Omega Polynomial in Hyperdiamonds

Abstract

Hyperdiamonds are covalently bonded carbon phases, more or less related to the diamond network, having a significant amount of sp³ carbon atoms and similar physical properties. Many of them have yet a hypothetical existence but a well-theorized description. Among these, the diamond D₅ was studied in detail, as topology, at TOPO GROUP CLUJ, Romania. The theoretical instrument used was the Omega polynomial, also developed in Cluj. It was computed in several 3D network domains and analytical formulas have been derived, not only for D₅ but also for the well-known diamond D₆ and other known networks.

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9.1 Introduction

Diamond D₆ (Fig. 9.1), the beautiful classical diamond, with all-hexagonal rings of sp³ carbon atoms crystallized in a face-centered cubic fcc network (space group Fd3m), has kept its leading interest among the carbon allotropes, even many “nano” varieties appeared (Decarli and Jamieson 1961; Aleksenski$\check{{i}}$ et al. 1997; Osawa 2007, 2008; Williams et al. 2007; Dubrovinskaia et al. 2006). Its mechanical characteristics are of great importance, and composites including diamonds may overpass the resistance of steel or other metal alloys. Synthetic diamonds can be produced by a variety of methods, including high pressure-high temperature HPHT, static or detonation procedures, chemical vapor deposition CVD (Lorenz 1995), ultrasound cavitation (Khachatryan et al. 2008), or mechano-synthesis (Tarasov et al. 2011), under electronic microscopy.

Fig. 9.1

Diamond D₆: adamantane D₆_10 (left), diamantane D₆_14 (middle), and diamond D₆_52 (a 222 net – right)

A relative of the diamond D₆, called lonsdaleite L₆ (Frondel and Marvin 1967), with a hexagonal network (space group P6₃/mmc – Fig. 9.2), was discovered in a meteorite in the Canyon Diablo, Arizona, in 1967. Several diamond-like networks have also been proposed (Diudea and Nagy 2007; Diudea et al. 2010a; Hyde et al. 2008).

Fig. 9.2

Lonsdaleite: L₆_12 (left), L₆_18 (middle), and L₆_48 (a 222 net – right)

Hyperdiamonds are covalently bonded carbon phases, more or less related to the diamond network, having a significant amount of sp³ carbon atoms. Their physical properties are close to that of the classical diamond, sometimes with exceeding hardness and/or endurance.

Design of several hypothetical crystal networks was performed by using our software programs (Diudea 2010a) CVNET and NANO-STUDIO. Topological data were provided by NANO-STUDIO, Omega, and PI programs.

This chapter is structured as follows. After the introductory part, the main networks, diamond D₅ and lonsdaleite L₅, are presented in detail. Next, two other nets, the uninodal net, called rhr, and the hyper boron nitride, are designed. Two sections with basic definitions in Omega polynomial and in Omega-related polynomials, respectively, are developed in the following. The topology of the discussed networks will be presented in the last part. Conclusions and references will close the chapter.

9.2 Structures Construction

9.2.1 Diamond D₅ Network

Diamond D ₅ , recently theorized by Diudea and collaborators (Diudea and Ilić 2011; Diudea 2010b; Diudea and Nagy 2012; Diudea et al. 2012), is a hyperdiamond, whose seed is the centrohexaquinane C₁₇ (Fig. 9.3). D₅ is the mtn crystal 4,4,4-c trinodal network, appearing in type II clathrate hydrates; it belongs to the space group Fd-3m and has point symbol net: {5^5.6}12{5^6}5 (Dutour Sikirić et al. 2010; Delgado-Friedrichs and O’Keeffe 2006, 2010). The hyper-structures, from ada- to dia- and a larger net are illustrated in Fig. 9.4, viewed both from C₂₀ (left column) and C₂₈ (right column) basis, respectively (Diudea et al. 2012).

Fig. 9.3

The seed of diamond D₅: C₁₇

Fig. 9.4

Hyper-adamantane: ada_20_158 and ada_28_213 (top), diamantane: dia_20_226_222 and dia_28_292_222 (middle), and diamond D₅_20_860_333 and D₅_28_ 1022_333 co-net (bottom)

The hyperdiamond D₅_20/28 mainly consists of sp³ carbon atoms building ada-like repeating units (C₂₀ cages including C₂₈ as hollows). The ratio C-sp³/C-total trends to 1 in a large enough network. As the content of pentagons R[5] per total rings trends to 90 % (see Table 9.3, entry 9), this yet hypothetical carbon allotrope is called the diamond D₅.

Energetic data, calculated at various DFT levels (Diudea and Nagy 2012; Diudea et al. 2012), show a good stability of the start and intermediate structures. Limited cubic domains of the D₅ networks have also been evaluated for stability, data proving a pertinent stability of D₅ diamond.

9.2.2 Lonsdaleite L₅ Network

By analogy to D₅_20/28, a lonsdaleite-like net was proposed (Diudea et al. 2012) (Fig. 9.5). The hyper-hexagons L₅_28_134 (Fig. 9.5, middle and right), whose nodes represent the C₂₈ fullerene, was used as the monomer (in the chair conformation). Its corresponding co-net L₅_20 was also designed. The lonsdaleite L₅_28/20 is partially superimposed on D₅_20/28 net. In crystallography, L₅ is known as the 7-nodal mgz-x-d net, with the point symbol: {5^5.6}12{5^6}5.

Fig. 9.5

Lonsdaleite: L₅_28_250 (side view, left), L₅_28_134 (side view, middle), and L₅_28_134 (top view, right)

9.2.3 Hyper Boron Nitride

Boron nitride is a chemical crystallized basically as the carbon allotropes: graphite (h-BN), cubic-diamond D₆ (c-BN), and lonsdaleite L₆ (wurtzite w-BN). Their physicochemical properties are also similar, with small differences.

Fullerene-like cages have been synthesized and several theoretical structures have been proposed for these molecules (Soma et al. 1974; Stephan et al. 1998; Jensen and Toftlund 1993; Mei-Ling Sun et al. 1995; Fowler et al. 1999; Oku et al. 2001; Narita and Oku 2001).

Based on B₁₂N₁₂ unit, with the geometry of truncated octahedron, we modeled three 3D arrays: a cubic domain, Fig. 9.6; a dual of cuboctahedron domain, Fig. 9.7; and an octahedral domain, Fig. 9.8.

Fig. 9.6

Boron nitride B₁₂N₁₂: truncated octahedron (left), a cubic (3,3,3)_432 domain built up from truncated octahedra joined by identifying the square faces (middle), and a corner view (right)

Fig. 9.7

Boron nitride B₁₂N₁₂: dual of cuboctahedron (left), a (3,3,3)_648 dual of cuboctahedron domain, constructed from truncated octahedra by identifying the square and hexagonal faces, respectively (middle), and its superposition with the cubic (3,3,3)_432 domain (right)

Fig. 9.8

Boron nitride B₁₂N₁₂: an octahedral (4,4,4)_480 domain: (1,1,0-left), (0,0,1-central), and (2,1,1-right) constructed from truncated octahedra by identifying the square and hexagonal faces, respectively

The topology of the above hyperdiamonds will be described by using the net parameter k, meaning the number of repeat units along the chosen 3D direction, and by the formalism of several counting polynomials, the largest part being devoted to Omega polynomial.

9.2.4 rhr Network

A uninodal 4-c net with a point {4².6².8²} was named rhr or sqc5544. In topological terms, its unit cell is a homeomorphic of cuboctahedron, one of the semi-regular polyhedra (Fig. 9.9). It can be obtained by making the medial operation on the cube or octahedron (Diudea 2010a). The net can be constructed by identifying the vertices of degree 2 in two repeating units, thus the resulting net will have all points of degree 4, as in the classical diamond D₆ (but the rings are both six- and eight-membered ones).

Fig. 9.9

The rhr unit (top) and network (444_1728, bottom: top view (left) and corner view (right))

9.3 Omega Polynomial

9.3.1 Relations co and op

Let G = (V(G),E(G)) be a connected graph, with the vertex set V(G) and edge set E(G). Two edges e = (u,v) and f = (x,y) of G are called co-distant (briefly: e co f ) if the notation can be selected such that (Diudea 2010a; John et al. 2007; Diudea and Klavžar 2010)

$$ e\;co\;f\Leftrightarrow d(v,x)=d(v,y)+1=d(u,x)+1=d(u,y) $$

(9.1)

where d is the usual shortest-path distance function. Relation co is reflexive, that is, e co e holds for any edge e of G and it is also symmetric: if e co f, then also f co e. In general, co is not transitive.

For an edge $e\in E(G)$, let $c(e):=\{f\in E(G);\;f\;co\;e\}$ be the set of edges co-distant to e in G. The set c(e) is called an orthogonal cut (oc for short) of G, with respect to e. If G is a co-graph then its orthogonal cuts $C(G)={c_1},\;{c_2},\ldots,{c_k}$ form a partition:

$$ E(G)={c_1}\cup {c_2}\cup \cdots \cup {c_k},\quad {c_i}\cap {c_j}=\emptyset,\;\;i\ne j $$

A subgraph $H\subseteq G$ is called isometric if ${d_H}(u,v)={d_G}(u,v)$, for any $(u,v)\in H$; it is convex if any shortest path in G between vertices of H belongs to H. The n-cube ${Q_n}$ is the graph whose vertices are all binary strings of length n, two strings being adjacent if they differ in exactly one position (Harary 1969). A graph G is called a partial cube if there exists an integer n such that G is an isometric subgraph of ${Q_n}$.

For any edge e = (u,v) of a connected graph G, let n _uv denote the set of vertices lying closer to u than to v: ${n_{uv }}=\left\{ {w\in V(G)|d(w,u)<d(w,v)} \right\}$. By definition, it follows that ${n_{uv }}=\left\{ {w\in V(G)|d(w,v)=d(w,u)+1} \right\}$. The sets (and subgraphs) induced by these vertices, n _uv and n _vu , are called semicubes of G; these semicubes are opposite and disjoint (Diudea and Klavžar 2010; Diudea et al. 2008; Diudea 2010c).

A graph G is bipartite if and only if, for any edge of G, the opposite semicubes define a partition of G: ${n_{uv }}+{n_{vu }}=v=\left| {V(G)} \right|$ .

The relation co is related to the ∼ (Djoković 1973) and $\Theta$ (Winkler 1984) relations:

$$ e\;\Theta\;f\Leftrightarrow d(u,x)+d(v,y)\ne d(u,y)+d(v,x) $$

(9.2)

Lemma 9.1

In any connected graph, co = ∼.

In general graphs, we have $\sim \subseteq \Theta$ and in bipartite graphs $\sim =\Theta$. From this and the above lemma, it follows (Diudea and Klavžar 2010)

Proposition 9.1

In a connected graph, co = ∼; if G is also bipartite, then $co =~{\sim}~=~\Theta$.

Theorem 9.1

In a bipartite graph, the following statements are equivalent (Diudea and Klavžar 2010):

• (i) G is a co-graph.

• (ii) G is a partial cube.

• (iii) All semicubes of G are convex.

• (iv) Relation $\Theta$ is transitive.

Equivalence between (i) and (ii) was observed in Klavžar (2008), equivalence between (ii) and (iii) is due to Djoković (1973), while the equivalence between (ii) and (iv) was proved by Winkler (1984).

Two 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. Then e co f holds by assuming the faces are isometric. Note that relation co involves distances in the whole graph while op is defined only locally (it relates face-opposite edges). A partial cube is also a co-graph but the reciprocal is not always true. There are co-graphs which are non-bipartite (Diudea 2010d), thus being non-partial cubes.

Relation op partitions the edge set of G into opposite edge strips ops: any two subsequent edges of an ops are in op-relation, and any three subsequent edges of such a strip belong to adjacent faces.

Lemma 9.2

If G is a co-graph, then its opposite edge strips ops {s _k } superimpose over the orthogonal cuts ocs {c _k }.

Proof

Recall the co-relation is defined on parallel equidistant edges relation (9.1). The same is true for the op-relation, with the only difference (9.1) is limited to a single face. Suppose e ₁ and e ₂ are two consecutive edges of ops; by definition, they are topologically parallel and also co-distant (i.e., belong to ocs). By induction, any newly added edge of ops will be parallel to the previous one and also co-distant. Because, in co-graphs, co-relation is transitive, all the edges of ops will be co-distant, thus ops and ocs will coincide.

Corollary 9.1

In a co-graph, all the edges of an ops are topologically parallel.

Observe that the relation co is a particular case of the edge equidistance eqd relation. The equidistance of two edges e = (uv) and f = (xy) of a connected graph G includes conditions for both (i) topologically parallel edges (relation (9.1)) and (ii) topologically perpendicular edges (in the Tetrahedron and its extensions – relation (9.3)) (Diudea et al. 2008; Ashrafi et al. 2008a):

$$ e\;eqd\;f({\it ii})\Leftrightarrow d(u,x)=d(u,y)=d(v,x)=d(v,y) $$

(9.3)

The ops strips can be either cycles (if they start/end in the edges e _even of the same even face f _even) or paths (if they start/end in the edges e _odd of the same or different odd faces f _odd).

Proposition 9.2

Let G be a planar graph representing a polyhedron with the odd faces insulated from each other. The set of ops strips $S(G)=\{{s_1},{s_2},\ldots,{s_k}\}$ contains a number of op paths opp which is exactly half of the number of odd face edges e _odd/2.

Proof of Proposition 9.2 was given in Diudea and Ilić (2009).

Corollary 9.2

In a planar bipartite graph, representing a polyhedron, all ops strips are cycles.

The ops is maximum possible, irrespective of the starting edge. The choice is about the maximum size of face/ring searched, and mode of face/ring counting, which will decide the length of the strip.

Definitions 9.1

Let G be an arbitrary connected graph and ${s_1},{s_2},\ldots,{s_k}$ be its op strips. Then ops form a partition of E(G) and the Ω-polynomial (Diudea 2006) of G is defined as

$$ \Omega (x)=\sum\limits_{i=1}^k {{x^{{|{s_i}|}}}} $$

(9.4)

Let us now consider the set of edges co-distant to edge e in G, c(e). A$\Theta$-polynomial (Diudea et al. 2008), counting the edges equidistant to every edge e, is written as

$$ \Theta (x)=\sum\limits_{{e\in E(G)}} {{x^{|c(e)| }}} $$

(9.5)

Suppose now G is a co-graph, when |c _k | = |s _k |, then (Diudea and Klavžar 2010)

$$ \Theta (x)=\sum\limits_{{e\in E(G)}} {{x^{|c(e)| }}} =\sum\limits_{i=1}^k {\sum\limits_{{e\in {S_i}}} {{x^{|c(e)| }}} =\sum\limits_e {|c(e)|} {x^{|c(e)| }}=\sum\limits_{i=1}^k {|{s_i}|} {x^{{|{s_i}|}}}} $$

(9.6)

Let us simplify a little the above notations: note by m(s) or simply m the number of ops of length s = |s _k | and rewrite the Omega polynomial as (Diudea 2010a; Ashrafi et al. 2008b; Khadikar et al. 2002; Diudea et al. 2010b)

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

(9.7)

Next we can write Theta and other two related polynomials, as follows:

$$ \Theta (x)=\sum\limits_s {ms\cdot {x^s}} $$

(9.8)

$$ \Pi (x)=\sum\limits_s {ms\cdot {x^{e-s }}} $$

(9.9)

$$ \mathrm{Sd}(x)=\sum\limits_s {m\cdot {x^{e-s }}} $$

(9.10)

The polynomial Θ(x) counts equidistant edges while Π(x) counts non-equidistant edges. The Sadhana polynomial, proposed by Ashrafi et al. (2008b) in relation with the Sadhana index Sd(G) proposed by Khadikar et al. (2002), counts non-opposite edges in G. Their first derivative (in x = 1) provides single-number topological descriptors also termed topological indices (Diudea 2010a):

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

(9.11)

$$ \Theta^{\prime}(1)=\sum\limits_s {m\cdot {s^2}} =\theta (G) $$

(9.12)

$$ \Pi^{\prime}(1)=\sum\limits_s {ms\cdot (e-s)} =\Pi (G) $$

(9.13)

$$ \mathrm{S}\mathrm{d}^{\prime}(1)=\sum\limits_s {m\cdot (e-s)} =e(\mathrm{Sd}(1)-1)=\mathrm{Sd}(G) $$

(9.14)

Note $\mathrm{Sd}(1)=\Omega (1)$, then the first derivative given in (9.14) is the product of the number of edges e = |E(G)| and the number of strips $\Omega (1)$ less one.

On Omega polynomial, the Cluj-Ilmenau index (Ashrafi et al. 2008a) CI = CI(G) was defined as

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

(9.15)

A polynomial related to Π(x) was defined by Ashrafi et al. (2008b) as

$$ \mathrm{P}{{\mathrm{I}}_e}(x)=\sum\limits_{{e\in E(G)}} {{x^{n(e,u)+n(e,v) }}} $$

(9.16)

where n(e,u) is the number of edges lying closer to the vertex u than to the v vertex. Its first derivative (in x = 1) provides the PI _e (G) index proposed by Khadikar (2000) and developed by Ashrafi et al. (2006).

Proposition 9.3

In any bipartite graph, $\Pi (G)=\mathrm{P}{{\mathrm{I}}_e}(G)$.

Proof

Ashrafi defined the equidistance of edges by considering the distance from a vertex z to the edge e = uv as the minimum distance between the given point and the two endpoints of that edge (Ashrafi et al. 2006, 2008a):

$$ d(z,e)= \min \{d(z,u),d(z,v)\} $$

(9.17)

Then, for two edges e = (uv) and f = (xy) of G,

$$ e\;eqd\;f({\it iii})\Leftrightarrow d(x,e)=d(y,e)\quad \mathrm{and}\quad d(u,f)=d(v,f) $$

(9.18)

In bipartite graphs, relations (9.1) and (9.3) superimpose over relations (9.17) and (9.18), then in such graphs, $\Pi (G)=\mathrm{P}{{\mathrm{I}}_e}(G)$. In general graphs, this is, however, not true.

Proposition 9.4

In co-graphs, the equality $\mathrm{CI}(G)=\Pi (G)$ holds.

Proof

By definition, one calculates

$$ \begin{aligned}[b] \mathrm{CI}(G) & ={{\left( {\sum\limits_{i=1}^k {|{s_i}|} } \right)}^2}-\left( {\sum\limits_{i=1}^k {|{s_i}|} +\sum\limits_{i=1}^k {|{s_i}|(|{s_i}|-1)} } \right) \\ & =|E(G){|^2}-\sum\limits_{i=1}^k {{{{\left( {\left| {{s_i}} \right|} \right)}}^2}} =\Pi^{\prime}(G,1)=\Pi (G) \end{aligned} $$

(9.19)

Relation (9.19) is valid only when assuming $\left| {{c_k}} \right|=\left| {{s_k}} \right|$, k = 1,2,…, thus providing the same value for the exponents of Omega and Theta polynomials; this is precisely achieved in co-graphs. In general graphs, however, $\left| {{s_i}} \right|\ne \left| {{c_k}} \right|$ and as a consequence, $\mathrm{CI}(G)\ne \Pi (G)$ (Diudea 2010a).

In partial cubes, which are also bipartite, the above equality can be expanded to the triple one:

$$ \mathrm{CI}(G)=\Pi (G)=\mathrm{P}{{\mathrm{I}}_e}(G) $$

(9.20)

a relation which is not obeyed in all co-graphs (e.g., in non-bipartite ones).

There is also a vertex-version of PI index, defined as (Nadjafi-Arani et al. 2009; Ilić 2010)

$$ \mathrm{P}{{\mathrm{I}}_v}(G)=\mathrm{P}{{\mathrm{I}^{\prime}}_v}(1)=\sum\limits_{e=uv } {{n_{u,v }}+{n_{v,u }}} =\left| V \right|\cdot \left| E \right|-\sum\limits_{e=uv } {{m_{u,v }}} $$

(9.21)

where n _u,v , n _v,u count the non-equidistant vertices with respect to the endpoints of the edge e = (u,v) while m(u,v) is the number of equidistant vertices vs u and v. However, it is known that, in bipartite graphs, there are no equidistant vertices vs. any edge, so that the last term in (9.21) will miss. The value of PI _v (G) is thus maximal in bipartite graphs, among all graphs on the same number of vertices; this result can be used as a criterion for checking whether the graph is bipartite (Diudea 2010a).

9.3.2 Omega Polynomial of Diamond D₆ and Lonsdaleite L₆

Topology of the classical diamond D₆ and lonsdaleite L₆ is listed in Table 9.1 (Diudea et al. 2011). Along with Omega polynomial, formulas to calculate the number of atoms in a cuboid of dimensions (k,k,k) are given. Above, k is the number of repeating units along the edge of such a cubic domain. One can see that the ratio C(sp³)/v(G) approaches the unity; this means that in a large enough net almost all atoms are tetra-connected, a basic condition for a structure to be diamondoid. Examples of calculus are given in Table 9.2.

Table 9.1 Omega polynomial in diamond D₆ and lonsdaleite L₆ nets, function of the number of repeating units along the edge of a cubic (k,k,k) domain

Table 9.2 Examples, Omega polynomial in diamond D₆ and lonsdaleite L₆ nets

9.3.3 Omega Polynomial of Diamond D₅ and Lonsdaleite L₅

Topology of diamond D₅ and lonsdaleite L₅, in a cubic (k,k,k) domain, is presented in Tables 9.3, 9.4, 9.5, 9.6, 9.7, and 9.8 (Diudea et al. 2011). Formulas to calculate Omega polynomial, number of atoms, number of rings, and the limits (to infinity) for the ratio of sp³ C atoms over total number of atoms and also the ratio R[5] over the total number of rings as well as numerical examples are given.

Table 9.3 Omega polynomial in diamond D₅_20 net function of k = no. ada_20 units along the edge of a cubic (k,k,k) domain

Table 9.4 Examples, Omega polynomial in D₅_20 net

Table 9.5 Omega polynomial in D₅_28 co-net function of k = no. ada_20 units along the edge of a cubic (k,k,k) domain

Table 9.6 Examples, Omega polynomial in D₅_28 co-net

Table 9.7 Omega polynomial in Lonsdaleite-like L₅_28 and L₅_20 nets function of k = no. repeating units along the edge of a cubic (k,k,k) domain

Table 9.8 Examples, Omega polynomial in L₅_28 and L₅_20 nets

9.3.4 Omega Polynomial of Boron Nitride Nets

Topology of boron nitride nets is treated similarly to that of D₅ and L₅ and is presented in Tables 9.9, 9.10, 9.11, 9.12, 9.13, 9.14, 9.15, 9.16, 9.17, and 9.18 (Diudea et al. 2011). Formulas to calculate Omega polynomial, number of atoms, number of rings, and the limits (to infinity) for the ratio of sp³ C atoms over total number of atoms are given, along with numerical examples. Formulas for Omega polynomial are taken as the basis to calculate the above four related polynomials in these bipartite networks. Formulas are derived here not only for a cubic domain (in case of c_B₁₂N₁₂) but also for a dual of cuboctahedron domain (case of COD_B₁₂N₁₂) and for an octahedral domain (case of Oct_B₁₂N₁₂).

Table 9.9 Omega polynomial in c_B₁₂N₁₂ net, (designed by Le(C_n)_all) function of k = no. repeating units along the edge of a cubic (k,k,k) domain

Table 9.10 Examples, Omega polynomial in c_B₁₂N₁₂ cubic (k,k,k) net

Table 9.11 Theta $\Theta$, Pi $\Pi$, Sadhana Sd, and PI_v polynomials in c_B₁₂N₁₂ cubic (k,k,k) net

Table 9.12 Examples, Theta $\Theta$, Pi $\Pi$, Sadhana Sd, and PI_v indices in c_B₁₂N₁₂ cubic (k,k,k) net

Table 9.13 Omega polynomial in B₁₂N₁₂ net function of k = no. repeating units along the edge of a Du(Med(Cube)) COD (k_all) domain

Table 9.14 Examples, Omega polynomial in COD_B₁₂N₁₂ (k_all) net

Table 9.15 Theta, Pi, Sadhana, and PI _v polynomials in COD_B₁₂N₁₂ (k_all) net

Table 9.16 Examples, Theta, Pi, Sadhana, and PI _v polynomials in COD_B₁₂N₁₂ (k_all) net

Table 9.17 Omega polynomial in B₁₂N₁₂ net function of k = no. repeating units along the edge of an octahedral Oct (k_all) domain

Table 9.18 Examples, Omega polynomial in Oct_B₁₂N₁₂ (k_all) net

9.3.5 Omega Polynomial of rhr Network

Formulas for Omega polynomial are derived here for a cubic domain (k,k,k) of the rhr network. The results are listed in Table. 9.19.

Table 9.19 Omega polynomial in the rhr net function of k

9.4 Conclusions

Design of several hypothetical crystal networks was performed by using original software programs CVNET and NANO-STUDIO, developed at TOPO GROUP CLUJ. The topology of the networks was described in terms of the net parameters by several counting polynomials, calculated by our NANO-STUDIO, Omega and PI software programs.

Hyperdiamonds are structures related to the classical diamond, having a significant amount of sp³ carbon atoms and covalent forces to join the consisting fullerenes in crystals. Design of several hypothetical crystal networks was performed by using original software programs CVNET and NANO-STUDIO, developed at TOPO GROUP CLUJ. The topology of the networks was described in terms of the net parameters and several counting polynomials, calculated by our NANO-STUDIO, Omega, and PI software programs.