. Using the relation op, we can partition the edge set of G into opposite edge strips, ops. An ops is a quasi-orthogonal cut qoc, since ops is not transitive.

Let G be a connected graph and \( {S_1},{S_2},\ldots,{S_k} \) be the ops strips of G. Then, the ops strips form a partition of E(G). The length of ops is taken as maximum. It depends on the size of the maximum-fold face/ring F max/R max considered so that any result on Omega polynomial will have this specification.

Denote by m(G,s) the number of ops of length s and define the Omega polynomial as (Diudea 2006; Diudea et al. 2006b, 2008, 2009; Ashrafi et al. 2008; Diudea and Klavžar 2010; Vizitiu et al. 2007)

$$ \Omega (G,x)=\sum\limits_s {m(G,s)\cdot {x^s}} $$
(8.8)

Its first derivative (in x = 1) equals the number of edges in the graph:

$$ {\Omega}^{\prime}\left( {G,1} \right)=\sum\limits_s {m\left( {G,s} \right)\cdot s=e=\left| {E(G)} \right|} $$
(8.9)

On Omega polynomial, the Cluj-Ilmenau index (John et al. 2007), CI = CI(G) was defined:

$$ \mathrm{CI}(G)=\{{{[{\Omega}^{\prime}(G,1)]}^2}-[{\Omega}^{\prime}(G,1)+\Omega^{\prime \prime}(G,1)]\} $$
(8.10)

Formulas to calculate Omega polynomial and CI index in three infinite networks, f cc-BTA_48 (Fig. 8.11), p-BCZ_48 (Fig. 8.10, top), and p-BCA_96 (Fig. 8.10, bottom), are listed in Table 8.5. Formulas were derived from the numerical data calculated on cuboids of (k, k, k) dimensions by the Nano Studio software (Nagy and Diudea 2009). Omega polynomial was calculated at R max(8) and R max(12), respectively; examples are given in view of an easy verification of the general formulas. Also, formulas for the number of atoms, edges, and rings (R 6, R 8 and R 12) are included in this table (Szefler and Diudea 2012).

Table 8.5 Omega polynomial and net parameters in polybenzene networks
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Table 8.5 (continued)
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Formulas to calculate Omega polynomial and CI index in the two infinite networks BMTA20k and BMTZ20k, designed on the ground of BMTA1_48 and BMTZ1_24 units, are presented in Tables 8.6 and 8.7. Formulas were derived from the numerical data calculated on rods consisting of k units BMTX20. Omega polynomial was calculated at R max = R 8. Formulas for the number of atoms, edges, and rings (R 6, R 8, and R 15, the last one being the simple ring of the hyper-ring BMTACy5) are included in Tables 8.6 and 8.7. Numerical examples are also given (Diudea and Szefler 2012).

Table 8.6 Formulas for Omega polynomial and net parameters in linear periodic BMTA20_k network
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Table 8.7 Formulas for Omega polynomial and net parameters in linear periodic BMTZ20_k network
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Omega polynomial description can be looked as an alternative to the crystallographic description, helping in understanding the topology of these networks.

8.7 Conclusions

Spanning fullerenes can be obtained by deleting some atoms or bonds from the graph of closed fullerenes, thus resulting in open structures which can further join to atoms of the same or different repeating units in construction of crystal- or quasicrystal-like networks. A variety of spanning fullerenes, designed either by cage opening or by sequences of map operations, has been used to build nano-dendrimers and crystal- and quasicrystal-like structures. Energetics of some open fullerenes has been calculated on optimized structures at Hartree-Fock and/or DFT level of theory. All of the discussed open fullerenes have shown less strained structures in comparison to C60 reference fullerene, while the overall stability, as suggested by the total energy and HOMO-LUMO gap, is comparable or even better to the reference fullerene, thus being candidates to the status of real molecules. Omega polynomial was used to describe the topology of some periodic networks, designed by using benzene as the simplest patch.